Physics

### An electron (mass m) with an initial velocity . If = , it’s de Broglie wavelength at time t is given bya) b) c) d ) The correct answer is c)$\frac{\lambda_{0}}{\sqrt{1+\frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}$ The de Broglie equation h=mv describes the relationship between a moving particle's momentum...

### An electron (mass m ) with an initial velocity It’s de Broglie wavelength at time t is given bya) b) c) d) The correct answer is a) $\frac{\lambda_{0}}{1+\frac{e E_{0}}{m} \frac{t}{v_{0}}}$ The wave associated with moving particle is called matter wave or de-Broglie wave and it propagates in the form of...

### A helicopter of mass 2000 kg rises with a vertical acceleration of 15 m/s2. The total mass of the crew and passengers is 500 kg. Give the magnitude and direction of the a) force on the floor of the helicopter by the crew and passengers b) action of the rotor of the helicopter on the surrounding air c) force on the helicopter due to the surrounding air

Given, M = 2000 kg helicopter mass m = 500 kg m = 500 kg m = 500 kg m = 500 kg m = 500 kg m = 500 kg Helicopter acceleration with crew and passengers = 15 m/s2 a) Force exerted by the crew and...

### A rectangular box lies on a rough inclined surface. The coefficient of friction between the surface and the box is μ. Let the mass of the box be m. a) at what angle of inclination θ of the plane to the horizontal will the box just start to slide down the plane? b) what is the force acting on the box down the plane, if the angle of inclination of the plane is increased to a > θ c) what is the force needed to be applied upwards along the plane to make the box either remain stationary or just move up with uniform speed? d) what is the force needed to be applied upwards along the plane to make the box move up the plane with acceleration a.

a) As the box starts to slide down the plane, $\mu=\tan \theta$ $\theta=\tan ^{-1}(\mu)$ b) If $a>\theta$, the angle of inclination will be the angle of repose and the net force acting will be...

### There are four forces acting at a point P produced by strings as shown in the figure, which is at rest. Find the forces F1 and F2.

Because the point is at rest with a = 0, the resulting forces on the point are zero. As a result, the X and Y axis net components will be zero. It's difficult to resolve all of the forces along the...

### A cricket bowler releases the ball in two different ways a) giving it only horizontal velocity and b) giving it horizontal velocity and a small downward velocity. The speed vs at the time of release is the same. Both are released at a height H from the ground. Which one will have greater speed when the ball hits the ground? Neglect air resistance.

a) $\frac{1}{2} v_{z}^{2}=g H \Rightarrow v_{z}=\sqrt{2 g H}$ Speed at ground is given as: $\sqrt{v_{s}^{2}+v_{z}^{2}}=\sqrt{v_{s}^{2}+2 g H}$ b)$\frac{1}{2} m v_{s}^{2}+m g H$ is the total energy...

### The displacement vector of a particle of mass m is given by a) show that the trajectory is an ellipse b) show that a) $\begin{array}{l} r(t)=\hat{i} A \operatorname{coswt}+\hat{j} B \sin \omega t \\ x=A \cos \omega t \\ y=B \sin \omega t \\ x / A=\cos \omega t \\ y / B=\sin \omega t \end{array}$ Squaring both...

### Block A of weight 100N rests on a frictionless inclined plane of slope angle 30o. A flexible cord attached to A passes over a frictionless pulley and is connected to block B of weight W. Find the weight W for which the system is in equilibrium.

Equilibrium between $A$ or $B$, Then we know that, $\mathrm{mg} \sin 30^{\circ}=\mathrm{F}$ $\begin{array}{l} 1 / 2 \mathrm{mg}=\mathrm{F} \\ \mathrm{F}=(1 / 2)(100)=50 \end{array}$ Therefore,...

### Why does a child feel more pain when falls down on a hard cement floor, than when she falls on the soft muddy ground in the garden?

When a kid falls on a hard cement floor, the child feels more pain than when the child falls on soft muddy ground in the yard, since the time it takes the child to stop on the cemented ground is...

### Why are porcelain objects wrapped in paper or straw before packing for transportation?

Because porcelain artefacts are delicate, they are wrapped in paper or straw before being packed for shipment. When the acceleration (v – u)/t drops while carrying these items, the shift in velocity...

### A block placed on a rough horizontal surface is pulled by a horizontal force F. Let f be the force applied by the rough surface on the block. Plot a graph of f versus F.

F1 is the force exerted on the heavy box, which is equal to F1 and is resisted by the lesser frictional force f1. F = Fs, which is the maximum static frictional force, is required for the box to...

### The velocity of a body of mass 2 kg as a function of t is given by Find the momentum and the force acting on it at time t = 2 sec.

m=2kg $\underset{v}{\rightarrow}(t)=2 t \hat{i}+t^{2} \hat{j}$ $\vec{v} \text { at } 2 \mathrm{sec}, \underset{v}{\vec{v}}(2 t)=2(2 t) \hat{i} 2^{2} \hat{j}, v(2)=4 \hat{i}+4 \hat{j}$ \$ \text {...

### Consider a light beam incident from air to a glass slab at Brewster’s angle as shown in the figure.A polaroid is placed in the path of the emergent ray at point P and rotated about an axis passing through the centre and perpendicular to the plane of the polaroid (a) For a particular orientation, there shall be darkness as observed through the polaroid (b) The intensity of light as seen through the polaroid shall be independent of the rotation (c) The intensity of light as seen through the Polaroid shall go through a minimum but not zero for two orientations of the polaroid (d) The intensity of light as seen through the polaroid shall go through a minimum for four orientations of the polaroid

The correct answer is c) The intensity of light as seen through the Polaroid shall go through a minimum but not zero for two orientations of the polaroid

### Consider sunlight incident on a slit of width 104 A. The image seen through the slit shall (a) be a fine sharp slit white in colour at the centre (b) a bright slit white at the centre diffusing to zero intensities at the edges (c) a bright slit white at the centre diffusing to regions of different colours (d) only be a diffused slit white in colour

The correct answer is (a) be a fine sharp slit white in colour at the centre

### In a Young’s double-slit experiment, the source is white light. One of the holes is covered by a red filter and another by a blue filter. In this case (a) there shall be alternate interference patterns of red and blue (b) there shall be an interference pattern for red distinct from that for blue (c) there shall be no interference fringes (d) there shall be an interference pattern for red mixing with one for blue

The correct answer is c) there shall be no interference fringes

### The figure shows a standard two-slit arrangement with slits S1, S2. P1, P2 are the two minima points on either side of P. At P2 on the screen, there is a hole and behind P2 is a second 2- slit arrangement with slits S3, S4 and a second screen behind them;(a) There would be no interference pattern on the second screen but it would be lighted (b) The second screen would be totally dark (c) There would be a single bright point on the second screen (d) There would be a regular two-slit pattern on the second screen

The correct answer is d) there would be a regular two-slit pattern on the second screen

### Two sources S1 and S2 of intensity I 1 and I 2 are placed in front of a screen in the figure (a). The pattern of intensity distribution seen in the central portion is given by the figure (b). In this case which of the following statements are true;(a) S1 and S2 have the same intensities (b) S1 and S2 have a constant phase difference (c) S1 and S2 have the same phase (d) S1 and S2 have the same wavelength.

The correct answer is a) S1 and S2 have the same intensities b) S1 and S2 have a constant phase difference c) S1 and S2 have the same phase

### Consider sunlight incident on a pinhole of width 103A. The image of the pinhole seen on a screen shall be (a) a sharp white ring (b) different from a geometrical image (c) a diffused central spot, white in colour (d) diffused coloured region around a sharp central white spot

The correct answer is b) different from a geometrical image d) diffused coloured region around a sharp central white spot

### Consider the diffraction pattern for a small pinhole. As the size of the hole is increased (a) the size decreases (b) the intensity increases (c) the size increases (d) the intensity decreases

The correct answer is a) the size decreases b) the intensity increases

### For light diverging from a point source (a) the wavefront is spherical (b) the intensity decreases in proportion to the distance squared (c) the wavefront is parabolic (d) the intensity at the wavefront does not depend on the distance

The correct answer is a) the wavefront is spherical b) the intensity decreases in proportion to the distance squared

### Is Huygen’s principle valid for longitudinal sound waves?

For longitudinal sound waves, Huygen's concept holds true.

### Consider a point at the focal point of a convergent lens. Another convergent lens of short focal length is placed on the other side. What is the nature of the wavefronts emerging from the final image?

Wavefront and ray orientations are perpendicular to each other. L1 is the source of parallel rays that produce I2 at the focal length of the lens in the diagram above. The created picture serves as...

### What is the shape of the wavefront on earth for sunlight?

The distance between the sun and the earth is known to be enormous. We also know that the sun is spherical and that it may be thought of as a point source of light that can be seen from a great...

### Why is the diffraction of sound waves more evident in daily experience than that of lightwave?

Sound waves have a frequency of 20 Hz to 20,000 Hz and a wavelength of 15 m to 15 mm, respectively. When the size of the slit is similar to the wavelength of the sound, it causes diffraction....

### The human eye has an approximate angular resolution of φ = 5.8 × 10–4 rad and a typical photo printer prints a minimum of 300 dpi (dots per inch, 1 inch = 2.54 cm). At what minimal distance z should a printed page be held so that one does not see the individual dots.

Angular separation, φ = 5.8 × 10–4 rad The average distance between the two dots = 2.54/300 = 0.85 × 10-2 cm At distance z cm, the angle subtended = 0.85 × 10-2/z Resolution angle for human = 0.85 ×...

### A polaroid (I) is placed in front of a monochromatic source. Another polaroid (II) is placed in front of this polaroid (I) and rotated till no light passes. A third polaroid (III) is now placed in between (I) and (II). In this case, will light emerge from (II)? Explain.

A monochromatic source is placed in front of polaroid (I), and polaroid (II) is positioned in front of polaroid (I) (I). The light is unaltered as it travels through the polaroid (II). The Polaroid...

### For the same objective, find the ratio of the least separation between two points to be distinguished by a microscope for light of 5000 Ao and electrons accelerated through 100V used as the illuminating substance.

5000 Ao = 5000 × 10-10 m 1/d = 2 sin β/1.22λ dmin = 1.22 λ/ 2 sin β λd = 1.22/10 × 10-10 m When the 100V light is used, d’min = 1.22 λd/ 2 sin β d’min = 1.22 × 1.22 × 10-10/2 sin β The required...

### Consider a two-slit interference arrangements such that the distance of the screen from the slits is half the distance between the slits. Obtain the value of D in terms of λ such that the first minima on the screen falls at a distance D from the centre O.

The minima will occur when ∆x = S2P – S1P = (2n-1/2)λ S1P = √D2 + (D – x)2 S2P = √D2 + (D + x)2 T2P = D + x T1P = D – x [D2 + (D+x)2]-1/2 – [D2+(D-x)2]1/2 = λ/2   D =...

### Four identical monochromatic sources A, B, C, D as shown in the figure produce waves of the same wavelength λ and are coherent. Two receiver R1 and R2 are at great but equal distances from B. (i) Which of the two receivers picks up the larger signal? (ii) Which of the two receivers picks up the larger signal when B is turned off? (iii) Which of the two receivers picks up the larger signal when D is turned off? (iv) Which of the two receivers can distinguish which of the sources B or D has been turned off?

i) R2 is the larger signal. ii) When B is switched off, R1 is y = a cos ꞷt IR1 = a2/2 R2 is y = a cos ꞷt IR2 = a2/2 So, it can be said that R1 and R2 are the same. iii) When D is switched off, R1 is...

### The optical properties of a medium are governed by the relative permittivity (εr) and relative permeability (µr ). The refractive index is defined as √µr εr = n. For ordinary material εr > 0 and µr > 0 and the positive sign is taken for the square root. In 1964, a Russian scientist V. Veselago postulated the existence of material with εr < 0 and µr < 0. Since then such ‘metamaterials’ have been produced in the laboratories and their optical properties studied. For such materials n = -√µrεr. As light enters a medium of such refractive index the phases travel away from the direction of propagation. (i) According to the description above show that if rays of light enter such a medium from the air (refractive index =1) at an angle θ in 2nd quadrant, them the refracted beam is in the 3rd quadrant (ii) Prove that Snell’s law holds for such a medium

BC = √μrεr (CD-AE) BC >0; CD > AE -√μrεr AE = BC – √μrεr CD BC = √μrεr (CD – AE) AE > CD Therefore, BC < 0 ii) BC = AC sin θi CD – AE = AC sin θr BC = √μrεr AC sin θi = √μrεr AC sin θr...

### To ensure almost 100 per cent transmissivity, photographic lenses are often coated with a thin layer of dielectric material. The refractive index of this material is intermediated between that of air and glass (which makes the optical element of the lens). A typically used dielectric film is MgF2 (n = 1.38). What should the thickness of the film be so that at the centre of the visible spectrum (5500 Ao) there is maximum transmission?

IA is incident at point A, forming an incident angle I between the air and the film surface. The reflected and refracted rays are AR1 and AD, respectively. D is the point where the partial...

### Consider a 20 W bulb emitting light of wavelength 5000 A ° and shining on a metal surface kept at a distance 2m. Assume that the metal surface has work function of 2 eV and that each atom on the metal surface can be treated as a circular disk of radius 1.5 A ° . (i) Estimate no. of photons emitted by the bulb per second. [Assume no other losses] (ii) Will there be photoelectric emission? (iii) How much time would be required by the atomic disk to receive energy equal to work function (2 eV)? (iv) How many photons would atomic disk receive within time duration calculated in (iii) above? (v) Can you explain how photoelectric effect was observed instantaneously?

Given, P = 20W λ = 5000 Ao = 5000 × 10-10 m d = 2m ϕ0 = 2 eV r = 1.5 × 10-10 m No.of photon emitted per second by the bulb = n’ = dN/dt i) No.of photon emitted by bulb per second = n’ = p λ/hc = 5 ×...

### A particle A with a mass mA is moving with a velocity v and hits a particle B (mass mB) at rest (one dimensional motion). Find the change in the de Broglie wavelength of the particle A. Treat the collision as elastic.

According to the law of conservation of momentum, mAv + mB0 = mAv1 + mBv2 mA(v-v1) = mBv2 1/2 mAv2 = 1/2 mAv12 + 1/2mBv22 mA(v-v1)(v+v1) = mBv22 v1 = (mA-mB/mA+mB)v v2 = (2mA/mA+mB)v λinitial =...

### A student performs an experiment on the photoelectric effect, using two materials A and B. A plot of Vstop vs ν is given in the figure. (i) Which material A or B has a higher work function? (ii) Given the electric charge of an electron = 1.6 × 10–19 C, find the value of h obtained from the experiment for both A and B. Comment on whether it is consistent with Einstein’s theory

i) The threshold frequency of A is vOA = 5 × 1014 Hz The threshold frequency of B is vOB = 10 × 1014 Hz Work function is ϕ = hv0 ϕOA/ ϕOB = 5 × 1014/10 × 1014 < 1 ϕOA <...

### Consider an electron in front of the metallic surface at a distance d (treated as an infinite plane surface). Assume the force of attraction by the plate is given as 2 2 0 1 4 4 q πε d Calculate work in taking the charge to an infinite distance from the plate. Taking d = 0.1nm, find the work done in electron volts. [Such a force law is not valid for d < 0.1nm].

An electron is progressively moved in the picture. The distance travelled by the electron is x, and this is accomplished by the application of an external force. F = q2/4×4πε0d2 d = 0.1 nm = 10-10 m...

### Consider a thin target ( m square, m thickness) of sodium, which produces a photocurrent of 100µA when a light of intensity 100W/m2 (λ = 660nm) falls on it. Find the probability that a photoelectron is produced when a photon strikes a sodium atom. [Take density of Na = 0.97 kg/m3].

Area = 10-4 m2 Thickness = 10-3 m Current = 10-4 A Intensity = 100 W/m2 Mass = (volume)(density) = 0.97 × 10-4 gm No.of target atoms = 0.254 × 1019 Total energy = nhv Where n = 3.3 × 1016 The...

### A neutron beam of energy E scatters from atoms on a surface with a spacing d = 0.1nm. The first maximum of intensity in the reflected beam occurs at θ = 30°. What is the kinetic energy E of the beam in eV?

From Bragg’s law, 2d sin θ = nλ p = h/λ = 6.6 × 10-24 kg m/s E = 1/2 mv2 = p2/2m E = 0.085 eV

### Two particles A and B of de Broglie wavelengths λ1 and λ2 combine to form a particle C. The process conserves momentum. Find the de Broglie wavelength of the particle C. (The motion is one dimensional).

By de-Broglie wavelengths, λ=h/p p=h/ λ p1 = h/ λ1 p2 = h/ λ2 p3 = h/ λ3 There are 4 possible cases and they are: Case 1: When p1 and p2 are positive, then λ3 = λ1 λ1/ λ1+ λ2 Case 2: When p1 and p2...

### Two monochromatic beams A and B of equal intensity I, hit a screen. The number of photons hitting the screen by beam A is twice that by beam B. Then what inference can you make about their frequencies?

Let nA represent the number of photons that fall per second from beam A, and nB represent the number of photons that fall per second from beam B. Given, A has twice as many photons reaching the...

### Assuming an electron is confined to a 1nm wide region, find the uncertainty in momentum using the Heisenberg Uncertainty principle. You can assume the uncertainty in position ∆x as 1nm. Assuming p = ∆p, find the energy of the electron in electron volts.

As the electrons rotate in a circular path, ∆r = 1 nm = 10-9 m ∆p = h/∆x ∆p = (331/314) × 10-25 E = 1/2 mv2 = ∆p2/2m E = 3.8 × 10-2 eV

### Consider a metal exposed to light of wavelength 600 nm. The maximum energy of the electron doubles when light of wavelength 400 nm is used. Find the work function in eV.

Let K1 and K2 be the maximum energy emitted by the electrons when 600 and 400 nm wavelength is used. K2 = 2K1 Therefore, the work function, ϕ = 1.03 eV

### Consider the figure given for photoemission. How would you reconcile with momentum-conservation? Note light (photons) have momentum in a different direction than the emitted electrons.

When photons collide with a metal surface, momentum is transferred to the atoms on the metal surface, resulting in a reduction in photon speed. Photons transmit their momentum to the nucleus and...

### There are two sources of light, each emitting with a power of 100 W. One emits X-rays of wavelength 1nm and the other visible light at 500 nm. Find the ratio of a number of photons of X-rays to the photons of visible light of the given wavelength?

Ex = hvx Ev = hvv Let nx and nv be the no.of photons from x-rays and visible light are of equal energies and they emit 100 W power. nxhvx = nvhvv nx/nv = vv/vx = λx/λv nx/nv = 1 nm/500 nm nx:nv =...

### Do all the electrons that absorb a photon come out as photoelectrons?

No, not every electron absorbed as a photon emerges as a photoelectron.

### There are materials which absorb photons of shorter wavelength and emit photons of longer wavelength. Can there be stable substances which absorb photons of larger wavelength and emit light of shorter wavelength?

When the frequency of a photon drops, the wavelength of the photon rises. There are two alternative scenarios: case one, in which the photons have a shorter wavelength and the energy is consumed...

### (i) In the explanation of the photoelectric effect, we assume one photon of frequency ν collides with an electron and transfers its energy. This leads to the equation for the maximum energy Emax of the emitted electron as Emax = hν – φ0 where φ0 is the work function of the metal. If an electron absorbs 2 photons (each of frequency ν ) what will be the maximum energy for the emitted electron? (ii) Why is this fact (two-photon absorption) not taken into consideration in our discussion of the stopping potential?

i)According to the question, the electron absorbs two protons with frequencies of v and v'= 2v, where v' is the frequency of the released electron. Emax = hv – ϕ0 ii) There is no emission since the...

### A proton and an α-particle are accelerated, using the same potential difference. How are the de Broglie wavelengths λp and λa related to each other?

The proton and α-particle are accelerated at the same potential difference. λ = h/√2mqv λ is proportional to 1/√mq λp/ λa = √maqa/mpqp = √8 Hence,  proton wavelength is √8 times α-particle....

### A particle moves in a closed orbit around the origin, due to a force which is directed towards the origin. The de Broglie wavelength of the particle varies cyclically between two values λ1, λ2 with λ1>λ2. Which of the following statement are true? (a) The particle could be moving in a circular orbit with origin as centre (b) The particle could be moving in an elliptic orbit with origin as its focus (c) When the de Broglie wavelength is λ1, the particle is nearer the origin than when its value is λ2 (d) When the de Broglie wavelength is λ2, the particle is nearer the origin than when its value is λ1

The correct answers are b) The particle could be moving in an elliptic orbit with origin as its focus d) When the de Broglie wavelength is λ2, the particle is nearer the origin than when its value...

### Photons absorbed in matter are converted to heat. A source emitting n photon/sec of frequency ν is used to convert 1kg of ice at 0°C to water at 0°C. Then, the time T taken for the conversion (a) decreases with increasing n, with ν fixed (b) decreases with n fixed, ν increasing (c) remains constant with n and ν changing such that nν = constant (d) increases when the product nν increases

The correct answers are a) decreases with increasing n, with ν fixed b) decreases with n fixed, ν increasing c) remains constant with n and ν changing such that nν = constant

### The de Broglie wavelength of a photon is twice the de Broglie wavelength of an electron. The speed of the electron is ve = c/100. Then (a) (b) (c) (d) The correct answers are b) Ee/Ep = 10-2 c) Pe/mec = 10-2

### Two particles A1 and A2 of masses m1, m2 (m1 > m2) have the same de Broglie wavelength. Then (a) their momenta are the same (b) their energies are the same (c) energy of A1 is less than the energy of A2 (d) energy of A1 is more than the energy of A2

The correct answers are a) their momenta are the same c) energy of A1 is less than the energy of A2

### Relativistic corrections become necessary when the expression for the kinetic energy 1/2 mv2 , becomes comparable with mc2, where m is the mass of the particle. At what de Broglie wavelength will relativistic corrections become important for an electron? (a) λ =10nm (b) nm (c) nm (d) nm

The correct answers are c) λ =10–4nm d) λ =10–6nm

### An electron is moving with an initial velocity and is in a magnetic field Then it’s de Broglie wavelength (a) remains constant (b) increases with time (c) decreases with time (d) increases and decreases periodically

The correct answer is a) remains constant

### A proton, a neutron, an electron and an α-particle have the same energy. Then their de Broglie wavelengths compare as (a) λp = λn > λe > λα (b) λα < λp = λn < λe (c) λe < λp = λn > λα (d) λe = λp = λn = λα

The correct answer is b) λα < λp = λn < λe

### Consider Fig. 11.7 in the NCERT textbook of physics for Class XII. Suppose the voltage applied to A is increased. The diffracted beam will have the maximum at a value of θ that (a) will be larger than the earlier value (b) will be the same as the earlier value (c) will be less than the earlier value (d) will depend on the target

The correct answer is c) will be less than the earlier value

### Consider a beam of electrons (each electron with energy E0) incident on a metal surface kept in an evacuated chamber. Then (a) no electrons will be emitted as only photons can emit electrons (b) electrons can be emitted but all with an energy, E0 (c) electrons can be emitted with any energy, with a maximum of E0 – φ (φ is the work function) (d) electrons can be emitted with any energy, with a maximum of E0

The correct answer is d) electrons can be emitted with any energy, with a maximum of E0

### The wavelength of a photon needed to remove a proton from a nucleus which is bound to the nucleus with 1 MeV energy is nearly (a) 1.2 nm (b) nm (c) nm (d) nm

The correct answer is b) 1.2 × 10–3 nm

### A particle is dropped from a height H. The de Broglie wavelength of the particle as a function of height is proportional to (a) H (b) H ½ (c) H0 (d) The correct answer is d) H–1/2

### A person driving a car suddenly applies the brakes on seeing a child on the road ahead. If he is not wearing a seat belt, he falls forward and hits his head against the steering wheel. Why?

If a person is not wearing a seat belt and abruptly hits the brakes, he will fall forward and bang his head against the steering wheel because his upper body will continue to move in the same...

### The position time graph of a body of mass 2 kg is as given in the figure. What is the impulse on the body at t – 0 sec and t = 4 sec.

Mass of body, m = 2 kg Time, t = 0 Initial velocity, v1 = 0 From graph, we know that t ≥ 0 to t ≤ 4 which is a straight line. The velocity of the body is constant v2 = tan θ = 3/4 = 0.75 m/s At t ≥...

### A person of mass 50 kg stands on a weighing scale on a lift. If the lift is descending with a downward acceleration of 9 m/s2 what would be the reading of the weighing scale?

When the lift lowers with an acceleration a, the apparent weight on the weighing scale decreases. W' denotes the apparent weight. Therefore, W' = R = (mg – ma) = m(g – a) As a result, W' = 50(10-9)...

### A girl riding a bicycle along a straight road with a speed of 5 m/s throws a stone of mass 0.5 kg which has a speed of 15 m/s with respect to the ground along her direction of motion. The mass of the girl and the bicycle is 50 kg. Does the speed of the bicycle change after the stone is thrown? What is the change in speed, if so?

Given, m1 = 50 kg m2 = 0.5 kg u1 = 5 m/s u2 = 5 m/s v1 = ? v2 = 15 m/s The law of conservation of momentum states that Final momentum equals initial momentum. u1 = m1v1 + m2v2 (m1 + m2) We obtain by...

### A body of mass 10 kg is acted upon by two perpendicular forces, 6N and 8N. The resultant acceleration of the body is a) 1 m/s2 at an angle of tan-1 (4/3) w.r.t 6N force b) 0.2 m/s2 at an angle of tan-1 (4/3) w.r.t 6N force c) 1 m/s2 at an angle of tan-1(3/4) w.r.t 8N force d) 0.2 m/s2 at an angle of tan-1(3/4) w.r.t 8N force

The correct answers are a) 1 m/s2 at an angle of tan-1 (4/3) w.r.t 6N force c) 1 m/s2 at an angle of tan-1(3/4) w.r.t 8N force

### Two billiard balls A and B, each of mass 50 g and moving in opposite directions with speed of 5 m/s each, collide and rebound with the same speed. If the collision lasts for 10-3 seconds, which of the following statements are true? a) the impulse imparted to each ball is 0.25 kg.m/s and the force on each ball is 250N b) the impulse imparted to each ball is 0.25 kg.m/s and the force exerted on each ball is 25 × 10-5 N c) the impulse imparted to each ball is 0.5 Ns d) the impulse and the force on each ball are equal in magnitude and opposite in direction

The correct answer is: c) the impulse imparted to each ball is 0.5 Ns d) the impulse and the force on each ball are equal in magnitude and opposite in direction

### In the figure, a body A of mass m slides on a plane inclined at angle θ1 to the horizontal and μ1 is the coefficient of friction between A and the plane. A is connected by a light string passing over a frictionless pulley to another body B, also of mass m, sliding on a frictionless plane inclined at angle θ2 to the horizontal. Which of the following statements are true?;a) A will never move up the plane b) A will just start moving up the plane when c) for A to move up the plane, θ2 must always be greater than θ1 d) B will always slide down with constant speed

The correct answer is: b) A will just start moving up the plane when $$\mu =\frac{\sin {{\theta }_{2}}-\sin {{\theta }_{1}}}{\cos {{\theta }_{1}}}$$ c) for A to move up the plane, θ2 must always be...

### Mass m1 moves on a slope making an angle θ with the horizontal and is attached to mass m2 by a string passing over a frictionless pulley as shown in the figure. The coefficient of friction between m1 and the sloping surface is μ. Which of the following statements are true?;a) if m2 > m1 sin θ, the body will move up the plane b) if m2 > m1(sin θ + μ cos θ), the body will move up the plane c) if m2 < m1 (sin θ + μ cos θ), the body will move up the plane d) if m2 < m1 (sin θ – μ cos θ), the body will move down the plane

The correct answer is: b) if m2 > m1(sin θ + μ cos θ), the body will move up the plane d) if m2 < m1 (sin θ – μ cos θ), the body will move down the plane

### In figure, the coefficient of friction between the floor and the body B is 0.1. The coefficient of friction between the bodies B and A is 0.2. A force F is applied as shown on B. The mass of A is m/2 and of B is m. Which of the following statements are true?;a) the bodies will move together if F = 0.25 mg b) the body A will slip with respect to B if F = 0.5 mg c) the bodies will move together if F = 0.5 mg d) the bodies will be at rest if F = 0.1 mg e) the maximum value of F for which the two bodies will move together is 0.45 mg

The correct answer is: a) the bodies will move together if F = 0.25 mg b) the body A will slip with respect to B if F = 0.5 mg d) the bodies will be at rest if F = 0.1 mg e) the maximum value of F...

### The motion of a particle of mass m is given by x = 0 for t < 0 sec, x(t) = A sin 4p t for 0 < t < (1/4) sec, and x = 0 for t > (1/4) sec. Which of the following statements is true? a) the force at t = (1/8) sec on the particle is -16π2Am b) the particle is acted upon by on impulse of magnitude 4π2Am at t = 0 sec and t = (1/4) sec c) the particle is not acted upon by any force d) the particle is not acted upon by a constant force e) there is no impulse acting on the particle

The correct answer is: a) the force at t = (1/8) sec on the particle is -16π2Am b) the particle is acted upon by on impulse of magnitude 4π2Am at t = 0 sec and t = (1/4) sec d) the particle is not...

### A car of mass m starts from rest and acquires a velocity along the east a in two seconds. Assuming the car moves with uniform acceleration, the force exerted on the car is a) mv/2 eastward and is exerted by the car engine b) mv/2 eastward and is due to the friction on the tyres exerted by the road c) more than mv/2 eastward exerted due to the engine and overcomes the friction of the road d) mv/2 exerted by the engine

The correct answer is b) mv/2 eastward and is due to the friction on the tyres exerted by the road

### A body with mass 5 kg is acted upon by a force F=-3i+4j N. If its initial velocity at t = 0 is v=6i-12j m/s, the time at which it will just have a velocity along the y-axis is a) never b) 10 s c) 2 s d) 15 s

The correct answer is b) 10 s

### A body of mass 2 kg travels according to the law x(t) = p(t) + qt2 + rt3 where p = 3 m/s, q = 4 m/s2, and r = 5 m/s3. The force acting on the body at t=2 seconds is a) 136 N b) 134 N c) 158 N d) 68 N

The correct answer is a) 136 N

### A hockey player is moving northward and suddenly turns westward with the same speed to avoid an opponent. The force that acts on the player is a) frictional force along westward b) muscle force along southward c) frictional force along south-west d) muscle force along south-west

The correct answer is c) frictional force along the south-west

### Conservation of momentum in a collision between particles can be understood from a) conservation of energy b) Newton’s first law only c) Newton’s second law only d) both Newton’s second and third law

The correct answer is d) both Newton’s second and third law

### In the previous problem, the magnitude of the momentum transferred during the hit is a) zero b) 0.75 kg.m/s c) 1.5 kg.m/s d) 14 kg.m/s

The correct option is c) 1.5 kg.m/s

### A cricket ball of mass 150 g has an initial velocity u=3i+4j m/s and a final velocity v=-(3i+4j) m/s after being hit. The change in momentum a) zero (b)-(0.45i+0.6j) (c)-(0.9i+1.2j)) (d)-5(i+j)

The correct answer is  (c) -(0.9i+1.2j)

### A metre scale is moving with uniform velocity. This implied a) the force acting on the scale is zero, but a torque about the centre of mass can act on the scale b) the force acting on the scale is zero and the torque acting about the centre of mass of the scale is also zero c) the total force acting on it need not be zero but the torque on it is zero d) neither the force nor the torque needs to be zero

The correct answer is b) the force acting on the scale is zero and the torque acting about the centre of mass of the scale is also zero

### A ball is travelling with uniform translator motion. This means that a) it is at rest b) the path can be a straight line or circular and the ball travels with uniform speed c) all parts of the ball have the same velocity and the velocity is constant d) the centre of the ball moves with constant velocity and the ball spins about its centre uniformly

The correct option is c) all parts of the ball have the same velocity and the velocity is constant

### A balloon filled with helium rises against gravity increasing its potential energy. The speed of the balloon also increases as it rises. How do you reconcile this with the law of conservation of mechanical energy? You can neglect the viscous drag of air and assume that the density of air is constant.

The net buoyant force Equals vpg when the dragging viscous force of the air on the balloon is ignored. Where v is the volume of air that has been displaced. The upward net density is denoted by p....

### Two identical steel cubes collide head-on face to face with a speed of 10 cm/s each. Find the maximum compression of each. Young’s modulus for steel = Y = 2 × 1011 N/m2.

Y = stress/strain Y = FL/A∆L WD = F∆L KE = 5 × 10-4 J WD = KE ∆L = 5 × 10-7 m

### A rocket accelerates straight up by ejecting gas downwards. In a small time interval ∆t, it ejects a gas of mass ∆m at a relative speed u. Calculate KE of the entire system at t + ∆t and t and show that the device that ejects gas does work = (1/2) ∆m u2 in this time interval.

M is the rocket's mass at any given moment t. The rocket's velocity is v. The mass of the gas expelled during the time interval t is m. As a result,  K = 1/2 u2∆m

### A curved surface as shown in the figure. The portion BCD is free of friction. There are three spherical balls of identical radii and masses. Balls are released from one by one from A which is at a slightly greater height than C. with the surface AB, ball 1 has large enough friction to cause rolling down without slipping; ball 2 has a small friction and ball 3 has a negligible friction. a) for which balls is total mechanical energy conserved? b) which ball can reach D? c) for balls which do not reach D, which of the balls can reach back A?

a) For ball 1 the total mechanical energy is conserved b) Ball 1 reaches D c) Ball 3 reaches back A

### A block of mass 1 kg is pushed up a surface inclined to horizontal at an angle of 30o by a force of 10 N parallel to the inclined surface. The coefficient of friction between the block and the incline is 0.1. If the block is pushed up by 10 m along the incline, calculate a) work done against gravity b) work done against the force of friction c) increase in potential energy d) increase in kinetic energy e) work done by an applied force

a) Work against gravity equals mgh 5 m= h 50 J WD against gravity b) The work done against the friction force is fs = 53 J. d) WD against gravity = 50 J increase in PE d) The system's increase in KE...

### On complete combustion, a litre of petrol gives off heat equivalent to 3 × 107 J. In a test drive a car weighing 1200 kg, including the mass of driver, runs 15 km per litre while moving with a uniform speed on a surface and air to be uniform, calculate the force of friction acting on the car during the test drive, if the efficiency of the car engine were 0.5.

car engine Efficiency = 0.5 Energy given by the car with 1 litre of petrol = 1.5 × 107 WD = 1.5 × 107 f = 103 N

### An adult weighing 600 N raises the centre of gravity of his body by 0.25 m while taking each step of 1 m length in jogging. If he jogs for 6 km, calculate the energy utilized by him in jogging assuming that there is no energy loss due to friction of ground and air. Assuming that the body of the adult is capable of converting 10% of energy intake in the form of food, calculate the energy equivalents of food that would be required to compensate energy utilized for jogging.

The energy used up is given as = mgh mg = 600 N h = 0.25m No.of steps in 6 km = 6000 steps Energy used in 6000 m = (6000)(600)(0.25)J Energy utilized in jogging = 9 × 104 J

### An engine is attached to a wagon through a shock absorber of length 1.5 m. The system with a total mass of 50,000 kg is moving with a speed of 36 km/h when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by 1.0 m. If 90% of the energy of the wagon is lost due to friction, calculate the spring constant.

KE = 1/2 mv2 m = 50000 kg v = 10 m/s KE = 2500000J KE of spring = 10% of the KE wagon K = 5 × 105 N/m

### Suppose the average mass of raindrops is 3.0 × 10^(-5) kg and their average terminal velocity 9 m/s. Calculate the energy transferred by rain to each square meter of the surface at a place which receives 100 cm of rain in a year.

Energy transferred by the rain to the surface of the earth = 1/2 mv2 The velocity of the rain = 9 m/s Mass = (volume)(density) = 1000 kg Energy transferred by 100 cm rainfall = 1/2 mv2 =...

### A raindrop of mass 1.00 g falling from a height of 1 km hits the ground with a speed of 50 m/s. Calculate a) the loss of PE of the drop b) the gain in KE of the drop c) is the gain in KE equal to loss of PE? If not why?

a) PE at the highest point = 10 J b) Gain in KE = 1/2 mv2 = 1.250 J c) Gain in KE is not equal to the PE

### The bob A of a pendulum released from horizontal to the vertical hits another bob B of the same mass at rest on a table as shown in the figure. If the length of the pendulum is 1 m, calculate a) the height to which bob A will rise after collision b) the speed with which bob B starts moving. Neglect the size of the bobs and assume the collision to be elastic.

a) After the impact, bob A does not rise much because the PE of bob A is converted to KE and the momentum is transferred to bob B. (B) The speed of bob B is calculated as the sum of bob A's KE and...

### Consider a one-dimensional motion of a particle with total energy E. There are four regions A, B, C, and D in which the relation between potential energy V, kinetic energy (K) and total energy is as given below: Region A: V > E Region B: V < E Region C: K > E Region D: V > K State with reason in each case whether a particle can be found in the given region or not.

E = V + K and V > E for area A, implying that the KE is negative and therefore this is not feasible. K = E – V and V E for area B, implying that both energies are larger than zero. V = E – K and...

### A ball of mass m, moving with a speed 2v0 collides inelastically with an identical ball at rest. Show that a) for a head-on collision, both the balls move forward b) for a general collision, the angle between the two velocities of scattered balls is less than 90o.

a) Let v1 and v2 be the velocities of the two balls after the collision. According to the law of conservation of momentum, mv0 = mv1 + mv2 v2 = v1 + 2ev0 e < 1 b) Using the law of conservation of...

### A graph of potential energy V(x) versus x is shown in the figure. A particle of energy E0 is executing motion in it. Draw graph of velocity and kinetic energy versus x for one complete cycle AFA.

From the given graph of KE versus x From the below graph of velocity versus x

### A bob of mass m suspended by a light string of length L is whirled into a vertical circle as shown in the figure. What will be the trajectory of the particle if the string is cut at a) point B b) point C c) point X

a) When the string is severed at point B, the particle's tangential velocity will be vertically downward, and the bob will travel in the same direction. b) When the string is severed at point C, the...

### Two bodies of unequal mass are moving in the same direction with equal kinetic energy. The two bodies are brought to rest by applying retarding force of the same magnitude. How would the distance moved by them before coming to rest compare?

KE1 = KE2 WD1 = WD2 F1s1 = F2s2 F1 = F2 s1 = s2

### Give an example of a situation in which an applied force does not result in a change in kinetic energy.

The kinetic energy of work done in a circular motion remains unchanged.

### The average work done by a human heart while it beats once is 0.5 J. Calculate the power used by heart if it beats 72 times in a minute.

P = WD/time WD is one beat of heart = 0.5 J WD in 72 beats = 36 J P = WD/t = 0.6 W

### Calculate the power of a crane in watts, which lifts a mass of 100 kg to a height of 10 min 20 sec.

P = WD/time = Fs cos θ/t = mgh cos θ/t h = 10 m t = 20 sec F = mg = 1000 Therefore, P = 500 Watts

### In an elastic collision of two billiard balls, which of the following quantities remain conserved during the short time of collision of the balls a) kinetic energy b) total linear momentum Give a reason for your answer in each case.

Because there is no non-conservative force, the kinetic energy and total linear momentum of the billiard balls are preserved.

### A body is moved along a closed loop. Is the work done in moving the body necessarily zero? If not, state the condition under which work done over a closed path is always zero.

When the conservative force acts on the body during motion, the work done by the moving body is zero. When a non-conservative force acts on a moving body, the work done by the body is not zero.

### A body falls towards earth in the air. Will its total mechanical energy be conserved during the fall? Justify.

The free-falling body's total mechanical energy is not preserved since it is utilised to overcome the frictional force of the air molecules.

### Calculate the work done by a car against gravity in moving along a straight horizontal road. The mass of the car is 400 kg and the distance moved is 2m.

WD = Fs cos θ WD = Fs cos 90o = 0 Hence, the work done by the car against the gravity is zero.

### A body is being raised to a height h from the surface of the earth. What is the sign of work done by a) applied force b) gravitational force

a) The applied force produces positive work. b) The gravitational pull produces negative work.

### Why is electrical power required at all when the elevator is descending? Why should there be a limit on the number of passengers in this case?

In the event of an elevator falling, the number of passengers is limited because it is not a free fall and descends at a constant speed.

### A rough inclined plane is placed on a cart moving with a constant velocity u on horizontal ground. A block of mass M rests on the incline. Is any work done by a force of friction between the block and incline? Is there then a dissipation of energy?

The block is clearly tilted on the plane in the illustration above. There is no work done since there is no displacement and no waste of energy.

### Two blocks M1 and M2 having equal mass are free to move on a horizontal frictionless surface. M2 is attached to a massless spring as shown in the figure. Initially, M2 is at rest and M1 is moving toward M2 with speed v and collides head-on with M2.;a) while spring is fully compressed all the KE of M1 is stored as PE of spring b) while spring is fully compressed the system momentum is not conserved, though final momentum is equal to the initial momentum c) if spring is massless, the final state of the M1 is the state of rest d) if the surface on which blocks are moving has friction, then a collision cannot be elastic

c) if spring is massless, the final state of the M1 is a state of rest d) if the surface on which blocks are moving has friction, then a collision cannot be elastic

### A bullet of mass m fired at 30o to the horizontal leaves the barrel of the gun with a velocity v. The bullet hits a soft target at a height h above the ground while it is moving downward and emerges out with half the kinetic energy it had before hitting the target. Which of the following statements are correct in respect of bullet after it emerges out of the target? a) the velocity of the bullet will be reduced to half its initial value b) the velocity of the bullet will be more than half of its earlier velocity c) the bullet will continue to move along the same parabolic path d) the bullet will move in a different parabolic path e) the bullet will fall vertically downward after hitting the target f) the internal energy of the particles of the target will increase

b) the velocity of the bullet will be more than half of its earlier velocity d) the bullet will move in a different parabolic path f) the internal energy of the particles of the target will...

### A man, of mass m, standing at the bottom of the staircase, of height L, climbs it and stands at its top. a) work done by all forces on man is zero b) work done by all the force on man is zero c) work done by the gravitational force on man is mgL d) the reaction force from a step does not do work because the point of application of the force does not move while the force exists

b) work done by all the force on man is zero d) the reaction force from a step does not do work because the point of application of the force does not move while the force exists

c) 1.05 × 104 N

d) 155.0 J

a) 250π2

b) 50 J

\b)

b) V = E, K = O

### Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track as shown in the figure. Which of the following statement is correct?a) both the stones reach the bottom at the same time but not with the same speed b) both stone reach the bottom with the same speed and stone I reaches the bottom earlier than stone II c) both the stones reach the bottom with the same speed and stone II reaches the bottom earlier than stone I d) both the stones reach the bottom at different times and with different speeds

c) both the stones reach the bottom with the same speed and stone II reaches the bottom earlier than stone I

### During inelastic collision between two bodies, which of the following quantities always remain conserved? a) total kinetic energy b) total mechanical energy c) total linear momentum d) speed of each body

c) total linear momentum

### A body is falling freely under the action of gravity alone in a vacuum. Which of the following quantities remain constant during the fall? a) kinetic energy b) potential energy c) total mechanical energy d) total linear momentum

c) total mechanical energy

c) zero

### A man squatting on the ground gets straight up and stand. The force of reaction of ground on the man during the process is a) constant and equal to mg in magnitude b) constant and greater than mg in magnitude c) variable but always greater than mg d) at first greater than mg, and later becomes equal to mg

d) at first greater than mg, and later becomes equal to mg

### A proton is kept at rest. A positively charged particle is released from rest at a distance d in its field. Consider two experiments; one in which the charged particles is also a proton and in another, a positron. In the same time t, the work done on the two moving charged particles is a) the same as the same force law is involved in the two experiments b) less for the case of a positron, as the positron moves away more rapidly and the force on it weakens c) more for the case of a positron, as the positron moves away from a larger distance d) same as the work done by charged particle on the stationary proton

c) more for the case of a positron, as the positron moves away from a larger distance

### An electron and a proton are moving under the influence of mutual forces. In calculating the change in the kinetic energy of the system during motion, one ignores the magnetic force of one on another. This is because, a) the two magnetic forces are equal and opposite, so they produce no net effect b) the magnetic forces do no work on each particle c) the magnetic forces do equal and opposite work on each particle d) the magnetic forces are necessarily negligible

b) the magnetic forces do no work on each particle

### Five long wires A, B, C, D, and E each carrying I are arranged to form edges of a pentagonal prism as shown in the figure. Each carries current out of the plane of paper. a) what will be magnetic induction at a point on the axis O> Axis is at a distance R from each wire b) what will be the field if current in one of the wires is switched off c) what if current in one of the wire A is reversed

(a) Because A, B, C, D, and E are perpendicular to the plane of paper at the specified places, the magnetic induction at a point on the axis will be zero, which is represented by R. (b) The field...

### A multirange current meter can be constructed by using a galvanometer circuit shown in the figure. We want a current meter that can measure 10 mA, 100 mA, and 1 A using a galvanometer of resistance 10Ω and that produces maximum deflection for a current of 1 mA. Find S1, S2, and S3 that have to be used.

I1 is measured as = 10 mA = IGG = (I1 – IG)(S1 + S2 + S3) I2 is measured as = 100 mA = IG(G+S1)=(I2-IG)(S2-S3) I3 is measured as = 1 A = IG(G+S1+S2)=(I3-IG)(S3) S1 = 1 Ω S2 = 0.1 Ω S3 = 0.01...

### Consider a circular current-carrying loop of radius R in the x-y plane with centre at the origin. Consider the line integral a) show that monotonically increases with L b) use an appropriate Amperian loop to that where I is the current in the wire c) verify directly the above result d) suppose we replace the circular coil by a square coil of sides R carrying the same current I. What can you say about a) A circular current-carrying loop's magnetic field is given as $$\Im (L)=\int_{-L}^{+L}{Bdl}=2Bl$$ It is a L function that increases monotonically. b) The Amperian loop is defined as follows:...

### A uniform conducting wire of length 12a and resistance R is wound up as a current-carrying coil in the shape of i) an equilateral triangle of side a; ii) a square if sides a and iii) a regular hexagon of sides a. The coil is connected to a voltage source V0. Find the magnetic moment of the coils in each case.

a) A triangle with side a that is equilateral. Number of loops = 4 A = √3/4 a2 m = Ia2√3 is the magnetic moment. b) In the case of a square with sides of a A = a2 Number of loops = 3 m =...

### An electron and a positron are released from (0, 0, 0) and (0, 0, 1.5R) respectively, in a uniform magnetic field each with an equal momentum of magnitude p = eBR. Under what conditions on the direction of momentum will the orbits be non-intersecting circles?

The circular orbits of electron and positron should not overlap when the centres are bigger than 2R. Let the distance between Cp and Ce be denoted by d. Then  d2 = 4R2 + 9/4R2 – 6R2 cosθ Because d...

### A rectangular conducting loop consists of two wires on two opposite sides of length l joined together by rods of length d. The wires are each of the same material but with cross-sections differing by a factor of 2. The thicker wire has a resistance R and the rods are of low resistance, which in turn are connected to a constant voltage source Vo. The loop is placed in uniform a magnetic field B at 45oto its plane. Find τ, the torque exerted by the magnetic field on the loop about an axis through the centres of rods.

F1 = i1l B sin 90o = V0/2R lB τ1= d/2√2 F1 = V0ldB/2√2 R τ = 1/4√2 V0AB/R

### A 100 turn rectangular coil ABCD is hung from one arm of a balance. A mass 500 g is added to the other arm to balance the weight of the coil. A current 4.9 A passes through the coil and a constant magnetic field of 0.2 T acting inward is switched on such that only arm CD of length 1 cm lies in the field. How much additional mass ‘m’ must be added to regain the balance?

When t = 0, the external magnetic field is off. Mgl = Wcoil l 0.5 gl = Wcoil l Wcoil = 0.5 9.8 N Let m be the mass that is added to restore equilibrium. The magnetic field is activated when the...

### A long straight wire carrying a current of 25 A rests on a table as shown in the figure. Another wire PQ of length 1 m, mass 2.5 g carries the same current but in the opposite direction. The wire PQ is free to slide up and down. To what height will PQ rise?

The magnetic field produced by a long straight current-carrying wire is given as B = μ0I/2πh Magnetic force on the small conductor is F = BIl sin θ = BIl F = mg = μ0I2l/2πh h = 0.51...

### A multirange voltmeter can be constructed by using a galvanometer circuit as shown in the figure. We want to construct a voltmeter that can measure 2V, 20V, and 200V using galvanometer of resistance 10Ω and that produces maximum deflection for a current of 1 mA. Find R1, R2, and R3 that have to be used.

iG(G+R1) = 2 for 2V range iG(G+R1+R2) = 20 for 20V range iG(G+R1+R2+R3) = 200 for 200V range Solving the above, we get R1 = 1990 Ω R2 = 18kΩ R3 = 180 kΩ

### Do magnetic forces obey Newton’s third law. Verify for two current elements located at the origin and located at (0,R,0). Both carry current I.

If there is no current flowing in the conductors that are parallel to each other, the magnetic forces do not obey Newton's third law.

### An electron enters with a velocity v = v0i into a cubical region in which there are uniform electric and magnetic fields. The orbit of the electron is found to spiral down inside the cube in the plane parallel to the x-y plane. Suggest a configuration of fields E and B that can lead to it.

The spiral route is formed by the fields E and B in their current configuration.

### A charged particle of charge e and mass m is moving in an electric field E and magnetic field B. Construct dimensionless quantities and quantities of dimension [T]-1.

mv2/R = evB eB/m = v/R = ꞷ B = F/ev = [MA-1T-2] [ꞷ] = [eB/m]=[v/R] = [T-1]

### A current-carrying loop consists of 3 identical quarter circles of radius R, lying in the positive quadrants of the x-y, y-z, and z-x planes with their centres at the origin, joined together. Find the direction and magnitude of B at the origin.

The quarter's vector sum of the magnetic field at the origin is given as $${{\vec{B}}_{net}}=\frac{1}{4}\left( \frac{{{\mu }_{0}}I}{2R} \right)(\widehat{i}+\widehat{j}+\widehat{k})$$

### Describe the motion of a charged particle in a cyclotron if the frequency of the radio frequency (rf) field were doubled.

The time period of the radio frequency is halved when the frequency is doubled, resulting in a half revolution of the charges.

### The magnetic force depends on v which depends on the inertial frame of reference. Does then the magnetic force differ from inertial frame to frame? Is it reasonable that the net acceleration has a different value in different frames of reference?

Because velocity varies depending on frame of reference, the net acceleration might have a different value in different frames of reference.

### Show that a force that does no work must be a velocity dependent force.

$$dW=\vec{F}\cdot dl=0$$ $$\vec{F}\cdot \vec{v}=0$$

### Verify that the cyclotron frequency ꞷ = eB/m has the correct dimensions of [T]-1.

In a cyclotron, the particle follows a circular path, with magnetic force acting as a centripetal force. mv2/R = evB eB/m = v/R = ꞷ B = F/ev = [MLT-2]/[AT][LT-1] = [MA-1T-2] [ꞷ] = [eB/m] = [v/R] =...

### A charged particle would continue to move with a constant velocity in a region wherein, a) E = 0, B ≠ 0 b) E ≠ 0, B ≠ 0 c) E ≠ 0, B = 0 d) E = 0, B = 0

a) E = 0, B ≠ 0 b) E ≠ 0, B ≠ 0 d) E = 0, B = 0

### A cubical region of space is filled with some uniform electric and magnetic fields. An electron enters the cube across one of its faces with velocity v and a positron enters via opposite face with velocity –v. At this instant, a) the electric forces on both the particles cause identical acceleration b) the magnetic forces on both the particles cause equal accelerations c) both particles gain or lose energy at the same rate d) the motion of the centre of mass (CM) is determined by B alone

b) the magnetic forces on both the particles cause equal accelerations c) both particles gain or lose energy at the same rate d) the motion of the centre of mass (CM) is determined by B alone

### Two identical current-carrying coaxial loops, carry current I in an opposite sense. A simple amperian loop passes through both of them once. Calling the loop as C, a) b) the value of c) there may be a point on C where B and dl are perpendicular d) B vanishes everywhere on C

b) the value of $$\oint\limits_{c}{B.dl$$ is independent of sense of C c) there may be a point on C where B and dl are perpendicular

### Consider a wire carrying a steady current, I placed un a uniform magnetic field B perpendicular to its length. Consider the charges inside the wire. It is known that magnetic forces do not work. This implies that a) motion of charges inside the conductor is unaffected by B since they do not absorb energy b) some charges inside the wire move to the surface as a result of B c) if the wire moves under the influence of B, no work is done by the force d) if the wire moves under the influence of B, no work is done by the magnetic force on the ions, assumed fixed within the wire

b) some charges inside the wire move to the surface as a result of B d) if the wire moves under the influence of B, no work is done by the magnetic force on the ions, assumed fixed within the wire

### The gyro-magnetic ratio of an electron in an H-atom, according to Bohr model is a) independent of which orbit it is in b) negative c) positive d) increases with the quantum number n

a) independent of which orbit it is in b) negative

d) zero

### In a cyclotron, a charged particle a) undergoes acceleration all the time b) speeds up between the dees because of the magnetic field c) speeds up in a dee d) slows down within a dee and speeds up between dees

a) undergoes acceleration all the time

### An electron is projected with uniform velocity along the axis of a current-carrying long solenoid. Which of the following is true? a) the electron will be accelerated along the axis b) the electron path will be circular about the axis c) the electron will experience a force at 45o to the axis and hence execute a helical path d) the electron will continue to move with uniform velocity along the axis of the solenoid

d) the electron will continue to move with uniform velocity along the axis of the solenoid

### A current circular loop of radius R is placed in the x-y plane with centre at the origin. Half of the lop with x > 0 is now bent so that it now lies in the y-z plane. a) the magnitude of magnetic moment now diminishes b) the magnetic moment does not change c) the magnitude of B at (0,0,z),z >> R increases d) the magnitude of B at (0,0,z),z >> R is unchanged

a) the magnitude of magnetic moment now diminishes

a) B ┴ v

### Two charged particles traverse identical helical paths in an opposite sense in a uniform magnetic field a) they have equal z-components of momenta b) they must have equal charges c) they necessarily represent a particle-antiparticle pair d) the charge to mass ratio satisfy: (e/m)1 + (e/m)2 = 0

d) the charge to mass ratio satisfy: (e/m)1 + (e/m)2 = 0

### There are two current-carrying planar coils made each from identical wires of length L. C1 is circular and C2 is square. They are so constructed that they have the same frequency of oscillation when they are placed in the same uniform B and carry the same current. Find a in terms of R.

The circular coil C1 has a radius of R, a length of L, and a number of turns per unit length of n1 = L/2R. The square C2 has a side, a perimeter, and a number of turns per unit length of n2 = L/4a....

### Consider the plane S formed by the dipole axis and the axis of the earth. Let P be a point on the magnetic equator and in S. Let Q be the point of intersection of the geographical and magnetic equators. Obtain the declination and dip angle at P and Q.

The declination is zero, P is in the plane, S is in the north, and P is in the plane. The declination for point P is 0 since it is in the plane S created by the dipole axis and the earth's axis....

### Assume the dipole model for earth’s magnetic field B which is given by Bv = vertical component of magnetic field = μ0/4π 2m cos θ/r3, BH = horizontal component of magnetic field = μ0/4π 2m sin θm/r3, θ = 90o latitude as measured from magnetic equator. Find loci of points for which i) |B| is minimum ii) dip angle is zero, and iii) dip angle is ±45o.

a) |B| is minimum at the magnetic equator. b) Angle of dip is zero when θ = π/2 c) When dip angle is ±45o θ = tan-1 is the locus.

### What are the dimensions of χ, the magnetic susceptibility? Consider an H-atom. Guess an expression for χ, up to a constant by constructing a quantity of dimensions of χ, out of parameters of the atom: e, m, v, R and μ0. Here, m is the electronic mass, v is electronic velocity, R is Bohr radius. Estimate the number so obtained and compare with the value of | χ| equivalent to 10-5 for many solid materials.

χm = I/H = intensity of magnetisation/magnetising force χ is dimensionless as I and H has the same units χ = 10-4

### Verify the Ampere’s law for the magnetic field of a point dipole of dipole moment Take C as the closed curve running clockwise along i) the z-axis from z = a > 0 to z = R; ii) along the quarter circle of radius R and centre at the origin, in the first quadrant of x-z plane; iii) along the x-axis from x = R to x = a and iv) along the quarter circle of radius a and centre at the origin in the first quadrant of the x-z plane.

Magnetic field = 0M/4(1/a2-1/R2) along the z-axis b) On the circular arc, the magnetic field at point A is = 0m/4R2. c) (d) The magnetic moment is 0

### Use i) the Ampere’s law for H and ii) continuity of lines of B, to conclude that inside a bar magnet a) lines of H run from the N pole to S pole, while b) lines of B must run from the S pole to N pole.

The amperian loop is denoted by the letter C. We can find the angle between the two points by solving the above equation. cos is negative because it is greater than 90o.

### A bar magnet of magnetic moment m and moment of inertia I is cut into two equal pieces, perpendicular to length. Let T be the period of oscillations of the original magnet about an axis through the midpoint, perpendicular to the length, in a magnetic field B. What would be the similar period T’ for each piece?

T stands for the time period. The moment of inertia is me. The magnet's mass is m. B stands for magnetic field. T = 2I/MB M' = M/2 magnetic dipole moment T' = T/2 is the time period.

### Suppose we want to verify the analogy between electrostatic and magnetostatic by an explicit experiment. Consider the motion of i) electric dipole p in an electrostatic field E and ii) magnetic dipole m in a magnetic field B. Write down a set of conditions on E, B, p, m so that the two movements are verified to be identical.

pE sin θ = μB sin θ pE = μB E = cB pcB = μB p = μ/c

### Three identical bar magnets are riveted together at the centre in the same plane as shown in the figure. This system is placed at rest in a slowly varying magnetic field. It is found that the system of magnets does not show any motion. The north-south poles of one magnet is shown in the figure. Determine the poles of the remaining two.

As the system is in equilibrium, the net torque and net force will be equal to zero.

### Verify the Gauss’s law for magnetic field of a point dipole of dipole moment m at the origin for the surface which is a sphere of radius R.

P is the point at a distance r from O and OP, then the magnetic field is given as: dS is the elementary area of the surface P, then dS = r2 (r2 sin θ d θr) Solving the above we...

### A ball of superconducting material is dipped in liquid nitrogen and placed near a bar magnet. i) In which direction will it move? ii) What will be the direction of it’s magnetic moment?

I The superconducting material will repel the bar magnet. ii) The magnetic moment will be directed from left to right.

### From molecular view point, discuss the temperature dependence of susceptibility for diamagnetism, paramagnetism, and ferromagnetism.

The temperature has little effect on the temperature dependence of susceptibility for a diamagnetism. The temperature affects the temperature dependence of susceptibility for paramagnetism and...

### Explain quantitatively the order of magnitude difference between the diamagnetic susceptibility of N2 and Cu.

nitrogen Density = 28 g/ 22400 cc copper Density = 8 g/ 22400 cc Ratio  = 16 × 10-4 Diamagnetic susceptibility = density of nitrogen/density of copper = 1.6 × 10-4

### A permanent magnet in the shape of a thin cylinder of length 10 cm has M = 106 A/m. Calculate the magnetisation current Im.

Intensity = 106 A/m l = 0.1 m M = IM/l IM = Ml = 105 A

### A proton has spin and magnetic moment just like an electron. Why then its effect is neglected in magnetism of materials?

A comparison of a proton's and an electron's spinning is made by comparing their magnetic dipole moment, which is given as μp = eh/4πmp μe = eh/4πme μp/μe = me/mp = 1/1837 >> 1 μp << μe...

### Essential difference between electrostatic shielding by a conducting shell and magneto static shielding is due to a) electrostatic field lines can end on charges and conductors have free charges b) lines of B can also end but conductors cannot end them c) lines of B cannot end on any material and perfect shielding is not possible d) shells of high permeability materials can be used to divert lines of B from the interior region

a) electrostatic field lines can end on charges and conductors have free charges c) lines of B cannot end on any material and perfect shielding is not possible d) shells of high permeability...

### A long solenoid has 1000 turns per meter and carries a current of 1 A. It has a soft iron core of μr = 1000. The core is heated beyond the Curie temperature Tc a) the H field in the solenoid is unchanged but the B field decreases drastically b) the H and B fields in the solenoid are nearly unchanged c) the magnetisation in the core reverses direction d) the magnetisation in the core diminishes by a factor of about 108

a) the H field in the solenoid is unchanged but the B field decreases drastically d) the magnetisation in the core diminishes by a factor of about 108

### The primary origin(s) of magnetism lies in a) atomic currents b) Pauli exclusion principle c) polar nature of molecules d) intrinsic spin of electron

a) atomic currents d) intrinsic spin of electron

### S is the surface of a lump of magnetic material a) lines of B are necessarily continuous across S b) some lines of B must be discontinuous across S c) lines of H are necessarily continuous across S d) lines of H cannot all be continuous across S

a) lines of B are necessarily continuous across S d) lines of H cannot all be continuous across S

b) 2/3 Am-1

### Consider the two idealized systems: i) a parallel plate capacitor with large plates and small separation and ii) a long solenoid of length L >> R, the radius of the cross-section. In i) E is ideally treated as a constant between plates and zero outside. In ii) magnetic field is constant inside the solenoid and zero outside. These idealised assumptions, however, contradict fundamental laws as below: a) case (i) contradicts Gauss’s law for electrostatic fields b) case (ii) contradicts Gauss’s law for magnetic fields c) case (i) agrees withd) case (ii) contradicts

b) In the case of magnetic fields, instance (ii) violates Gauss's law.

### In a permanent magnet at room temperature a) magnetic moment of each molecule is zero b) the individual molecules have a non-zero magnetic moment which is all perfectly aligned c) domains are partially aligned d) domains are all perfectly aligned

d) all of the domains are exactly matched

### The magnetic field of the earth can be modelled by that of a point dipole placed at the centre of the earth. The dipole axis makes an angle of 11.3o with the axis of the earth. At Mumbai, declination is nearly zero. Then, a) the declination varies between 11.3o W to 11.3o E b) the least declination is 0o c) the plane defined by dipole axis and the earth axis passes through Greenwich d) declination average over the earth must be always negative

a) The declination ranges from 11.3 degrees West to 11.3 degrees East.

### A toroid of n turns, mean radius R and cross-sectional radius a carries current I. It is placed on a horizontal table taken as an x-y plane. Its magnetic moment m a) is non-zero and points in the z-direction by symmetry b) points along the axis of the toroid c) is zero, otherwise, there would be a field falling as 1/r3 at large distances outside the toroid d) is pointing radially outwards

c) is zero; otherwise, a field dropping as 1/r3 at great distances outside the toroid would exist.

### A long solenoid ‘S’ has ‘n’ turns per meter, with diameter ‘a’. At the centre of this coil, we place a smaller coil of ‘N’ turns and diameter ‘b’ (where b < a). If the current in the solenoid increases linearly, with time, what is the induced emf appearing in the smaller coil. Plot graph showing nature of variation in emf, if current varies as a function of mt2 + C.

The solenoid's changing magnetic field is represented as: onI = B1(t) (t) The second coil's magnetic flux is 2 = onI(t).b2 As a result of the solenoid's changing magnetic field, the induced emf in...

### A metallic ring of mass m and radius l (ring being horizontal) is falling under gravity in a region having a magnetic field. If z is the vertical direction, the z-component of the magnetic field is Bz = Bo (1+λ z). If R is the resistance of the ring and if the ring falls with a velocity v, find the energy lost in the resistance. If the ring has reached a constant velocity, use the conservation of energy to determine v in terms of m, B, λ and acceleration due to gravity g.

v = mgR/B02π2λ2l4

### Find the current in the sliding rod AB (resistance = R). B is constant and is out of the paper. Parallel wires have no resistance. v is constant. Switch S is closed at time t = 0.

The angle between A and B = 0o Therefore, the emf is Bvd. -LdI(t)/dt + Bvd = IR LdI(t)/dt + RI (t) = Bvd Solving the equation, we get I = Bvd/R [1-e-Rt/2]