The advantage of using thick metallic strips is that the resistance of these strips is negligible.

### Nuclei with magic no. of proton and magic no. of neutrons and are found to be very stable

(i) Verify this by calculating the proton separation energy Sp for and . The proton separation energy for a nuclide is the minimum energy required to separate the least tightly bound proton from a nucleus of that nuclide. It is given by . Given ,

(ii) What does the existence of magic number indicate?

i) The proton separation energy is given by, $\mathrm{SpSn}=(\mathrm{M} 119.70+\mathrm{Mh}-\mathrm{M} 120.70) \mathrm{c}^{2}=0.0114362 \mathrm{c}^{2}$ Similarly we have, $\mathrm{SpSp}=(\mathrm{M}...

### The deuteron is bound by nuclear forces just as -atom is made up of and e bound by electrostatic forces. If we consider the force between neutron and proton in deuteron as given in the form of a Coulomb potential but with an effective charge estimate the value of given that the binding energy of a deuteron is .

The binding energy of $\mathrm{H}$ atom is given as $\mathrm{E}=13.6 \mathrm{eV}$ The reduced $\mathrm{m}^{\prime}$ is given as $918 \mathrm{~m}$ The mass of a neutron or a proton is given as...

### Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive. An example is

Assume that we start with 1000 38S nuclei at time . The number of is of count zero at and will again be zero at . At what value of , would the number of counts be a maximum?

Let the disintegration constants for $S^{38}$ and $Cl^{38}$ be $\lambda_ 1$ and $\lambda_ 2$ respectively. $\mathrm dN_1 / \mathrm{dt}=-\lambda N_{1}$ $\mathrm dN_2 / \mathrm{dt}=$ rate of decay of...

### Consider a radioactive nucleus A which decays to a stable nucleus through the following sequence: Here is an intermediate nuclei which is also radioactive. Considering that there are atoms of A initially, plot the graph showing the variation of number of atoms of and versus time.

At $\mathrm{t}=0,$ $\mathrm{~N}_{\mathrm{A}}=\mathrm{N}_{0}$ As time passes, $N_A$ decreases exponentially, while the number of atoms in B increases, reaches its maximum, and then decays to...

### In pair annihilation, an electron and a positron destroy each other to produce gamma radiation. How is the momentum conserved?

An electron and a positron kill one other to produce gamma radiation in pair annihilation. Also, their momentum is conserved because they move in opposing directions.

### Which sample, A or B shown in the figure has shorter mean-life?

Solution: At t=0, $(d N / d t) A=(d N / d t) B$ $d N / d t=-\lambda N$ $\left(N_{0}\right) A=\left(N_{0}\right) B$ $\lambda_ A N_ A=\lambda_ B N_ B$ $N_ a>N_ B$ $\lambda_ B>\lambda_ A$

### and nuclei have the same mass number. Do they have the same binding energy?

Despite the fact that $H e_{2}^{3}$ and $H e_{1}^{3}$ have the same mass number, their binding energies are different. Because the number of protons and neutrons in both nuclei differs, the binding...

### Tritium is an isotope of hydrogen whose nucleus Triton contains 2 neutrons and 1 proton. Free neutrons decay into . If one of the neutrons in Triton decays, it would transform into He3 nucleus. This does not happen. This is because

(a) Triton energy is less than that of a nucleus

(b) the electron created in the beta decay process cannot remain in the nucleus

(c) both the neutrons in triton have to decay simultaneously resulting in a nucleus with 3 protons, which is not a nucleus

(d) because free neutrons decay due to external perturbations which is absent in a triton nucleus

The correct option is: (a) Triton energy is less than that of a $\mathrm{He}^{3}$ nucleus

### When a nucleus in an atom undergoes radioactive decay, the electronic energy levels of the atom

(a) do not change for any type of radioactivity

(b) change for α and β radioactivity but not for γ-radioactivity

(c) change for α-radioactivity but not for others

(d) change for β-radioactivity but not for others

The correct option is: (b) change for α and β radioactivity but not for γ-radioactivity

### Suppose we consider a large number of containers each containing initially 10000 atoms of a radioactive material with a half-life of 1 year. After 1 year

(a) all the containers will have 5000 atoms of the material

(b) all the containers will contain the same number of atoms of the material but that number will only be approximately 5000

(c) the containers will, in general, have different numbers of the atoms of the material but their average will be close to 5000

(d) none of the containers can have more than 5000 atoms

The correct option is: (c) the containers will, in general, have different numbers of the atoms of the material but their average will be close to 5000

### A plane EM wave travelling along z-direction is described by and Show that

i) the average energy density of the wave is given by

ii) the time-averaged intensity of the wave is given by

i) The energy density due to electric field $E$ is given as $\mathrm{uE}=1 / 2 \varepsilon_{0} \mathrm{E}^{2}$ The energy density due to magnetic field $B$ is givena as $\mathrm{uB}=1 / 2...

### A long straight cable of length is placed symmetrically along the z-axis and has radius . The cable consists of a thin wire and a co-axial conducting tube. An alternating current sin flows down the central thin wire and returns along the co-axial conducting tube. The induced electric field at a distance from the wire inside the cable is . In ,

compare the conduction current 10 with the displacement current

The displacement will be, $I_{0}^{\mathrm{d}} / \mathrm{I}_{0}=(\mathrm{am} / \lambda)^{2}$

### Seawater at frequency has permittivity , permeability and resistivity . Imagine a parallel plate capacitor immersed in seawater and driven by an alternating voltage source What fraction of the conduction current density is the displacement current density?

The separation between the plates of the capacitor is given as $V(t)=V_{0} \sin (2 \pi v t)$ Ohm's law for the conduction of current density is given as $\mathrm{J}_{0}{...

### Even though an electric field E exerts a force qE on a charged particle yet the electric field of an EM wave does not contribute to the radiation pressure. Explain.

Despite the fact that an electric field E imposes a force qE on a charged particle, the electric field of an EM wave has no effect on radiation pressure because radiation pressure is the product of...

### Show that the radiation pressure exerted by an EM wave of intensity I on a surface kept in vacuum is I/c.

Energy received by the surface per second is given as $E = IA$ Number of photons received by the surface per second is given as $N$ The perfect absorbing can be expressed as $h/ \lambda$ Hence,...

### Show that the average value of radiant flux density S over a single period is given by

Radiant flux density is given as $\vec{S}=\frac{1}{\mu_{0}}\left(\vec{E} \times \vec B\right)=c^{2} \epsilon_{0}\left({\vec{E}} \times \vec B\right)$ $\mathrm{E}=\mathrm{E}_{0} \cos...

### Show that the magnetic field at a point in between the plates of a parallel plate capacitor during charging is

Let $I_{d}$ be the displacement current in the magnetic field region between two parallel plate capacitor plates. The magnetic field induction at a point between two capacitor plates at a...

### A variable frequency a.c source is connected to a capacitor. How will the displacement current change with a decrease in frequency?

Capacitive reaction is given as, $\mathrm{Xc}=1 / 2 \mathrm{mf} \mathrm{C}$ As the frequency decreases, $X_ c$ rises, and the conduction current becomes inversely proportional to $X_ c$.

### Why does a microwave oven heats up a food item containing water molecules most efficiently?

Because the microwave's frequency and the resonant frequency of the water molecules are the same, the microwave oven cooks a food item containing water molecules most efficiently.

### An EM wave of intensity I falls on a surface kept in vacuum and exerts radiation pressure on it. Which of the following are true?

a) radiation pressure is if the wave is totally absorbed

b) radiation pressure is if the wave is totally reflected

c) radiation pressure is if the wave is totally reflected

d) radiation pressure is in the range for real surface

The correct options are: a) radiation pressure is $\mathrm{l} / \mathrm{c}$ if the wave is totally absorbed c) radiation pressure is $2 \mathrm{l} / \mathrm{c}$ if the wave is totally reflected d)...

### A charged particle oscillates about its mean equilibrium position with a frequency of . The electromagnetic waves produced:

a) will have a frequency of

b) will have a frequency of

c) will have a wavelength of

d) fall in the region of radiowaves

The correct options are: a) will have a frequency of $10^{9} \mathrm{~Hz}$ c) will have a wavelength of $0.3 \mathrm{~m}$ d) fall in the region of radiowaves

### An alternating current generator has an internal resistance and an internal reactance . It is used to supply power to a passive load consisting of a resistance and a reactance . For maximum power to be delivered from the generator to the load, the value of is equal to

(a) zero.

(b) .

(c) .

(d) .

(c) $-X_{g}$.

### Is the momentum conserved when charge crosses a junction in an electric circuit? Why or why not?

Answer: When a charge traverses a junction in an electric circuit, the momentum of the charge does not conserve. This is due to the fact that the drift velocity is directly proportional to the...

### A uniform square plate S and a uniform rectangular plate R have identical areas and masses. Show that

Area of square $=$ area of rectangle $I_{z R}-I_{z S}=m\left(d_{R} / 2\right)^{2}-m\left(d_{S} / 2\right)^{2}$ On solving the above equation, we can get $I_{zR}/I_{zS}>1$

### Two cylindrical hollow drums of radii R and 2R and of a common height h, are rotating with angular velocities ω (anti-clockwise) and ω (clockwise) respectively. Their axes, fixed are parallel and in a horizontal plane separated by (3R + δ). They are now brought in contact

What would be the ratio of final angular velocities when friction ceases?

The anticlockwise and clockwise angular velocities of the drum are $\omega_{1}$ and $\omega_{2}$ respectively. When the velocities are equal, there is no force of friction and it is given as...

### A disc of radius R is rotating with an angular speed about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is .

a) what condition should be satisfied for rolling to being?

b) calculate the time taken for the rolling to being

a) The condition that needs to be satisfied is $v_{cm}=\omega_{o}R$ b) Frictional force is responsible for allowing rolling motion to occur without the disc slipping.

### A disc of radius R is rotating with an angular speed about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is .

a) what was the velocity of its centre of mass before being brought in contact with the table?

b) what happens to the linear velocity of a point on its rim when placed in contact with the table?

a) Before coming into contact with the table, the disc was rotating, and $v_{cm}=0$ was the rest. b) When the revolving disc comes into contact with the table, the linear velocity of the disc...

### Two discs of moments of inertia and about their respective axes and rotating with angular speed and are brought into contact face to face with their axes of rotation coincident.

a) does the law of conservation of angular momentum apply to the situation? why?

b) find the angular speed of the two-disc system

a) Because there is no external torque on the system and the gravitational and normal reactions to external forces have net torque zero, the equation of conservation of angular momentum can be...

### (n-1) equal point masses each of mass m are placed at the vertices of a regular n-polygon. The vacant vertex has a position vector concerning the centre of the polygon. Find the position vector of the centre of mass.

$r_{c m}=\frac{(n-1) m b+m a}{(n-1) m+m}$ is given Where, ${r_{cm}}$ The position of mass $m$ at the $n^{th}$ vertex is called. $r_{\mathrm{cm}}=0$ $\frac{(n-1) m b+m a}{(n-1) m+m}=0$ $(n-1) m b+m...

### A wheel in uniform motion about an axis passing through its centre and perpendicular to its plane is considered to be in mechanical equilibrium because no net external force or torque is required to sustain its motion. However, the particles that constitute the wheel do experience a centripetal acceleration directed towards the centre. How do you reconcile this fact with the wheel being in equilibrium? How would you set a half-wheel into uniform motion about an axis passing through the centre of mass of the wheel and perpendicular to its plane? Will you require external forces to sustain the motion?

A wheel is a stiff elastic body with a consistent motion that passes through its center, perpendicular to the wheel's plane. Due to elastic force, every particle of the wheel receives a centripetal...

### A uniform cube of mass m and side a is placed in a frictionless horizontal surface. A vertical force F is applied to the edge as shown in the figure.

Match the following $\begin{array}{|l|l|} \hline \text { a) } \mathrm{mg} / 4<\mathrm{F}<\mathrm{mg} / 2 & \text { i) cube will move up } \\ \hline \text { b) } \mathrm{F}>\mathrm{mg} /...

### Why does a solid sphere have a smaller moment of inertia than a hollow cylinder of same mass and radius, about an axis passing through their axes of symmetry?

The moment of inertia is related to the square of the distance between the mass and the rotational axis. The distribution of mass in a solid sphere occurs from the sphere's centre to its radius. In...

### Regarding figure, of a cube of edge a and mass n, state whether the following are true or false.

a) the moment of inertia of cube about the z-axis is

b) the moment of inertia of cube about z’ is

c) the moment of inertia of cube about z’’ is

d)

Correct answers are: b) the moment of inertia of cube about z’ is $I_{z}^{\prime}=I_{z}+\frac{m a^{2}}{2}$ d) $I_{x}=I_{y}$

### Choose the correct alternatives:

a) for a general rotational motion, angular momentum L and angular velocity ω need not be parallel

b) for a rotational motion about a fixed axis, angular momentum L and angular velocity ω are always parallel

c) for a general translational motion, momentum p and velocity v is always parallel

d) for a general translational motion, acceleration a and velocity v are always parallel

a) Angular momentum L and angular velocity ω do not have to be parallel for a general rotating motion. c) Momentum p and velocity v are always parallel in a typical translational motion.

### A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind. The moment of inertia about the z-axis is then

a) increased

b) decreased

c) the same

d) changed in an unpredicted manner

Correct answer is b) decreased

### For which of the following does the centre of mass lie outside the body?

a) a pencil

b) a shotput

c) a dice

d) a bangle

Correct answer is d) a bangle

### Einstein’s mass-energy relation emerging out of his famous theory of relativity relates mass (m) to energy (E) as , where c is the speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at nuclear level is usually measured in MeV where , the masses are measured in unified equivalent of 1u is 931.5 MeV.

a) Show that the energy equivalent of 1 u is 931.5 MeV.

b) A student writes the relation as 1 u = 931.5 MeV. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.

a) The energy that is comparable to a given mass can be computed using Einstein's mass-energy relation. $1amu=1u=1.67\times 10^{-27}kg$ On Applying $E=mc^{2}$ we get, E = 931.5 MeV b) As $E=mc^{2}$...

### In an experiment to estimate the size of a molecule of oleic acid, 1mL of oleic acid is dissolved in 19mL of alcohol. Then 1mL of this solution is diluted to 20mL by adding alcohol. Now, 1 drop of this diluted solution is placed on water in a shallow trough. The solution spreads over the surface of water forming one molecule thick layer. Now, lycopodium powder is sprinkled evenly over the film we can calculate the thickness of the film which will give us the size of oleic acid molecule.

Read the passage carefully and answer the following questions:

a) What would be the volume of oleic acid in each mL of solution prepared?

b) How will you calculate the volume of n drops of this solution of oleic.

a) 1 mL of oleic acid is found in every 20 mL of oleic acid. This signifies that 1/20 mL of oleic acid is present in each mL of solution. Adding alcohol dilutes 1 mL of this solution to 20 mL. As a...

### If the velocity of light c, Planck’s constant h and gravitational constant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.

We will us Principle of homogeneity for solving this problem. $\begin{array}{l} {[\mathrm{h}]=\left[\mathrm{ML}^{2}...

### During a total solar eclipse-the moon almost entirely covers the sphere of the sun. Write the relation between the distances and sizes of the sun and moon.

$R_{me}=$ distance of the moon from the earth $R_{se}=$ distance of the sun from the moon $A_{sun}=$ area of the sun $A_{moon}=$ area of the moon...

### Photon is quantum of radiation with energy E = hv where v is frequency and h is Planck’s constant. The dimensions of h are the same as that of:

a) linear impulse

b) angular impulse

c) linear momentum

d) angular momentum

Correct options are b) angular impulse and d) angular momentum

### On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct:

a) y = a sin 2πt/T

b) y = a sin vt

c) y = a/T sin (t/a)

d) y = a√2 [sin (2 πt/T) – cos (2πt/T)]

Correct answers are b) y = a sin vt and c) y = a/T sin (t/a)

### Which of the following measurements is most precise?

a) 5.00 mm

b) 5.00 cm

c) 5.00 m

d) 5.00 km

Correct answer is a) 5.00 mm

### The number of significant figures in 0.06900 is:

a) 5

b) 4

c) 2

d) 3

The correct answer is b) 4 The number of zeroes to the left of a non-zero integer is not considered relevant, whereas the number of zeroes to the right of a non-zero number is.

### 7.2. Which of the following points is the likely position of the centre of mass of the system shown in the figure. a) A b) B c) C d) D

Answer: The correct answer is option c) C Explanation: Centre of mass represents the mean position of the matter of a system. In the given figure half of the sphere is hollow whereas the second half...

### Let the magnetic field on the earth be modelled by that of a point magnetic dipole at the centre of the earth. The angle of dip at a point on the geographical equator a) is always zero b) can be zero at specific points c) can be positive or negative d) is bounded

b) can be zero at specific points c) can be positive or negative d) is bounded

### 7.1. The centre of mass of which of the following is located outside the body? a) a pencil b) a shotput c) a dice d) a bangle

Answer: The correct option is d) a bangle. Explanation: The mass of a bangle is distributed along the circumference of the ring shape of the bangle as it is a hollow shape. The center of mass is...