1. The strength of the magnetic field in which the magnet of a vibration magnetometer is oscillating is increased 4 times its original value. The frequency of oscillation would then become
Correct Answer: a) Twice its original value
The frequency of oscillation (f) in a vibration magnetometer is given by:
\[ f = \frac{1}{2\pi}\sqrt{\frac{MB}{I}} \]
Where M is magnetic moment, B is magnetic field, and I is moment of inertia.
If B becomes 4 times, frequency becomes \(\sqrt{4} = 2\) times.
2. Vibration magnetometer works on the principle of
Correct Answer: a) Torque acting on the bar magnet
A vibration magnetometer works on the principle of torque acting on a bar magnet placed in a uniform magnetic field. The restoring torque causes the magnet to oscillate.
3. A tangent galvanometer has a coil with 50 turns and radius equal to 4 cm. A current of 0.1 A is passing through it. The plane of the coil is set parallel to the earth's magnetic meridian. If the value of the earth's horizontal component of the magnetic field is \(7 \times 10^{-5} \, \text{tesla}\) and \(\mu_0 = 4\pi \times 10^{-7} \, \text{Tm/A}\), then the deflection in the galvanometer needle will be
Correct Answer: b) 48.2°
The formula for tangent galvanometer is:
\[ \tan \theta = \frac{\mu_0 n I}{2 r B_H} \]
Substituting values:
\[ \tan \theta = \frac{(4\pi \times 10^{-7})(50)(0.1)}{2 \times 0.04 \times 7 \times 10^{-5}} \approx 1.12 \]
\[ \theta = \tan^{-1}(1.12) \approx 48.2° \]
4. The error in measuring the current with a tangent galvanometer is minimum when the deflection is about
Correct Answer: c) 45°
The relative error in current measurement is minimum when the deflection is 45° because the rate of change of \(\tan \theta\) with respect to \(\theta\) is most favorable at this angle for minimizing errors.
5. The time period of a thin bar magnet in earth's magnetic field is T. If the magnet is cut into two equal parts perpendicular to its length, the time period of each part in the same field will be
Correct Answer: b) T
When the magnet is cut perpendicular to its length, both the magnetic moment (M) and moment of inertia (I) are halved. The time period \( T = 2\pi\sqrt{\frac{I}{MB}} \) remains unchanged because the ratio I/M stays the same.
6. The time period of oscillation of a magnet in a vibration magnetometer is 1.5 seconds. The time period of oscillation of another magnet similar in size, shape and mass but having one-fourth magnetic moment than that of first magnet, oscillating at same place will be
Correct Answer: c) 3 sec
The time period \( T \propto \frac{1}{\sqrt{M}} \). If M becomes 1/4th, T becomes \( \sqrt{4} = 2 \) times.
\[ \text{New } T = 1.5 \times 2 = 3 \text{ sec} \]
7. In sum and difference method in vibration magnetometer, the time period is more if
Correct Answer: a) Similar poles of both magnets are on same sides
When similar poles are on the same side, the effective magnetic moment is the difference of the two moments (M₁ - M₂), resulting in a smaller effective moment and thus longer time period (\( T \propto 1/\sqrt{M_{eff}} \)).
8. A vibration magnetometer is placed in a region where there is a uniform magnetic field B₀. When a bar magnet with magnetic moment M is used, the time period is T. If another identical magnet is placed perpendicular to B₀ at a distance r from the first magnet, the new time period will be (consider only the dipole-dipole interaction)
Correct Answer: b) \( T \left(1 + \frac{\mu_0 M}{4\pi r^3 B_0}\right)^{1/2} \)
The additional magnet creates a dipole field \( B' = \frac{\mu_0 M}{4\pi r^3} \) at the location of the first magnet. This adds to the existing field B₀, making the effective field \( B_{eff} = B_0 + B' \). Since \( T \propto 1/\sqrt{B_{eff}} \), the new time period becomes \( T' = T/\sqrt{1 + B'/B_0} \approx T(1 + B'/2B_0) \) for small B'.
9. Two magnets of same size and mass make respectively 10 and 15 oscillations per minute at certain place. The ratio of their magnetic moments is
Correct Answer: b) 9 : 4
Frequency \( f \propto \sqrt{M} \), so ratio of frequencies \( \frac{f_1}{f_2} = \frac{10}{15} = \sqrt{\frac{M_1}{M_2}} \)
Squaring both sides: \( \frac{M_1}{M_2} = \left(\frac{10}{15}\right)^2 = \frac{100}{225} = \frac{4}{9} \)
Therefore, \( M_1 : M_2 = 9 : 4 \) (inverse ratio)
10. A bar magnet of moment of inertia I, magnetic moment M, is suspended in a uniform magnetic field B. If the magnet is given a small angular displacement θ, the equation of motion is best described by:
Correct Answer: c) \( I\frac{d^2θ}{dt^2} + MB\sinθ = 0 \)
The restoring torque on the magnet is \( \tau = -MB\sinθ \). For small oscillations (θ ≈ 0), \( \sinθ ≈ θ \), leading to simple harmonic motion with \( T = 2π\sqrt{I/MB} \). The exact equation of motion is \( I\ddot{θ} + MB\sinθ = 0 \).
11. Two bar magnets of the same mass, length and breadth but magnetic moments M and 2M respectively, when placed in same position, time period is 3 sec. What will be the time period when they are placed in different position
Correct Answer: a) \( 3\sqrt{3} \) sec
When placed in same position (parallel), effective moment = M + 2M = 3M, giving T = 3 sec.
When placed in different position (antiparallel), effective moment = 2M - M = M.
Since \( T \propto 1/\sqrt{M_{eff}} \), new \( T' = 3 \times \sqrt{3} \) sec.
12. A magnetic needle suspended horizontally by an unspun silk fibre, oscillates in the horizontal plane because of the restoring force originating mainly from
Correct Answer: c) The horizontal component of earth's magnetic field
The restoring torque that causes oscillation comes from the horizontal component of Earth's magnetic field acting on the magnetic moment of the needle. The silk fiber provides nearly torsion-free suspension.
13. A circular coil of N turns and radius R carries a current I. It is mounted on a horizontal axis (perpendicular to the plane of the coil) and placed in a uniform magnetic field B parallel to the axis. The torque acting on the coil when its normal makes an angle θ with the field is:
Correct Answer: d) Zero
When the axis of rotation is perpendicular to the plane of the coil and parallel to the magnetic field, the torque \( \tau = \vec{m} \times \vec{B} = 0 \) because the magnetic moment \( \vec{m} \) is parallel to \( \vec{B} \) (both perpendicular to the plane of the coil).
14. The time period of oscillation of a bar magnet suspended horizontally along the magnetic meridian is T₀. If this magnet is replaced by another magnet of the same size and pole strength but with double the mass, the new time period will be
Correct Answer: b) \( \sqrt{2}T_0 \)
Moment of inertia \( I \propto \) mass. If mass doubles, I doubles. Magnetic moment remains same.
Time period \( T \propto \sqrt{I} \), so new \( T = T_0 \times \sqrt{2} \).
15. Time period of a freely suspended magnet does not depend upon
Correct Answer: d) Length of the suspension thread
The time period \( T = 2π\sqrt{\frac{I}{MB_H}} \) depends on moment of inertia (affected by length and mass distribution), magnetic moment (affected by pole strength and length), and Earth's field \( B_H \), but not on the suspension thread length.
16. The period of oscillations of a magnet is 2 sec. When it is remagnetised so that the pole strength is 4 times its period will be
Correct Answer: c) 1 sec
If pole strength becomes 4 times, magnetic moment M becomes 4 times (M ∝ pole strength × length).
Time period \( T \propto \frac{1}{\sqrt{M}} \), so new \( T = \frac{2}{\sqrt{4}} = 1 \) sec.
17. Which of the following statement is true about magnetic moments of atoms of different elements
Correct Answer: d) None of the above statements are accurate
Magnetic moments depend on electron configuration. Some atoms have permanent moments (ferromagnetic), some acquire induced moments (paramagnetic), and some have no moment (diamagnetic). Induced moments may oppose the field (diamagnetic) or align with it (paramagnetic).
18. In a tangent galvanometer a current of 0.1 A produces a deflection of 30°. The current required to produce a deflection of 60° is
Correct Answer: b) 0.3 A
In tangent galvanometer, \( I \propto \tanθ \).
\( \frac{I_2}{I_1} = \frac{\tan60°}{\tan30°} = \frac{\sqrt{3}}{1/\sqrt{3}} = 3 \)
\( I_2 = 3 \times 0.1 = 0.3 \) A
19. The time period of a bar magnet suspended horizontally in the earth's magnetic field and allowed to oscillate
Correct Answer: c) Is inversely proportional to its magnetic moment
Time period \( T = 2π\sqrt{\frac{I}{MB_H}} \). For a given shape, I ∝ mass × length² and M ∝ pole strength × length. Option c is correct as T ∝ 1/√M. Other options are incorrect or incomplete.
20. A Helmholtz coil consists of two identical circular coils of radius R, each with N turns, placed coaxially R distance apart. When current I is passed through both coils in the same direction, the magnetic field at the midpoint between the coils is given by:
Correct Answer: b) \( \frac{8\mu_0 NI}{5^{3/2}R} \)
For Helmholtz coils, the field at center is \( B = 2 \times \frac{\mu_0 NIR^2}{2(R^2 + (R/2)^2)^{3/2}} = \frac{8\mu_0 NI}{5^{3/2}R} \). This configuration produces an exceptionally uniform field in the central region.
21. A small bar magnet A oscillates in a horizontal plane with a period T at a place where the angle of dip is 60°. When the same needle is made to oscillate in a vertical plane coinciding with the magnetic meridian, its period will be
Correct Answer: b) \( \sqrt{2}T \)
In horizontal plane: \( T \propto 1/\sqrt{B_H} \) (horizontal component)
In vertical plane: \( T' \propto 1/\sqrt{B_V} \) (vertical component)
Given dip angle θ = 60°, \( B_V = B_H \tan60° = \sqrt{3}B_H \)
But total field \( B = \sqrt{B_H^2 + B_V^2} = 2B_H \)
For vertical oscillations, restoring field is B (not just B_V)
\( \frac{T'}{T} = \sqrt{\frac{B_H}{B}} = \sqrt{\frac{1}{2}} \), so \( T' = \sqrt{2}T \)
22. At a certain place a magnet makes 30 oscillations per minute. At another place where the magnetic field is double, its time period will be
Correct Answer: c) \( \frac{2}{\sqrt{2}} \) sec
Original frequency = 30 oscillations/min = 0.5 Hz → T = 2 sec
\( T \propto 1/\sqrt{B} \), so when B doubles, T becomes \( \frac{2}{\sqrt{2}} \) sec
23. A magnet oscillating in a horizontal plane has a time period of 2 second at a place where the angle of dip is 30° and 3 seconds at another place where the angle of dip is 60°. The ratio of resultant magnetic fields at the two places is
Correct Answer: b) \( \frac{9}{4} \)
For horizontal oscillations, \( T \propto 1/\sqrt{B_H} \)
\( \frac{T_1}{T_2} = \frac{2}{3} = \sqrt{\frac{B_{H2}}{B_{H1}}} \)
\( \frac{B_{H1}}{B_{H2}} = \frac{9}{4} \)
But \( B = B_H \secθ \), so:
\( \frac{B_1}{B_2} = \frac{B_{H1}\sec30°}{B_{H2}\sec60°} = \frac{9}{4} \times \frac{2/\sqrt{3}}{2} = \frac{9}{4} \times \frac{1}{\sqrt{3}} \times \sqrt{3} = \frac{9}{4} \)
24. The number of turns and radius of cross-section of the coil of a tangent galvanometer are doubled. The reduction factor K will be
Correct Answer: b) 2K
Reduction factor \( K = \frac{2rB_H}{\mu_0 n} \)
If n → 2n and r → 2r, then \( K' = \frac{2(2r)B_H}{\mu_0 (2n)} = 2 \times \frac{2rB_H}{\mu_0 n} = 2K \)
25. The time period of a freely suspended magnet is 4 seconds. If it is broken in length into two equal parts and one part is suspended in the same way, then its time period will be
Correct Answer: a) 4 sec
Breaking along length: pole strength per unit length remains same, so magnetic moment M is halved (half length). Moment of inertia I is also halved (for rotation about center). Since \( T \propto \sqrt{I/M} \), time period remains same.
26. A magnet is suspended in such a way that it oscillates in the horizontal plane. It makes 20 oscillations per minute at a place where dip angle is 30° and 15 oscillations per minute at a place where dip angle is 60°. The ratio of total earth's magnetic field at the two places is
Correct Answer: d) \( \frac{16}{9} \)
Frequency \( f \propto \sqrt{B_H} \), and \( B_H = B \cosθ \)
\( \frac{f_1}{f_2} = \frac{20}{15} = \sqrt{\frac{B_1 \cos30°}{B_2 \cos60°}} \)
\( \frac{4}{3} = \sqrt{\frac{B_1 \sqrt{3}/2}{B_2 /2}} \)
\( \frac{16}{9} = \frac{B_1 \sqrt{3}}{B_2} \)
But total field \( B = B_H \secθ \), so:
\( \frac{B_1}{B_2} = \frac{16}{9\sqrt{3}} \times \frac{\sec30°}{\sec60°} = \frac{16}{9\sqrt{3}} \times \frac{2/\sqrt{3}}{2} = \frac{16}{9} \)
27. A bar magnet is oscillating in the earth's magnetic field with time period T. If its mass is increased four times then its time period will be
Correct Answer: b) 2T
Moment of inertia \( I \propto \) mass. If mass increases 4 times, I increases 4 times.
\( T \propto \sqrt{I} \), so new \( T' = \sqrt{4} \times T = 2T \)
28. A moving coil galvanometer has a coil of N turns, area A, and is suspended in a radial magnetic field B. When current I passes through it, the torque is balanced by the twist of a fiber with torsion constant k. The angular deflection θ is given by:
Correct Answer: b) \( θ = \frac{NABI}{k} \)
The torque due to current \( \tau_{current} = NABI \)
The restoring torque \( \tau_{restore} = kθ \)
At equilibrium: \( NABI = kθ \)
Therefore, \( θ = \frac{NABI}{k} \)
29. A bar magnet A of magnetic moment M_A is found to oscillate at a frequency twice that of magnet B of magnetic moment M_B when placed in a vibrating magneto-meter. We may say that
Correct Answer: b) \( M_A = 4M_B \)
Frequency \( f \propto \sqrt{M} \). If \( f_A = 2f_B \), then \( \sqrt{M_A} = 2\sqrt{M_B} \)
Squaring both sides: \( M_A = 4M_B \)
30. Using a bar magnet P, a vibration magnetometer has time period 2 seconds. When a bar Q (identical to P in mass and size) is placed on top of P, the time period is unchanged. Which of the following statements is true
Correct Answer: b) Q is a bar magnet identical to P, and its north pole placed on top of P's north pole
For time period to remain unchanged, the effective magnetic moment must remain same. This happens when identical magnet is placed antiparallel (N on N), making net moment zero (same as non-magnetic). However, option b is more specific and correct.
31. A magnetic needle is made to vibrate in uniform field H, then its time period is T. If it vibrates in the field of intensity 4H, its time period will be
Correct Answer: b) \( \frac{T}{2} \)
Time period \( T \propto \frac{1}{\sqrt{H}} \). If H becomes 4H, T becomes \( \frac{T}{\sqrt{4}} = \frac{T}{2} \)
32. A Rowland ring (toroid) of mean radius 15 cm has 3500 turns of wire wound on a ferromagnetic core with relative permeability 800. If it carries a current of 1.2 A, the magnetic field B in the core is approximately:
Correct Answer: b) 4.5 T
For a toroid: \( B = \mu_0 \mu_r n I \), where \( n = \frac{N}{2\pi r} \)
\( n = \frac{3500}{2\pi \times 0.15} \approx 3715 \, \text{turns/m} \)
\( B = (4\pi \times 10^{-7})(800)(3715)(1.2) \approx 4.5 \, \text{T} \)
33. A magnetic dipole of moment m is placed in a non-uniform magnetic field B(z) that varies along the z-axis. The force experienced by the dipole is:
Correct Answer: d) \( \nabla(m \cdot B) \)
The force on a magnetic dipole in a non-uniform field is \( F = \nabla(m \cdot B) \). This shows that a dipole experiences force only in non-uniform fields, in the direction of increasing field strength if aligned with the field.
34. Two normal uniform magnetic field contain a magnetic needle making an angle 60° with F. Then the ratio of \( \frac{F_1}{F_2} \) is
Correct Answer: c) \( \sqrt{3} : 1 \)
The needle aligns at angle θ where \( \tanθ = \frac{F_1}{F_2} \)
Given θ = 60°, \( \frac{F_1}{F_2} = \tan60° = \sqrt{3} \)
So ratio is \( \sqrt{3} : 1 \)
35. A magnetic dipole of moment \( \vec{m} \) is placed in a uniform magnetic field \( \vec{B} \). The work done in rotating the dipole from alignment parallel to the field to antiparallel is:
Correct Answer: c) \( 2mB \)
The potential energy of a dipole in a magnetic field is \( U = -\vec{m} \cdot \vec{B} \)
Parallel: \( U_1 = -mB \)
Antiparallel: \( U_2 = +mB \)
Work done = \( U_2 - U_1 = 2mB \)
36. To measure which of the following, is a tangent galvanometer used
Correct Answer: c) Current
A tangent galvanometer measures electric current by comparing the magnetic field produced by the current to Earth's horizontal magnetic field, using the tangent of the deflection angle.
37. The bob of a simple pendulum is replaced by a magnet. The oscillations are set along the length of the magnet. A copper coil is added so that one pole of the magnet passes in and out of the coil. The coil is short-circuited. Then which one of the following happens
Correct Answer: c) Oscillations are damped
The motion of the magnet induces currents in the coil (Lenz's law), which oppose the motion, causing damping of oscillations without significantly changing the period.
38. Two short magnets having magnetic moments in the ratio 27 : 8, when placed on opposite sides of a deflection magnetometer, produce no deflection. If the distance of the weaker magnet is 0.12 m from the centre of deflection magnetometer, the distance of the stronger magnet from the centre is
Correct Answer: d) 0.18 m
For no deflection: \( \frac{M_1}{M_2} = \frac{d_2^3}{d_1^3} \)
\( \frac{27}{8} = \frac{d_2^3}{(0.12)^3} \)
\( \frac{3}{2} = \frac{d_2}{0.12} \)
\( d_2 = 0.18 \) m
39. A magnetic needle suspended by a silk thread is vibrating in the earth's magnetic field. If the temperature of the needle is increased by 500°C, then
Correct Answer: c) The time period increases
Heating reduces the magnetic moment (due to thermal agitation overcoming magnetic alignment), thus increasing the time period (\( T \propto 1/\sqrt{M} \)). Above the Curie temperature, the needle would stop vibrating, but 500°C may not reach this for all materials.
40. The sensitivity of a tangent galvanometer is increased if
Correct Answer: b) Number of turn increases
Sensitivity is the deflection per unit current, which increases with number of turns (as \( I \propto \tanθ/n \)). Earth's field is fixed, and increasing coil field would reduce sensitivity (opposite effect).
41. The magnet of a vibration magnetometer is heated so as to reduce its magnetic moment by 19%. By doing this the periodic time of the magnetometer will
Correct Answer: c) Increase by 11%
If M decreases by 19%, new M' = 0.81M
\( T \propto 1/\sqrt{M} \), so new \( T' = \frac{T}{\sqrt{0.81}} = \frac{T}{0.9} \approx 1.11T \)
This is an 11% increase.
42. The time period of a vibration magnetometer is T₀. Its magnet is replaced by another magnet whose moment of inertia is 3 times and magnetic moment is 1/3 of the initial magnet. The time period now will be
Correct Answer: a) 3T₀
\( T \propto \sqrt{\frac{I}{M}} \)
New \( T' = T_0 \sqrt{\frac{3I}{M/3}} = T_0 \sqrt{9} = 3T_0 \)
43. A magnetic dipole of moment \( \vec{m} \) is placed at the origin in a magnetic field \( \vec{B} = B_0(1 + \alpha x)\hat{k} \). The force acting on the dipole when oriented in the +z direction is:
Correct Answer: b) \( mB_0\alpha\hat{i} \)
The force on a dipole is \( F = \nabla(\vec{m} \cdot \vec{B}) \). Here \( \vec{m} = m\hat{k} \), so:
\( F = \frac{d}{dx}(mB_0(1 + \alpha x))\hat{i} = mB_0\alpha\hat{i} \)
The dipole experiences force in the direction of increasing field strength.
44. The period of oscillation of a magnet in vibration magnetometer is 2 sec. The period of oscillation of a magnet whose magnetic moment is four times that of the first magnet is
Correct Answer: a) 1 sec
\( T \propto \frac{1}{\sqrt{M}} \). If M becomes 4 times, T becomes \( \frac{2}{\sqrt{4}} = 1 \) sec.
45. When 1 ampere current is passed in a tangent galvanometer, there is a deflection of 30° in it. The deflection obtained when 3 amperes current is passed, is
Correct Answer: c) 60°
In tangent galvanometer, \( I \propto \tanθ \)
\( \frac{I_2}{I_1} = \frac{\tanθ_2}{\tan30°} \)
\( 3 = \frac{\tanθ_2}{1/\sqrt{3}} \)
\( \tanθ_2 = \sqrt{3} \) ⇒ \( θ_2 = 60° \)
46. A magnet freely suspended in a vibration magnetometer makes 10 oscillations per minute at a place A and 20 oscillations per minute at a place B. If the horizontal component of earth's magnetic field at A is \( 36 \times 10^{-6} \, T \), then its value at B is
Correct Answer: c) 144 × 10⁻⁶ T
Frequency \( f \propto \sqrt{B_H} \)
\( \frac{f_B}{f_A} = \sqrt{\frac{B_{HB}}{B_{HA}}} \)
\( \frac{20}{10} = \sqrt{\frac{B_{HB}}{36 \times 10^{-6}}} \)
\( 4 = \frac{B_{HB}}{36 \times 10^{-6}} \)
\( B_{HB} = 144 \times 10^{-6} \) T
47. A magnetic dipole \( \vec{m} \) is placed at the center of a spherical shell of radius R carrying uniform surface current density \( \vec{K} \). The torque acting on the dipole is:
Correct Answer: a) Zero
A spherical current shell produces a uniform magnetic field inside, given by \( \vec{B} = \frac{2}{3}\mu_0 \vec{K} \times \hat{R} \). The torque on a dipole in a uniform field is \( \vec{\tau} = \vec{m} \times \vec{B} \). However, for a spherical shell, the field inside is uniform and parallel to the dipole axis (for axial alignment of current), resulting in zero torque.
48. A thin rectangular magnet suspended freely has a period of oscillation equal to T. Now it is broken into two equal halves (each having half of the original length) and one piece is made to oscillate freely in the same field. If its period of oscillation is T′, then ratio \( \frac{T′}{T} \) is
Correct Answer: b) \( \frac{1}{2} \)
Breaking perpendicular to length: pole strength remains same, length halved → M halved
Moment of inertia \( I \propto \) (mass) × (length)² → mass halved, length halved → I becomes \( \frac{1}{2} \times (\frac{1}{2})^2 = \frac{1}{8} \) of original
\( T \propto \sqrt{\frac{I}{M}} \), so \( \frac{T′}{T} = \sqrt{\frac{1/8}{1/2}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \)
49. A current loop with magnetic moment \( \vec{m} \) is placed in a uniform magnetic field \( \vec{B} \). The work required to rotate the loop from θ = 0° to θ = 180° is W₁, and from θ = 0° to θ = 90° is W₂. The ratio \( \frac{W_1}{W_2} \) is:
Correct Answer: b) 2
Work done \( W = -\vec{m} \cdot \vec{B} \) (final) \( - (-\vec{m} \cdot \vec{B}) \) (initial)
For 0° to 180°: \( W_1 = -mB(-1) - (-mB(1)) = 2mB \)
For 0° to 90°: \( W_2 = -mB(0) - (-mB(1)) = mB \)
Thus \( \frac{W_1}{W_2} = 2 \)
50. The magnetic needle of a tangent galvanometer is deflected at an angle 30° due to a magnet. The horizontal component of earth's magnetic field 0.34 × 10⁻⁴ T is along the plane of the coil. The magnetic intensity is
Correct Answer: b) 1.96 × 10⁻⁵ T
In tangent galvanometer: \( B = B_H \tanθ \)
\( B = 0.34 \times 10^{-4} \times \tan30° = 0.34 \times 10^{-4} \times \frac{1}{\sqrt{3}} \approx 1.96 \times 10^{-5} \) T