Questions & explanations
1. A coil of self-inductance L carries a current that changes at a rate dI/dt. The back emf induced in the coil is:
- L dI/dt
- -L dI/dt
- L I
- -L I
Answer: -L dI/dt
According to Faraday's law of electromagnetic induction, the induced emf in a coil is given by ε = -L dI/dt, where the negative sign indicates that the induced emf opposes the change in current (Lenz's law). Option B correctly includes the negative sign and the rate of change of current. Option A omits the negative sign; options C and D involve current instead of its rate of change.
2. In an LR circuit, when a constant voltage is suddenly applied, the current through the inductor:
- Immediately reaches its maximum value.
- Rises exponentially to the maximum value with time constant L/R.
- Rises linearly with time.
- Remains zero indefinitely.
Answer: Rises exponentially to the maximum value with time constant L/R.
When a constant voltage is applied to an LR series circuit, the current grows exponentially as I(t) = I₀ (1 - e^{-t/τ}), where I₀ = V/R is the steady-state maximum current and τ = L/R is the time constant. The current does not jump instantly (option A), does not increase linearly (option C), and does not stay zero (option D). Hence option B is correct.
3. The self-inductance of a long solenoid of length l, cross-sectional area A, and total number of turns N is given by (μ0 is permeability of free space):
- μ0 N^2 A / l
- μ0 N A / l
- μ0 N^2 l / A
- μ0 N A l
Answer: μ0 N^2 A / l
For a long solenoid, the magnetic field inside is B = μ0 n I, where n = N/l is the number of turns per unit length. The total flux linkage is NΦ = N (BA) = μ0 N^2 A I / l. Hence self-inductance L = NΦ/I = μ0 N^2 A / l. Option A gives the correct formula. Options B, C, and D are missing factors or have incorrect combinations of parameters.
4. Which of the following correctly defines self-inductance of a coil and its SI unit?
- It is the property of a coil to oppose any change in current; unit: ohm.
- It is the ratio of magnetic flux linked with the coil to the current flowing through it; unit: henry.
- It is the product of current and magnetic flux; unit: henry.
- It is the rate of change of magnetic flux; unit: weber.
Answer: It is the ratio of magnetic flux linked with the coil to the current flowing through it; unit: henry.
Self-inductance L is defined as L = Φ/I, where Φ is the magnetic flux linked with the coil and I is the current. Its SI unit is henry (H). Option B gives the correct definition and unit. Option A incorrectly assigns ohm (unit of resistance); option C incorrectly defines it as product; option D describes induced emf, not self-inductance.
5. Mutual inductance between two coils is defined as:
- The ratio of induced emf in one coil to the rate of change of current in the other.
- The product of magnetic fluxes of the two coils.
- The sum of self-inductances of the two coils.
- The ratio of current in one coil to the emf induced in the other.
Answer: The ratio of induced emf in one coil to the rate of change of current in the other.
Mutual inductance M is defined as M = ε₂ / (dI₁/dt) (ignoring sign), where ε₂ is the induced emf in the second coil due to a change in current I₁ in the first coil. It is also given by M = N₂Φ₂ / I₁. Option A correctly describes this ratio. Options B, C, and D do not represent the definition of mutual inductance.
6. The energy density (energy per unit volume) associated with a magnetic field of magnitude B in free space is:
- B^2/(2μ0)
- μ0 B^2/2
- 1/2 B^2 μ0
- B/(2μ0)
Answer: B^2/(2μ0)
The magnetic energy density in free space is given by u = B^2/(2μ0), where μ0 is the permeability of free space. This is analogous to the electric energy density. Option A is the correct expression. Options B and C have μ0 in the numerator instead of denominator; option D has B instead of B^2.
7. Eddy currents are induced in:
- Insulators when placed in a changing magnetic field.
- Conductors when placed in a changing magnetic field.
- Dielectrics in a steady electric field.
- Magnetic materials only.
Answer: Conductors when placed in a changing magnetic field.
Eddy currents are circulating currents induced in bulk conductors by a changing magnetic flux (Faraday's law). Insulators and dielectrics do not conduct, and a steady field does not induce currents. They occur in any conductor, not just magnetic materials.
8. The energy stored in an inductor of inductance L when a steady current I flows through it is:
- 1/2 L I^2
- L I^2
- 1/2 L I
- L I
Answer: 1/2 L I^2
The energy stored in the magnetic field of an inductor is given by U = 1/2 L I^2. This expression is derived from the work done to establish the current. Option A is correct. Options B, C, and D have incorrect powers or missing factors.
9. Lenz's law is a manifestation of the principle of conservation of energy because:
- Induced current always flows in such a way to increase the magnetic flux.
- The work done against the opposing force is converted into electrical energy.
- Energy is created by the induced current.
- There is no energy transfer involved.
Answer: The work done against the opposing force is converted into electrical energy.
If Lenz's law were not satisfied, the induced current would aid the change, producing energy from nothing. Instead, external work is needed to overcome the opposing force, and that work appears as electrical energy—conserving energy.
10. A bar magnet is moved towards a closed loop. According to Lenz's law, the induced current in the loop will flow in a direction such that it:
- Opposes the motion of the magnet.
- Aids the motion of the magnet.
- Attracts the magnet.
- Has no effect on the magnet.
Answer: Opposes the motion of the magnet.
Lenz's law says the induced current opposes the change that produces it. As the magnet approaches, the loop acts like a magnet with its north pole facing the approaching north pole, thus repelling it.
11. A metal rod of length L is moved with velocity v perpendicular to a uniform magnetic field B. The motional EMF induced across its ends is given by:
- B L v
- B^2 L v
- B L v^2
- B L / v
Answer: B L v
Motional EMF arises because the moving charges in the rod experience a magnetic force F = q v B. The work per unit charge gives ε = (F/q)L = B L v, with the direction given by the right-hand rule.
12. According to Faraday's laws of electromagnetic induction, an EMF is induced in a circuit when:
- A steady current flows through it.
- The magnetic flux linked with the circuit changes with time.
- The circuit is placed in a uniform magnetic field.
- The circuit has a large resistance.
Answer: The magnetic flux linked with the circuit changes with time.
Faraday's laws state that the induced EMF is due to a change in magnetic flux linking the circuit. A steady current or a static field produces no change, and resistance does not cause induction.