Questions & explanations
1. A metal rod of mass m and length l slides without friction on parallel rails in a uniform vertical magnetic field B. The rails are connected by a resistor R. The rod is released from rest on an inclined plane of angle θ. What is the terminal speed of the rod?
- v = mgR sinθ / (B² l²)
- v = mgR / (B² l²)
- v = mgR sinθ / (B l)
- v = mgR / (B l)
Answer: v = mgR sinθ / (B² l²)
The induced emf is ε = B l v, current i = B l v / R. The magnetic force on the rod is F_mag = i l B = B² l² v / R, opposing motion. The component of gravity along the incline is mg sinθ. At terminal speed, net force is zero: mg sinθ = B² l² v_t / R, so v_t = mgR sinθ / (B² l²). This is consistent with Lenz's law and energy conservation: mechanical power equals electrical power dissipated.
2. A bar magnet with its north pole facing a coil is moved towards the coil. As seen from the magnet's side, the induced current in the coil is:
- zero
- anticlockwise
- first clockwise then anticlockwise
- clockwise
Answer: clockwise
As the north pole approaches, the flux through the coil increases to the right. Lenz's law says the induced current must oppose this increase, so it creates a magnetic field pointing left. Using the right-hand rule, thumb left gives fingers curling clockwise when viewed from the left (magnet's side). Hence current is clockwise.
3. A uniform magnetic field into the page is decreasing. The induced current in a circular loop lying in the plane of the page is:
- clockwise
- anticlockwise
- zero
- alternating
Answer: clockwise
The external field into the page is decreasing. Lenz's law says the induced current must oppose this decrease, so it creates a magnetic field also into the page. Using the right-hand curl rule, thumb into page gives fingers curling clockwise. Hence induced current is clockwise.
4. A rectangular loop is placed in the plane of a long straight wire. The current in the wire is increasing. What is the direction of induced current in the loop?
- Clockwise
- No induced current
- Anticlockwise
- Depends on the rate of increase
Answer: Anticlockwise
Using the right-hand thumb rule, the magnetic field due to the wire at the loop is into the page. As current increases, flux into the page increases. Lenz's law says induced current opposes this change, so it produces flux out of the page, which requires anticlockwise current.
5. Lenz's law is a consequence of which conservation principle?
- Conservation of charge
- Conservation of momentum
- Conservation of energy
- Conservation of mass
Answer: Conservation of energy
Lenz's law states that the induced current opposes the change in magnetic flux. This opposition requires extra work to be done, which is converted into electrical energy. Without this opposition, energy would be created from nothing, violating conservation of energy.
6. A rectangular metal plate moves into a uniform magnetic field directed into the page. What is the direction of eddy currents in the plate?
- Clockwise near the leading edge
- Anticlockwise near the leading edge
- No eddy currents are induced
- Clockwise near the trailing edge
Answer: Anticlockwise near the leading edge
As the plate enters the field, flux into the page increases. Lenz's law says induced current opposes this increase, so it produces a magnetic field out of the page. By right-hand rule, this requires anticlockwise current loops near the leading edge.
7. A long solenoid's current is increasing. What is the direction of the induced electric field inside the solenoid at a point off the axis?
- Radially outward from the axis
- Tangential, anticlockwise when viewed from the end where B points toward you
- Tangential, clockwise when viewed from the end where B points toward you
- Along the axis, opposite to the direction of B
Answer: Tangential, anticlockwise when viewed from the end where B points toward you
By Faraday's law, the induced electric field is tangential due to cylindrical symmetry. Lenz's law: induced E opposes the increase in B. For increasing B out of the page (toward you), the induced current is anticlockwise, so E is anticlockwise.
8. In Faraday's experiment, a bar magnet is moved toward a coil connected to a galvanometer. The galvanometer shows deflection only when the magnet is
- stationary inside the coil
- removed from the coil and kept far away
- placed at a fixed distance from the coil
- moving relative to the coil
Answer: moving relative to the coil
Faraday's experiment shows that induced current flows only when the magnetic flux through the coil changes. Moving the magnet changes the flux, causing deflection. When the magnet is stationary, flux is constant, so no current is induced.
9. A conducting ring is dropped from rest above a vertical bar magnet with north pole facing up. Which statement correctly describes the ring's motion?
- The ring falls with acceleration equal to g throughout.
- The ring falls with acceleration equal to g until it reaches the magnet, then stops.
- The ring falls with acceleration greater than g due to an attractive force.
- The ring falls with acceleration less than g due to an upward repulsive force.
Answer: The ring falls with acceleration less than g due to an upward repulsive force.
As the ring approaches the magnet, magnetic flux through it increases. By Lenz's law, induced current creates a magnetic field opposing this increase, producing an upward repulsive force on the ring. This reduces its acceleration below g.
10. A bar magnet with north pole facing downward is moved vertically towards a horizontal coil from above. The area vector points upward. What is the sign of induced emf?
- Zero
- Negative
- Positive
- Cannot be determined
Answer: Positive
As the north pole approaches, the downward magnetic flux through the coil increases. Since area vector points upward, the flux is negative and its magnitude increases, so dΦ/dt is negative. By Faraday's law, ε = -dΦ/dt, so ε is positive.
11. A rectangular loop of area 0.1 m² rotates in a uniform 0.5 T field with angular speed 10 rad/s. At t=0, the plane of the loop is perpendicular to the field. Which graph best shows the induced emf ε(t) for one full cycle?
- ε(t) = 0.5 cos(10t) V
- ε(t) = 0.5 sin(10t) V
- ε(t) = 0.25 sin(10t) V
- ε(t) = 0.25 cos(10t) V
Answer: ε(t) = 0.5 sin(10t) V
Using Faraday's law, flux Φ = BA cosθ with θ = ωt (since at t=0, plane ⊥ B so normal ∥ B, θ=0). Φ = 0.5×0.1×cos(10t) = 0.05 cos(10t) Wb. Induced emf ε = -dΦ/dt = 0.05×10 sin(10t) = 0.5 sin(10t) V. The negative sign gives a sine function.
12. A loop rotates with constant angular speed in a non-uniform magnetic field. Which statement about the induced emf is correct?
- It is periodic with period T = 2π/ω
- It is zero because the field is non-uniform
- It is constant because the rotation is uniform
- It is sinusoidal because the loop rotates uniformly
Answer: It is periodic with period T = 2π/ω
Even in a non-uniform field, the flux through the loop changes periodically as the loop rotates, so the induced emf is periodic with the same period as rotation. The waveform may not be sinusoidal, but it repeats every full rotation.