Electromagnetic Induction — Questions & Answers

Sample questions from Electromagnetic Induction

1. Q1. Two ideal inductors of 3 H and 6 H are connected in parallel, far enough apart that their mutual inductance is negligible. The equivalent inductance is:

   • A.9 H
   • B.2 H✓
   • C.4.5 H
   • D.18 H
   Solution

   In parallel the same EMF e appears across each: e = L1 dI1/dt = L2 dI2/dt, and dI/dt = dI1/dt + dI2/dt = e(1/L1 + 1/L2). So 1/L = 1/L1 + 1/L2 = 1/3 + 1/6 = 1/2, giving L = 2 H (like resistors). In series (no coupling) it would be 9 H.

2. Q2. For the flux law phi = (3t^2 - 4t + 1) mWb through a coil, the induced EMF reverses its direction at:

   • A.t = 1/3 s
   • B.t = 1 s
   • C.t = 2/3 s✓
   • D.It never reverses
   Solution

   The EMF is proportional to dphi/dt = 6t - 4, which changes sign when 6t - 4 = 0, i.e. at t = 2/3 s. The instants where phi itself is zero (the roots of the quadratic) are irrelevant to the sign of the EMF.

3. Q3. A superconducting ring (zero electrical resistance) has a persistent, unchanging current flowing in it, with no external EMF source. This is consistent with electromagnetic theory because:

   • A.Superconductors have infinite self-inductance, making current impossible in principle
   • B.Superconductors violate Faraday's law entirely
   • C.The current must actually be decaying, just imperceptibly slowly, and 'persistent current' is a misnomer
   • D.With zero resistance, once a current is established, it persists indefinitely without any driving EMF needed to overcome resistive losses (since there are none)✓
   Solution

   In an ordinary (resistive) circuit, a continuously applied EMF is needed to maintain a steady current against resistive (I^2 R) energy losses, but in a SUPERCONDUCTING ring (R=0 exactly), there are no resistive losses to overcome, so once a current is established (e.g. by briefly applying an external changing flux while the ring was still normal/resistive, then cooling it below its critical temperature to become superconducting, trapping that flux/current), the current can persist indefinitely with NO ongoing EMF needed, fully consistent with Ohm's law (V=IR=I x 0=0, requiring no driving voltage) and observed experimentally in real superconducting rings. A student who thinks Faraday's law is somehow violated by this persistent current picks B, but it is fully consistent -- no CHANGING flux is occurring in the steady persistent-current state, so there is simply no induced EMF being generated OR needed.

4. Q4. A coil is connected to a battery through a switch. A LARGE resistance is deliberately placed in parallel with the coil. Its purpose is:

   • A.To increase the steady current through the coil
   • B.To increase the time constant while the current grows
   • C.To provide a path for the current when the switch is opened, limiting the induced EMF and preventing a spark✓
   • D.To convert the coil into a transformer
   Solution

   On opening the switch the coil current cannot stop instantly; with no alternative path dI/dt would be enormous and the back-EMF L dI/dt would flash across the contacts. The parallel resistor lets the current decay smoothly with time constant L/R, keeping the induced EMF finite, and is made large so that it wastes little power in normal operation.

5. Q5. A transformer core is built from thin insulated sheets. For effective reduction of eddy current loss, the sheets must be stacked so that their planes are:

   • A.Perpendicular to the magnetic field
   • B.Parallel to the magnetic field✓
   • C.At 45° to the field
   • D.Orientation does not matter as long as they are insulated
   Solution

   Eddy currents circulate in planes PERPENDICULAR to the changing B. To interrupt these loops, the insulating layers must cut across them, i.e. the sheets must be stacked with their planes PARALLEL to B. Sheets perpendicular to B would leave each sheet as a full closed conducting plane for eddy currents.

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