Electromagnetic Induction: common doubts, answered
The questions students ask most often about Electromagnetic Induction, each with a short answer. For the full chapter, read the Electromagnetic Induction notes.
About the chapter
Is an emf induced in a loop that is not a closed circuit?
Yes. An emf is induced whenever the magnetic flux linked with a conductor changes, whether or not the circuit is closed, but a current flows only around a closed conducting path. A rod moving across a field develops an emf Blv between its ends with no current at all, and a coil with a gap shows a voltage across the gap. Closing the circuit lets that emf drive a current, I = ε/R.
What is the difference between motional emf and the emf induced by a changing field?
Motional emf arises because a conductor moves through a field and its charges feel qv × B; the other kind arises in a stationary loop because the field through it changes with time. Both obey Faraday's law, ε = −dΦB/dt, since in each case the flux linked with the circuit changes. In the moving rod, the changing area of the circuit supplies the change; in the still coil, a changing B does.
The experiments of Faraday and Henry
Read this section in the notes →Why is a current induced in a coil only while a magnet is moving near it?
Because induction needs a changing magnetic flux, and a magnet held still gives a steady flux. Faraday and Henry found that a galvanometer joined to a coil deflects only while the magnet or coil is moving, or while the current in a nearby coil is being switched on or off. Faster motion gives a larger deflection and reversing the motion reverses it. A steady field, however strong, induces nothing.
Magnetic flux
Read this section in the notes →Is magnetic flux a scalar or a vector, and how is the angle in BA cos θ measured?
Magnetic flux is a scalar, given by the dot product ΦB = B·A = BA cos θ. The angle θ is between B and the area vector, which points along the normal to the surface. If a question gives the angle between B and the plane of the coil instead, use the sine of that angle. Flux is measured in weber, with 1 Wb = 1 T m².
What is the change in flux when a coil facing a field is turned through 180°?
The change is 2BA, not zero. A coil facing the field has flux +BA; after a half turn the same field passes through it from the other side, so the flux is −BA. The change is −BA − BA, of size 2BA, or 2NBA for a coil of N turns. Treating flux as a bare magnitude hides this and gives a wrong answer of zero.
Faraday's law of induction
Read this section in the notes →Can the magnetic flux through a coil change without anything moving?
Yes. Flux ΦB = BA cos θ can change through B, through A or through θ. Changing the current in a nearby coil changes B with nothing moving at all, which is how a transformer works. Pulling a loop out of a field changes the area in the field, and turning a coil changes θ. Any of these changes induces an emf.
Why is the induced emf multiplied by N in Faraday's law?
Because every turn of a closely wound coil links the same changing flux, so the same emf is induced in each turn. The turns are joined one after another in series, so their emfs add up: ε = −N dΦB/dt. Forgetting N is one of the most common slips in numerical problems on induction.
What does the minus sign in Faraday's law mean?
It expresses Lenz's law: the induced emf acts to oppose the change in flux that causes it. In ε = −N dΦB/dt the size of the emf comes from how fast the flux changes, and the minus sign fixes its direction. If the flux is increasing, the induced current sets up a field against it; if the flux is decreasing, a field that tries to keep it going.
Lenz's law and conservation of energy
Read this section in the notes →How does Lenz's law follow from conservation of energy?
If the induced current helped the change instead of opposing it, a small push on a magnet would make it speed up by itself while also producing electrical energy, which is energy from nothing. Because the induced current opposes the motion, you have to do work against it, and that mechanical work is exactly what appears as electrical energy and heat in the circuit.
How do I use Lenz's law to find the direction of the induced current?
First decide which way the flux through the loop points and whether it is increasing or decreasing. The induced current must create a field that opposes this change: against the existing field if the flux grows, along it if the flux shrinks. Then use the right-hand rule to turn that required field direction into a clockwise or anticlockwise current in the loop.
Motional emf
Read this section in the notes →Why does an emf appear across a rod moving through a magnetic field?
Because the free charges in the rod move with it and feel a magnetic force qv × B along its length. This pushes electrons towards one end, so one end becomes negative and the other positive, until the electric field between the ends balances the magnetic push. The emf is ε = Blv when rod, velocity and field are mutually perpendicular. A rod moving along B gets no emf.
Why must a force be applied to keep a rod sliding at constant speed on rails in a magnetic field?
Because the induced current in the rod feels a magnetic force IlB that opposes its motion. Left alone, the rod would slow down. At steady speed the applied force exactly balances this retarding force, and the mechanical power supplied, force times speed, appears as electrical power and finally as heat in the circuit's resistance. This is Lenz's law and energy conservation working together.
Rotating rods and wheels
Read this section in the notes →Why is the emf of a rotating rod ½BωR² and not BωR²?
Because different points along the rod move at different speeds. A point at distance r from the pivot moves at ωr, rising from zero at the pivot to ωR at the tip. The average speed along the rod is ωR/2, so the emf between pivot and free end is B × R × ωR/2 = ½BωR². Using the tip speed for the whole rod doubles the answer.
Why don't the emfs of all the spokes of a rotating metal wheel add up?
Because the spokes are connected in parallel between the axle and the rim, not in series. Each spoke has the same emf, ½BωR², between its axle end and its rim end, and all the axle ends meet at one point while all the rim ends meet at the rim. Equal emfs in parallel give that same emf, so the wheel acts like a single spoke.
Mutual inductance
Read this section in the notes →Why is the mutual inductance of two coils the same whichever coil carries the current?
Because it depends only on the geometry of the pair and the medium, not on which coil is driven. For two long coaxial solenoids, working out the flux either way leads to the same result, M = μ₀n₁n₂πr₁²l, where r₁ is the inner radius. So M₁₂ = M₂₁, and a changing current in either coil induces ε = −M dI/dt in the other.
What factors does the mutual inductance of two coils depend on?
It depends on the numbers of turns, the sizes of the coils, their separation and relative orientation, and the material between them. Coils that are close together and coaxial share more flux and so have a larger M. A core of magnetic material multiplies M by its relative permeability. M does not depend on the currents themselves, only on how much of one coil's flux links the other.
Self-inductance and magnetic energy
Read this section in the notes →What is back emf in an inductor, and does it only oppose a rising current?
Back emf is the emf a coil induces in itself when its own current changes: ε = −L dI/dt. It opposes any change in current, not the current itself. When the current rises, the back emf fights the rise; when the current falls, it acts to keep it flowing. Breaking a circuit with a large inductor makes the current fall almost instantly, so L dI/dt becomes very large and can cause a spark at the switch.
Why is self-inductance compared to mass or inertia?
Because it resists changes in current the way mass resists changes in velocity. A large L means the current cannot be started or stopped suddenly. The stored energy, U = ½LI², even has the same form as kinetic energy ½mv², with L in place of m and I in place of v. This energy is held in the magnetic field, with energy density B²/2μ₀.
How does the self-inductance of a solenoid change if its turns are doubled?
It becomes four times larger, provided the length and area stay the same. For a long solenoid L = μrμ₀n²Al, which goes as the square of the turns per unit length. Doubling the turns doubles the field made by a given current and also doubles the number of turns that field threads, so the two effects multiply. An iron core raises L further by its relative permeability.
AC generator
Read this section in the notes →At what position of the coil is the emf of an AC generator maximum?
The emf is maximum when the plane of the coil is parallel to the field, where the flux through it is zero. That is where the flux is changing fastest. With ε = NBAω sin ωt and the angle ωt measured between B and the coil's normal, the emf peaks at 90°. When the coil faces the field squarely the flux is largest but momentarily steady, so the emf is zero.
How can the peak emf of an AC generator be increased?
By raising any factor in ε₀ = NBAω: more turns, a stronger field, a larger coil area or faster rotation. Faster rotation also raises the frequency, ν = ω/2π, so where the supply frequency must stay fixed the other factors are changed instead. In power stations the coil is turned by mechanical energy, for example from falling water or high-pressure steam driving a turbine.
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