Electromagnetic induction

Magnetic flux. Flux through a surface of area $\vec A$ in a field $\vec B$ is the component of $B$ perpendicular to the surface, multiplied by the area:

where: $B$ is the magnetic field strength (tesla, T), $A$ is the area of the surface (m²), and $\theta$ is the angle between $\vec B$ and the area vector (the normal to the surface). Flux is measured in weber (Wb); it is a scalar that depends on orientation, not just on how strong the field is.

Faraday's two laws. (1) Whenever the flux linked with a circuit changes, an EMF is induced in it. (2) The induced EMF equals the rate of change of flux linkage. For a coil of $N$ turns,

The induced current in a closed loop of resistance $R$ is simply $I = \varepsilon/R$ — once you know $\varepsilon$, this is just Ohm's law.

Lenz's law and energy. The minus sign above is not decoration — it says the induced EMF always drives a current whose magnetic effect opposes the change in flux that produced it. This is Lenz's law, and it is a direct restatement of conservation of energy: to change the flux against this opposition, an external agent must do work, and that work is exactly what shows up as electrical energy in the circuit. There is no way to get induction "for free."

Motional EMF. Flux can change because $B$, $A$, or $\theta$ changes — sliding a rod along rails changes $A$. A straight conductor of length $l$ moving with velocity $v$ perpendicular to a uniform field $B$ (and perpendicular to its own length) has its free charges pushed by the Lorentz force $\vec F = q(\vec v\times\vec B)$, until an internal electric field balances that push. This charge separation is exactly an EMF source, with magnitude $\varepsilon = Blv$ — derived formally below.

Eddy currents. When a bulk conductor (not just a wire loop) experiences a changing flux — a metal sheet swinging through a field, for instance — swirling currents called eddy currents are induced within its body. They dissipate energy as heat, obey Lenz's law just like loop currents, and are put to use in electromagnetic braking and induction furnaces, but they cause unwanted power loss in transformer cores, which is why cores are laminated (thin, insulated sheets) rather than solid blocks.

A metal rod of length 0.5 m moves at 4 m/s at right angles to a magnetic field of 0.6 T, with the rod, its velocity, and the field mutually perpendicular. The rod's own resistance is 2 Ω and it completes a closed circuit of total resistance 5 Ω. Find (a) the motional EMF, (b) the induced current, and (c) the power the external agent must supply to keep the rod moving at constant speed.

Step 1. Motional EMF: $\varepsilon = Blv = 0.6 \times 0.5 \times 4 = 1.2\ \text{V}$.

Step 2. Induced current from Ohm's law on the total circuit resistance: $I = \dfrac{\varepsilon}{R_{total}} = \dfrac{1.2}{5} = 0.24\ \text{A}$.

Step 3. By Lenz's law, this current in the field experiences a retarding force $F = BIl = 0.6 \times 0.24 \times 0.5 = 0.072\ \text{N}$, opposing the rod's motion. To keep the rod moving at constant $v$, an external agent must supply exactly enough mechanical power to balance this force: $P = Fv = 0.072 \times 4 = 0.288\ \text{W}$.

Step 4. Sanity check by computing the electrical power dissipated directly: $P_{elec} = I^2 R_{total} = (0.24)^2 \times 5 = 0.288\ \text{W}$. It matches the mechanical power exactly, confirming energy conservation — every joule the agent puts in against the magnetic braking force reappears as electrical (and ultimately heat) energy in the circuit. The agent must supply 0.288 W.

Competitive stretch

If the rod instead slides freely under a constant applied force (frictionless rails, circuit closed), the retarding force $F = B^2l^2v/R$ grows with speed, so the rod approaches — but never quite reaches — a terminal velocity where the applied force balances this drag, the same mathematics as terminal velocity under air resistance.

• Dropping the minus sign in Faraday's law and then guessing direction from intuition — always invoke Lenz's law explicitly to fix the polarity, don't skip straight to a number.
• Confusing flux $\phi_B = BA\cos\theta$ (a scalar that depends on orientation) with the field $B$ itself — a coil rotating in a constant field still has changing flux because $\theta$ changes, even though $B$ never does.
• Forgetting the factor $N$ (number of turns) when the question involves a coil rather than a single loop — the EMF scales linearly with $N$.
• Treating eddy currents as purely a nuisance — they are deliberately exploited in induction heating and electromagnetic braking; only in transformer cores are they an unwanted loss, which is why those cores are laminated.

Ask the tutor: if you'd like more practice with tilted-coil flux problems or a walkthrough of why a freely sliding rod never quite reaches a fixed terminal speed under magnetic braking, just ask.