Lenz's law

Lenz’s law states: the current induced in a loop by a changing magnetic flux flows in the direction that opposes the change. The induced current’s own magnetic field tries to maintain the existing flux — to resist whatever is happening externally.

This is the meaning of the minus sign in Faraday’s law:

$V_{\text{emf}}=-\frac{d\Phi }{dt}.$

If $\Phi$ is increasing, $V_{\text{emf}}<0$, driving current in the direction that produces opposing flux. If $\Phi$ is decreasing, $V_{\text{emf}}>0$, current direction reverses to maintain the flux.

Why it has to be this way

Lenz’s law is energy conservation in disguise. If the induced current’s field reinforced the change, you’d get a runaway: any small change would amplify itself, building flux and current indefinitely without an energy source. That’s a perpetual-motion machine.

By opposing the change, the induced current acts like inertia for $\Phi$ — it requires external work to keep changing the flux against the induced opposition. That work goes into the energy dissipated in the resistive part of the loop (Joule heating) or stored in the magnetic field.

Operational rule

1. Identify the current state of ${\mathbf{B}}_{\text{ext}}$ through the loop (direction and magnitude).
2. Decide whether ${\Phi }_{\text{ext}}$ is increasing or decreasing.
3. The induced current flows so its own ${\mathbf{B}}_{\text{ind}}$ opposes the change:
   • If ${\Phi }_{\text{ext}}$ is increasing, ${\mathbf{B}}_{\text{ind}}$ points against ${\mathbf{B}}_{\text{ext}}$ (to slow the increase).
   • If ${\Phi }_{\text{ext}}$ is decreasing, ${\mathbf{B}}_{\text{ind}}$ points with ${\mathbf{B}}_{\text{ext}}$ (to maintain the flux).
4. Use the right-hand rule to find the current direction: curl fingers in the sense of ${\mathbf{B}}_{\text{ind}}$, thumb gives current direction.

Worked example

A loop in the $xy$-plane with $\mathbf{B}(t)=\hat{\mathbf{z}}\cdot 0.3t$ T — increasing in $+\hat{\mathbf{z}}$.

• Flux $\Phi$ through the loop increases in $+\hat{\mathbf{z}}$.
• Lenz: induced current opposes, so ${\mathbf{B}}_{\text{ind}}$ inside the loop is in $-\hat{\mathbf{z}}$.
• Right-hand rule: curl right-hand fingers in the sense that produces $-\hat{\mathbf{z}}$ inside the loop → fingers go clockwise viewed from $+\hat{\mathbf{z}}$.
• So induced current flows clockwise viewed from above.

If $\mathbf{B}$ were instead in $-\hat{\mathbf{z}}$ and still increasing in magnitude, the induced current would flow counterclockwise.

Practical consequences

Eddy currents. Move a conducting plate through a non-uniform magnetic field, and Lenz’s law tells you induced currents flow in a way that opposes the motion — the plate experiences a drag force. This is the working principle of:

• Magnetic braking (no friction, no wear).
• Induction cooktops (eddy currents heat the cookware).
• Levitation rails (e.g., Maglev) — kind of, with refinements.

Transformer back-EMF. In a transformer’s primary winding, the changing flux it itself creates induces a back-emf in the primary. Without this self-induced back-emf, infinite current would flow from any voltage source connected to the primary. The back-emf nearly cancels the source voltage at no-load.

Skin effect. In an AC conductor, eddy currents oppose the changing field inside, pushing the current toward the surface. The depth where current density drops to $1/e$ is the skin depth $\delta =1/\sqrt{\pi f\mu \sigma }$, which determines AC resistance of wires at high frequencies.

Compared with Newton’s third law

Lenz’s law is often described as the “EM version of Newton’s third law” — every action provokes a reaction that opposes it. The analogy isn’t quite literal, but the intuition is right: physical systems resist changes to their state, and the resistance is energy-conserving.