Lenz's Law

Statement of Lenz's law

• H. F. E. Lenz (1804-1864) devised a rule for the direction of the induced current, arrived at independently of Faraday and Henry at almost the same time.
• Lenz's law: the direction of the induced current in a circuit is such that the magnetic field produced by the induced current opposes the change in the magnetic flux that induces the current. The direction of the induced emf is the same as that of the induced current.
• In short: the induced emf tends to set up a current the action of which opposes the change that causes it.
• The general statement of the same idea: every effect of induction acts in opposition to the cause that produces it.

Application 1: motion of a magnet towards a loop

• Consider the north pole of a bar magnet moving towards a conducting loop, with the loop's plane perpendicular to the magnet's axis.
• As the magnet approaches, a current is induced in the loop. This current makes the loop a magnetic dipole, oriented so that it opposes the motion of the magnet.
• The loop's north pole must therefore face the approaching north pole of the magnet, so as to repel it. The curled right-hand rule for a magnetic dipole then gives the direction of the induced current: it is anticlockwise as seen from the magnet.
• If the magnet is pulled away, the loop must have a south pole facing the retreating north pole so as to attract it and oppose the retreat. The induced current now flows clockwise.
• Diagram in words: a bar magnet with its N pole pointing at a circular loop, an arrow on the magnet showing motion towards the loop, and a curved arrow on the loop showing the anticlockwise induced current, with N marked on the near face of the loop.

Application 2: the jumping ring experiment

• A coil is wound around an iron core held vertically upright, and a metallic ring is placed over the top of the core. When the current in the coil is switched on, the ring jumps several metres into the air.
• Explanation: before the current is switched on, the flux through the ring is zero. When the current is switched on, an upward flux appears. This change of flux induces an emf, and hence a current, in the ring.
• By Lenz's law the induced current in the ring is opposite in direction to the current in the coil. Opposite (antiparallel) currents repel, so the ring flies off.

Energy conservation in Lenz's law

• The induced current may be caused either by (i) the motion of a conductor in a magnetic field, or (ii) a change of magnetic flux through a stationary circuit.
• In the first case, the direction of the induced current in the moving conductor is such that the thrust exerted on it by the magnetic field is opposite to the direction of its motion, and thus opposes that motion.
• In the second case, the induced current sets up a magnetic field of its own which, inside the area bounded by the circuit, is (a) opposite to the original field if that field is increasing, and (b) in the same direction as the original field if that field is decreasing.
• Thus it is the change in flux through the circuit, not the flux itself, which is opposed by the induced current.
• Lenz's law follows directly from the conservation of energy. Suppose the induced current flowed in the direction that helped the cause producing it. Then the induced current and the flux through the loop would grow without limit; the current, once started, would keep increasing indefinitely and produce Joule heating at no extra cost — a self-sustaining perpetual motion machine. That violates the law of conservation of energy. Lenz's law is therefore a necessary consequence of energy conservation.
• To have an induced current we must have a closed circuit. If the conductor is not part of a closed circuit, mentally construct a circuit between its two ends, apply Lenz's law to find the direction of the induced current, and the polarity of the open ends follows.

Lenz's law and Faraday's law: the meaning of the minus sign

• Consider $e = -\dfrac{d\phi}{dt}$ with attention to the negative sign.
• Let the area vector $\vec{A}$ of the loop, perpendicular to the plane of the loop, be fixed and parallel to $\vec{B}$ ($\theta = 0$), and let $B$ increase with time. Using $\phi = \vec{B}\cdot\vec{A}$,

$$e = -\dfrac{d}{dt}\left(\vec{B}\cdot\vec{A}\right) = -\lvert\vec{A}\rvert\dfrac{dB}{dt}$$

• The right-hand side is negative, because $\lvert\vec{A}\rvert$ is positive and $\dfrac{dB}{dt}$ is positive ($B$ is increasing).
• The screwdriver (right-handed screw) rule fixes the positive sense of circulation around the loop as clockwise. Since the induced current in the loop is actually anticlockwise, it is negative in that convention, and so is the induced emf. The left-hand side is therefore also negative, as it must be.
• Hence the negative sign in $e = -\dfrac{d\phi}{dt}$ is what incorporates Lenz's law into Faraday's law.
• Short board answer: the negative sign indicates that the emf induced in the coil is in such a direction that it opposes the change in magnetic flux linked with the coil.

Standard results

• Faraday's laws plus Lenz's law: $e = -N\dfrac{d\phi}{dt}$.
• Magnet approaching: induced current anticlockwise (loop presents a like pole, repels). Magnet receding: clockwise (loop presents an unlike pole, attracts).
• Lenz's law is a consequence of the law of conservation of energy.