# Lenz's law

Lenz's law (pronounced /ˈlɛnts/), named after the physicist Heinrich Lenz who formulated it in 1834, says:

The direction of current induced in a conductor by a changing magnetic field due to Faraday's law of induction will be such that it will create a field that opposes the change that produced it.

${\displaystyle {\mathcal {E}}=-{\frac {\partial \Phi }{\partial t}},}$

which indicates that the induced voltage (${\displaystyle {\mathcal {E}}}$) and the change in magnetic flux (${\displaystyle \partial \Phi }$) have opposite signs.[1] It is a qualitative law that specifies the direction of induced current but says nothing about its magnitude. Lenz's Law explains the direction of many effects in electromagnetism, such as the direction of voltage induced in an inductor or wire loop by a changing current, or why eddy currents exert a drag force on moving objects in a magnetic field.

Lenz's law can be seen as analogous to Newton's third law in classic mechanics.[2]

For a rigorous mathematical treatment, see electromagnetic induction and Maxwell's equations.

## Opposing currents

If a change in the magnetic field of current i1 induces another electric current, i2, the direction of i2 is opposite that of the change in i1. If these currents are in two coaxial circular conductors 1 and 2 respectively, and both are initially 0, then the currents i1 and i2 must counter-rotate. The opposing currents will repel each other as a result.

Lenz's law states that the current induced in a circuit due to a change or a motion in a magnetic field is so directed as to oppose the change in flux and to exert a mechanical force opposing the motion.

### Example

Currents bound inside the atoms of strong magnets can create counter-rotating currents in a copper or aluminum pipe. This is shown by dropping the magnet through the pipe. The descent of the magnet inside the pipe is observably slower than when dropped outside the pipe.

When a voltage is generated by a change in magnetic flux according to Faraday's Law, the polarity of the induced voltage is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the examples below, if the flux is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to oppose the change.

## Detailed interaction of charges in these currents

Aluminium ring moved by electromagnetic induction, thus demonstrating Lenz's law. Performed by Prof. Oliver Zajkov at the Physics Institute at the Ss. Cyril and Methodius University of Skopje, Macedonia.
Experiment showing Lenz's law with two aluminium rings on a scales-like device set up on a pivot so as to freely move in the horizontal plane. One ring is fully enclosed, while the other has an opening, not forming a complete circle. When we place a bar magnet near the fully enclosed ring, the ring is repulsed by it. However, when the system comes to a rest, and we remove the bar magnet, then the ring is attracted by it. In the first case, the induced current created in the ring resists the increase of magnetic flux caused by the proximity of the magnet, while in the latter, taking the magnet out of the ring decreases the magnetic flux, inducing such current whose magnetic field resists the decrease of flux. This phenomenon is absent when we repeat the experiment with the ring that isn't enclosed by inserting and removing the magnet bar. The induced currents in this ring can't enclose themselves in the ring, and have a very weak field that cannot resist the change of the magnetic flux. Performed by Prof. Oliver Zajkov at the Physics Institute at the Ss. Cyril and Methodius University of Skopje, Macedonia.

In electromagnetism, when charges move along electric field lines work is done on them, whether it involves storing potential energy (negative work) or increasing kinetic energy (positive work).

When net positive work is applied to a charge q1, it gains speed and momentum. The net work on q1 thereby generates a magnetic field whose strength (in units of magnetic flux density (1 tesla = 1 volt-second per square meter)) is proportional to the speed increase of q1. This magnetic field can interact with a neighboring charge q2, passing on this momentum to it, and in return, q1 loses momentum.

The charge q2 can also act on q1 in a similar manner, by which it returns some of the momentum that it received from q1. This back-and-forth component of momentum contributes to magnetic inductance. The closer that q1 and q2 are, the greater the effect. When q2 is inside a conductive medium such as a thick slab made of copper or aluminum, it more readily responds to the force applied to it by q1. The energy of q1 is not instantly consumed as heat generated by the current of q2 but is also stored in two opposing magnetic fields. The energy density of magnetic fields tends to vary with the square of the magnetic field's intensity; however, in the case of magnetically non-linear materials such as ferromagnets and superconductors, this relationship breaks down.

## Field energy

The electric field stores energy. The energy density of the electric field is given by:

${\displaystyle u={\frac {1}{2}}\varepsilon |\mathbf {E} |^{2}\,,}$

In general the incremental amount of work per unit volume ${\displaystyle \delta W}$ needed to cause a small change of magnetic flux density ${\displaystyle \delta }$B is:

${\displaystyle \delta W=\mathbf {H} \cdot \delta \mathbf {B} .}$

## Conservation of momentum

Momentum must be conserved in the process, so if q1 is pushed in one direction, then q2 ought to be pushed in the other direction by the same force at the same time. However, the situation becomes more complicated when the finite speed of electromagnetic wave propagation is introduced (see retarded potential). This means that for a brief period the total momentum of the two charges is not conserved, implying that the difference should be accounted for by momentum in the fields, as asserted by Richard P. Feynman.[3] Famous 19th century electrodynamicist James Clerk Maxwell called this the "electromagnetic momentum".[4] Yet, such a treatment of fields may be necessary when Lenz's law is applied to opposite charges. It is normally assumed that the charges in question have the same sign. If they do not, such as a proton and an electron, the interaction is different. An electron generating a magnetic field would generate an EMF that causes a proton to accelerate in the same direction as the electron. At first, this might seem to violate the law of conservation of momentum, but such an interaction is seen to conserve momentum if the momentum of electromagnetic fields is taken into account.

## References

1. ^ Giancoli, Douglas C. (1998). Physics: principles with applications (5th ed.). p. 624.
2. ^ Schmitt, Ron. Electromagnetics explained. 2002. Retrieved 16 July 2010.
3. ^ The Feynman Lectures on Physics: Volume I, Chapter 10, Page 9.
4. ^ Maxwell, James C. A treatise on electricity and magnetism, Volume 2. Retrieved 16 July 2010.