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# Lenz's law

Electromagnetism |
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**Lenz's law**/ˈlɛnts/ is a common way of understanding how electromagnetic circuits obey Newton's third law and the conservation of energy.

^{[1]}Lenz's law is named after Heinrich Lenz, and it says:

If an induced current flows, its direction is always such that it will oppose the change which produced it.

Lenz's law is shown with the negative sign in Faraday's law of induction:

- <math>\mathcal{E}=-\frac{\partial \Phi}{\partial t},</math>

which indicates that the induced voltage (ℰ) and the change in magnetic flux (∂Φ) have opposite signs.^{[2]} Lenz's Law is a qualitative law that refers to the direction of induced current in relation to the effect which produces it without quantitatively relating their magnitudes.

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

## Contents

## Opposing currents

If a change in the magnetic field of current *i*_{1} induces another electric current, *i*_{2}, the direction of *i*_{2} is opposite that of the change in *i*_{1}. If these currents are in two coaxial circular conductors *ℓ*_{1} and *ℓ*_{2} respectively, and both are initially 0, then the currents *i*_{1} and *i*_{2} 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

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 *q*_{1}, it gains speed and momentum. The net work on *q*_{1} 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 *q*_{1}. This magnetic field can interact with a neighboring charge *q*_{2}, passing on this momentum to it, and in return, *q*_{1} loses momentum.

The charge *q*_{2} can also act on *q*_{1} in a similar manner, by which it returns some of the momentum that it received from *q*_{1}. This back-and-forth component of momentum contributes to magnetic inductance. The closer that *q*_{1} and *q*_{2} are, the greater the effect. When *q*_{2} 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 *q*_{1}. The energy of *q*_{1} is not instantly consumed as heat generated by the current of *q*_{2} 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:

- <math> u = \frac{1}{2} \varepsilon |\mathbf{E}|^2 \, ,</math>

In general the incremental amount of work per unit volume *δW* needed to cause a small change of magnetic flux density *δ***B** is:

- <math>\delta W = \mathbf{H}\cdot\delta\mathbf{B}.</math>

## Conservation of momentum

Momentum must be conserved in the process, so if *q*_{1} is pushed in one direction, then *q*_{2} 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

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

## External links

- Eddy Currents and Lenz's Law (audio slideshow from the National High Magnetic Field Laboratory)
- MIT A brief video demonstrating Lenz's law
- A dramatic demonstration of the effect on YouTube with an aluminum block in an MRI
- Eddy currents produced by magnet and copper pipe.