Eddy Current

Right-hand Rule Eddy Currents

The Right-Hand Rule, also known as Fleming's Right-hand Rule, is a handy trick for remembering the direction of the Lorentz force on charges moving in a magnetic field.

Unlike the charges in an electric field, the strength and direction of this force is determined by the velocity \(\vec{v}\) of the charges. This force is determined by the vector cross product of the velocity and field\[\vec{F}=q(\vec{v}\times\vec{B})\]

The vector formed from the cross product of two vectors is always perpendicular to the initial vectors. Hence, in order to, keep track of which direction the Lorentz force is acting in, we use the right-hand rule. Consider a conducting plate moving to the right under a fixed magnetic field pointing downwards through the plate, as shown below.

With the right hand out, point the index finger in the direction of the plate's movement, to the right. This represents the velocity \(\vec{v}\) of the charges. Now, keeping the index finger as it is, take the second finger and point it in the direction of the magnetic field \(\vec{B}\) that is downwards. Now, pointing the thumb gives the direction of the Lorentz force on the charges, towards the back of the plate.

This force causes the charges to drift towards the back of the plate, which in turn changes their velocity \(\vec{v}\) and the direction of the Lorentz Force. Following the direction of the charges with the index finger, we'll find that this Lorentz Force causes them to circle around, forming closed loops. So, we see that in situations where charges in a conductor are moving perpendicularly to a magnetic field, eddy currents form due to the Lorentz force and the right-hand rule.

Eddy Current Loss

Having seen how these eddy currents can occur, let's now look at the consequences of such currents and the effect they can have on some key technologies.

One of the most troublesome effects of eddy currents is that they can be a huge source of power loss, for example in transformers using electromagnetic induction to change the voltage of an electric current.

Eddy currents cause power loss since no material is a perfect conductor (aside from super-conductors, but that's a different question). Most materials have a non-zero resistivity, this means that currents flowing through them experience a drag force due to the charges present in the atoms and molecules of the conductor itself.

It is dependent on the intrinsic resistance of the material \(R\), as well as the surface area of the material \(A\) and its length \(L\):\[\rho=\frac{R A}{L}.\]

So if eddy currents arise in a conductor with non-zero resistivity, they will expend some of their kinetic energy as heat in trying to overcome the resistive forces. This heating effect can be brilliantly exploited in modern induction hobs, which use electromagnetic induction to produce eddy currents in a pan instead of heating them with gas fires in the conventional way.

However, this heating effect can be a massive problem in the context of transformers, where as much energy as possible needs to be conserved when converting between different voltages. This is why in modern transformers the iron cores are made of hundreds of sheets of thin laminated iron. As the iron sheets are insulated from each other, eddy currents are prevented from spreading across the iron core. Similarly, by keeping the sheets as thin as possible, the resistivity of the transformer is kept as low as possible.

Eddy Current Braking

This heating effect isn't the only eddy current-based phenomena that has been exploited to produce ingenious technologies. Eddy currents can also be used to produce electromagnetic braking systems, often used in trains or rollercoaster carts. The physics behind this braking is truly fascinating, relying on the back-and-forth induction of electric and magnetic fields.

Eddy current brakes exist because of a law of electromagnetism known as Lenz's law.

Lenz's law is contained within Faraday's Law of Induction as the negative sign within the equation\[\mathcal{E}=-\frac{\mathrm{d}\Phi_B(t)}{\mathrm{d}t}\]

This negative sign ensures that the current is directed such that the magnetic field produced by this current acts against the initial change in flux. This opposing magnetic field can act as a mechanical force, decelerating magnetized objects.

For instance, if a magnet is dropped through a copper pipe, the changing magnetic flux creates eddy currents in the pipe. This current then produces a magnetic force which acts against the changing magnetic flux, slowing down the falling magnet. This simple experiment can be done in a school lab, showing that the magnet will fall noticeably slower through the copper pipe than it would in free fall.

Eddy current brakes work by having a rotating disc of conductive metal moving through a fixed magnetic field. As we've seen, this will produce eddy currents in the rotating disc. Thanks to Lenz's law these eddy currents will then produce magnetic fields which oppose the rotation of the disc slowing it down, causing the kinetic energy of the disc to be dissipated as heat. This ingenious form of breaking is often used on rollercoaster carts to smoothly slow down carts at the end of the ride. However, as eddy currents can only be produced when the disc is rotating, eddy current brakes can't be used to keep a train at a stop. Usually, then eddy current brakes will be used with normal mechanical brakes, most commonly on trains.

Eddy Current Testing

Eddy currents are also commonly used in materials testing, where they can be used to look for cracks or flaws. In eddy current testing (ECT), an electrified probe composed of a coil of conductive wire is run over the surface of a conductive material. The magnetic field created by this electrified probe then creates eddy currents in the conductive material, measuring these eddy currents indicates the electrical conductivity and magnetic permeability of the material. Fluctuations in the conductivity or permeability of the material as the probe moves along indicates flaws or cracks in the material, which may be far too small to see with the naked eye. Due to the electrical nature, ECT can only be performed on conductive materials. ECT is hugely important in the aerospace industry, where the smallest of cracks in the surface of a plane can quickly become disastrous once at high speeds and altitudes.