# Does one of Maxwell's equations describe this magnetic field?

## Main Question or Discussion Point

I have some copper enamel wire, winded up into something of a circle/ellipse, with about 20 turns, and the purpose for it is to give me a decent magnetic field (which it does) using a DC source. The magnetic field is picked up by a hall effect sensor on a nearby circuit, which is connected to some LEDs, which in turn light up when the field is detected.

I'm stuck between 2 of Maxwell's equations which ought to describe my system:
1. Faraday's law of induction (only problem is that this describes a time-varying magnetic field, which can only come about from an AC source..right?)
2. Ampere-Maxwell law (or Ampere's circuital law. But this doesn't explain how the LED's are turned on..it just says an electric current will give me a magnetic field).

I'm thinking 2 best describes my system? Yes I get a magnetic field, and that is picked up by the sensor...but isn't Faraday's law of induction specifically meant for wireless power, ie, turning on light bulbs and the likes?

Thanks.

## Answers and Replies

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jtbell
Mentor
The Hall effect is due to the magnetic force on the electrons moving through a conductor that is in a magnetic field. Maxwell's equations don't describe forces.

Maxwell's equations describe electric and magnetic fields, which in turn give rise to electric and magnetic forces; why else are they described as vectors, or vector fields? Plus the Lorentz Force is related to the hall effect, and it's also related to Faraday's law of induction.

The Hall effect sensor takes in a magnetic field as an input, and outputs a voltage, or electric field: doesn't this mean one of Maxwell's equations is at work? Ampere-Maxwell law seems to best describe it.

Or do I have this the wrong way around? Thanks.

Maxwell's equations describe the generation of electric and magnetic fields as a result of charges, currents, and time changes in the fields themselves.

The equations do not say how these fields affect charges and currents. That's Lorentz's law, which completes the picture.

Hmm I see then. Can my system still be described by Faraday's law of induction, or Ampere-Maxwell law, at least superficially?

No. You need Lorentz's law (and Newton's 2nd law). The voltage you find is precisely from a zero Lorentz force:

f = q(E + v cross B0)

With no motion, the second term is zero and the electrons distribute themselves so that the electric field is zero inside the conductor (steady state solution will have no net force). When there is motion, the electrons re-distribute themselves into a new steady state where qE precisely opposes q*v cross B. The electric field does indeed arise from Gauss's law, but the charge distribution you would need to use in Gauss's law cannot be found without doing a force balance, and the Lorentz expression is crucial to link forces to EM fields.

Thanks for that..kinda makes more sense.