Electronic – A question about Faraday’s law of induction for a closed path

faradays-lawinduction

As far as I understand, Faraday’s law states that a varying magnetic field through a closed path creates an E-field along the closed path. And the integral of this field we call electric potential or voltage.

And if the loop is an open at one end, I guess all the potential exhibits itself across this gap. Below I assume a varying magnetic field across the wire loops.

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Above the top loop is open at A and B. If we observe 6V across A and B, and then if we replace the same loop this time closed but with 3kΩ, 2kΩ and 1kΩ resistors; can we say that 3V will be induced across 3kΩ, 2V across 2kΩ and 1V across 1kΩ?

If so, what causes the higher voltage across bigger resistors? Is there any formula which shows such a relation, which uses E-field and resistance not the current? Because in the original Maxwell equations, Faraday’s law is about the E-field created by changing magnetic field.

Best Answer

The integral form of Faraday's law is \$\oint\vec{E}\cdot dl = -d/dt\oint\vec{B}\cdot d\vec{s}\$

Knowing the magnetic flux through your loop, you can calculate the voltage around the loop.

The continuity equation \$\nabla\cdot\vec{J} = -d\rho/dt\$ tells you that your current density is constant around your loop, since there is no charge being added or taken away.

Ohm's law says \$\vec{J} = \sigma \vec{E}\$

You would integrate over a distance to turn this into the more common form of Ohm's law \$I = V/R\$.

In your example \$\vec{J}\$ is constant, so if \$\sigma\$ decreases in a portion of the loop, \$\vec{E}\$ must increase. This is why the 3kOhm resistor has 3V across it while the 1kOhm only has 1V across it.