# Maxwell's third equation (Faraday's law of induction)

#### mentaaal

Joined Oct 17, 2005
451
Hey guys, very quick question based on Mawell's third equation which states that ∫E.DL = -∂/dt(θ) where θ is the magnetic flux intersecting the circuit and the integral is a closed line integral.

Lets say you have a very simple circuit of an inductor and a resistor connected together and the inductor was exposed to a linearly time varying magnetic field. There would then be an induced voltage across the inductor due to the changing magnetic field. This voltage would then be dropped across the resistor in accordance with Kirchoff's voltage law. At this time if you were to perform the closed line integral ∫E.DL on the circuit described would you not get zero? Surely this would be a contradiction of the maxwell's third equation. Clearly I am missing something here in that one of the above statements is erroneous.

#### studiot

Joined Nov 9, 2007
4,998
If it's time varying you need to include for time in the integral.

#### mentaaal

Joined Oct 17, 2005
451
Ok so lets say that the magnetic flux is changing linearly such that the voltage induced across the inductor is constant. For a small time frame, say before any nonlinear effects such as saturation in the unductor, could you not regard the inductor - resistor circuit as being equivalent to a simple cell and a resistor.

#### steveb

Joined Jul 3, 2008
2,436
.... could you not regard the inductor - resistor circuit as being equivalent to a simple cell and a resistor.
No you can't regard them as being equivalent. You can say they are similar, but not equivalent. You can regard both situations as an electro-motive-force (EMF) driving current through a resistor. However, the source of the EMF is different in each case.

Basically, a cell has a gradient of electric field: i.e. voltage. So, you can take a line integral through a battery and find the potential difference. However, the coil does not have an electric field to integrate, and the line integral of electric field is zero through the inductor. Still, there is still an electro-motive-force (EMF) generated in the inductor, due to the changing flux: i.e. -∂/dt(θ).

Hence, the line integral of electric field around a coil (with flux change) and a resistor is not zero, and Faraday's Law and Kirchoff's Law are both obeyed.

The idea of electric field around a closed path being equal to zero is just not a valid general law of nature. It's only true in special cases. This is a common misconception.

#### tmac

Joined Oct 28, 2008
10
Hi!

I hope I understood your question correctly. Here goes:

The line integral of the electric field around the circuit will not be zero, since everywhere along the wire there is a tangential electric field that pushes the electrons along. Otherwise the electrons would not move, since they are in a dissipative medium (non-zero resistance).

Near the inductor, the varying magnetic field creates a rotational electric field that pushes the electrons along. This is described by one of Maxwell's equations.

There is no contradiction between this and the fact that the integral of the voltage differential around the complete circuit is zero, since the relationship between the electric field and the electric potential is more complicated in the presence of a varying magnetic field. The time-dependent vector potential now contributes to the electric field.

#### russ_hensel

Joined Jan 11, 2009
825
Also remember that putting a conductor in the field induces currents that change the field. That is why a changing magnetic field cannot penetrate a super conductor. ( I think)