Why Does Calculating EMF Use a Line Integral Instead of a Surface Integral?

[randomafk]

Hi,

I'm currently reading griffiths and I'm a little bit confused in how it calculates emf.

EMF is initially introduced as a path integral of force/unit charge around a closed loop. This makes sense, but then he presents us with an example problem (ex 7.4)

Basically, a circuit resistor is attached to a rotating metal disc normal to a uniform magnetic field at its edge and center. (fig 7.14)

He calculates emf as the line integral of the $∫f_{mag}ds$ from 0 to R of the disc. However, doesn't the current flow across the entire disc? Why would we still take a line integral -- shouldn't it be a surface integral across the disc? He says this makes the flux rule difficult since it assumes current alonga well-defined path. But doesn't that apply to calculating emf as well?

When we calculated voltage with an electric field, we do a line integral but it becomes a volume integral w.r.t to the volume in question. Is there any sort of analog here?

Thanks!

 

[eljose79]

Hi there,

Thank you for bringing up this question about calculating EMF in the context of a rotating metal disc in a magnetic field. It's a great point and one that can be a bit confusing at first.

First, let's clarify what EMF is. EMF stands for electromotive force and it is essentially the force that drives the movement of electrons in a circuit. It is measured in volts and is calculated by taking the line integral of the magnetic force around a closed loop.

In the example problem you mentioned, the rotating metal disc is essentially acting as a generator, converting mechanical energy into electrical energy. The EMF is generated as the disc rotates through the magnetic field, creating a changing magnetic flux. This changing flux induces an electric field, which in turn creates a current in the circuit.

Now, to address your question about why we take a line integral instead of a surface integral in this situation. The reason for this is because the EMF is generated along the edge of the disc where it is in contact with the magnetic field. The current then flows through the entire disc, but the EMF is still generated at the edge. Therefore, we only need to consider the line integral around the edge of the disc to calculate the EMF.

To answer your question about an analog to calculating voltage with an electric field, the concept is similar in that we are still using a line integral. However, instead of a volume integral, we are using a line integral because the EMF is generated along a line (the edge of the disc) rather than a volume.

I hope this helps to clarify things a bit. If you have any further questions, please don't hesitate to ask. Keep up the great work with your studies!