Introduction to Electrodynamics/Chapter 7/1
Contents
7.1 Electromotive Force
7.1.1 Ohm's Law
<html>Electrostatics and magnetostatics apply whenever <math>\rho\;</math> and <math>\vec{J}\;</math> are independent of time.
With steady currents, the charge density <math>\rho\;</math> remains constant. So you can have both steady currents and static charges at the same time.
One exception: <math>\vec{E} = 0\;</math> in a conductor. If this were so, you could not have a steady current inside a conductor.
For most substances,
- <math>\vec{J} = \sigma \vec{f}\;</math>
- <math>\sigma\;</math> is the conductivity of the material.
- <math>\rho = 1/\sigma\;</math> is the resistivity of the material.
- Insulators have a very small conductivity / very large resistivity, typically factor of 1,000,000,000,000,000,000!
- perfect conductors have infinite conductivity / zero resistivity.
- <math>\vec{f}\;</math> is the force per unit charge.
- Could be ANY force, even gravity, etc... "trained ants with tiny harnesses" (haha)
- We care about electromagnetic forces: <math>\vec{f} = \vec{E} + \vec{v} \times \vec{B}\;</math>
- Normally, the magnetic force is too small: <math>\vec{f} = \vec{E}\;</math>
Ohm's Law is <math>\vec{J} = \sigma \vec{E}\;</math>, the current is proportional to the electric field.
<html>
Example 1
A cylinder of constant cross-section has a potential put across the ends. How do the currents and potential relate?
First, we're going to assume that the electric field inside the cylinder is going to be constant. Example 3 explains why this is true.
Next, we relate current to the electric field:
- <math>\vec{J} = \sigma \vec{E}\;</math>
This is multiplied by the cross-sectional area to get the total current:
- <math>I = A J = \sigma A E\;</math>
The potential difference is equal to the electric field times the distance:
- <math>I = \sigma A V / L\;</math>
TODO
<html>
Example 2
<html>
<html>
Example 3
<html>
Problems
7.1.2 Electromotive Forces
Problems
7.1.3 Motional emf
Problems
Example 4
