Advanced E&M Phys 350

Chapter 8

Energy, force, momentum

Flow

One of the best places to start is the conservation of charge equation:

J is the vector that describes the movement of charge. If you picture a small volume you can assign a velocity to the charge in that region and imagine this stuff moving. The divergence tells us whether based on this movement there is an overall outward or inward flow. The differential equation (ie the divergence) does not provide this insight directly. To see what is happening one is advised to integrate over a volume and use the divergence theorem to relate the sum of the divergence in a volume to the amount of stuff passing through the boundaries of the volume. Clearly if there is a conservation of charge then whatever flows in must add to the charge already there.

The equation illustrates the basic strategy. There is some stuff and one needs to describe its movement. For conserved quantities a relationship like the one above must be satisfied.

Stuff:

- Energy U
_{em },*u*_{em }, U_{mech },*u*_{mech} - energy stored in the field U
_{em }and the energy density*u*_{em}of the field. - mechanical energy (kinetic energy of moving charges) U
_{mech }with density*u*_{mech} - Momentum density
*P*_{em}*,P*_{mech}

Flow:

- Energy Flow (field energy only) ,
- Momentum Flow (field momentum only)
- Note

- Now

We discover a relationship between the flow of energy and the amount of energy in a volume (8.14). We discover the relationship between the flow of momentum and the momentum in a volume (8.31). (See chapter for these equations.)

These equations show how the field energy and momentum move through a region and how they accumulate in a volume. Once the energy or momentum leave the field the equations no longer tracks the quantities. As particles leave a volume they take energy and momentum with them. For this chapter there is no interest in the flow of mechanical energy and mechanical momentum. We only need to know when it absorbs or adds to the field quantities.

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PASSWORD: maxwell

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