E&M II Griffiths Chapter 8

Energy conservation gives us Poynting’s theorem The electric field energy contained in a finite volume of space is

Rate of work done on the charge q in dt is

Next we’ll get rid of these ugly things

Vector equation Scalar Equation

But some of that field momentum might be not be available to increase the momentum of the charges, if the field momentum flows out through the boundary first. We have to subtract that part. We subtract the negative of this, which is the amount that flows out.

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(Momentum per unit area per unit time)

(These quantities are all per unit volume, i.e. densities)

Example 8.4 Charged cylinder radius a length l inside solenoid

In the solenoid. B goes to zero. An E-field is induced. Field momentum goes away, but total momentum has to be conserved. Outside solenoid Inside solenoid

The induced E-field acts to produce a torque on the charged cylinders. On the outer cylinder The outer cylinder acquires mechanical angular momentum due to this torque

+ = Lem Angular momentum is conserved. Field momentum is fully converted to mechanical momentum.