Induced Electric Fields — Question 9

Question 9

A closed path around a flux region has nonzero electric work without a conducting wire

A prescribed changing flux has constant derivative F through a fixed closed contour C. A test charge q is carried once around C while F stays constant. No wire is present, and the test charge does not perturb the source.

1. Find electric work for one and for n signed traversals.

2. Explain why this does not contradict the zero work of a magnetic force on a moving charge.

Question 9 - Solution

A closed path around a flux region has nonzero electric work without a conducting wire

(a) Method and calculation

$$\oint_C\vec E\cdot d\vec\ell=-F\Rightarrow\boxed{W_1=-qF}.$$$$\boxed{W_n=-nqF}\quad\text{for }n\text{ traversals in the same sense}.$$

(b) Interpretation and further calculation

$$P_B=q\vec v\cdot(\vec v\times\vec B)=0.$$$$\boxed{P_E=q\vec E\cdot\vec v\text{ may be nonzero}}.$$

Physical check

The changing-field source supplies or absorbs the energy. Faraday circulation exists in space even without a conducting loop.