Electric Field Lines — Question 6

Question 6

A closed oriented field line is incompatible with electrostatic work conservation

A proposed electrostatic field line forms a smooth closed loop of length $\ell$, and the field is nonzero everywhere on it. Its arrows follow the loop traversal. Use the electrostatic property $\oint\mathbf E\cdot d\mathbf l=0$.

1. Use the field-line tangent condition to test the proposal.

2. For the special candidate of constant tangent field magnitude $E_0$ on a circle of radius R, calculate the loop integral and work on a charge q externally guided once around it.

Question 6 - Solution

A closed oriented field line is incompatible with electrostatic work conservation

(a) Method and calculation

$$d\mathbf l=\hat{\mathbf t}\,ds,\quad\mathbf E=|\mathbf E|\hat{\mathbf t}.$$$$\oint\mathbf E\cdot d\mathbf l=\oint|\mathbf E|\,ds>0,\quad\text{contradicting electrostatics}.$$

(b) Interpretation and further calculation

$$\oint\mathbf E\cdot d\mathbf l=2\pi R E_0,\quad W=q\oint\mathbf E\cdot d\mathbf l=2\pi RqE_0.$$$$E_0>0,R>0\Rightarrow2\pi R E_0\ne0.$$Such a circulation may belong to a non-electrostatic induced field, but it cannot describe the assumed static-charge field.

Physical check

The no-closed-line result here explicitly uses the electrostatic circulation condition.