Calculating the Potential from the Field — Question 7

Question 7

A rotational field fails the rectangle test for an electrostatic potential

A proposed static electric field is $\mathbf E=-\gamma y\hat{\mathbf x}+\gamma x\hat{\mathbf y}$, with $\gamma\ne0$. Compare paths from (0,0) to (a,b), where a,b>0.

1. Integrate first along x then y, and then along y followed by x.

2. Find the circulation around the counterclockwise rectangle and decide whether a single-valued electrostatic potential exists.

Question 7 - Solution

A rotational field fails the rectangle test for an electrostatic potential

(a) Method and calculation

$$I_{xy}=0+\int_0^b\gamma a\,dy=\gamma ab.$$$$I_{yx}=0+\int_0^a(-\gamma b)\,dx=-\gamma ab.$$

(b) Interpretation and further calculation

$$\oint\mathbf E\cdot d\mathbf l=I_{xy}-I_{yx}=2\gamma ab\ne0.$$$$\Delta V_{xy}=-\gamma ab\ne+\gamma ab=\Delta V_{yx}.$$No single-valued electrostatic potential can generate this field on the rectangle.

Physical check

A field driven by changing magnetic flux may have circulation, but that is outside an electrostatic scalar-potential description.