Calculating the Field from the Potential — Question 3

Question 3

Cylindrical angular variation requires the inverse-radius gradient factor

In cylindrical coordinates for s>0, $V(s,\phi,z)=A\ln(s/a)+B\cos\phi$. Use $\nabla V=\hat{\mathbf s}\partial_sV+\hat{\boldsymbol\phi}s^{-1}\partial_\phi V+\hat{\mathbf z}\partial_zV$.

1. Find the field components.

2. Find the circulation along a full circle of radius s and the voltage difference along a semicircle from angle 0 to $\pi$.

Question 3 - Solution

Cylindrical angular variation requires the inverse-radius gradient factor

(a) Method and calculation

$$E_s=-A/s,\quad E_\phi=B\sin\phi/s,\quad E_z=0.$$$$\mathbf E=-\frac A s\hat{\mathbf s}+\frac{B\sin\phi}s\hat{\boldsymbol\phi}.$$

(b) Interpretation and further calculation

$$\oint\mathbf E\cdot d\mathbf l=\int_0^{2\pi}B\sin\phi\,d\phi=0.$$$$V(\pi)-V(0)=-\int_0^\pi B\sin\phi\,d\phi=-2B.$$

Physical check

The physical arc element is s dphi, which cancels the inverse-radius factor in the tangential field.