Potential Due to an Electric Dipole — Question 5

Question 5

An equatorial zero voltage retains a nonzero transverse derivative

An ideal dipole points along positive z. Its potential is $V(x,y,z)=kpz/(x^2+y^2+z^2)^{3/2}$. Consider P=(R,0,0), R>0.

1. Evaluate V at P and calculate $E_z=-\partial V/\partial z$ there.

2. Find the other field components and compare this result with the axial field magnitude at the same radius.

Question 5 - Solution

An equatorial zero voltage retains a nonzero transverse derivative

(a) Method and calculation

$$V_P=0,\quad\frac{\partial V}{\partial z}=kp\left(r^{-3}-3z^2r^{-5}\right).$$$$E_z(P)=-kp/R^3.$$

(b) Interpretation and further calculation

$$E_x=3kpxz/r^5=0,\quad E_y=3kpyz/r^5=0\quad\text{at P}.$$$$E_{\mathrm{axis}}=2kp/R^3=2|\mathbf E_P|.$$

Physical check

A function can vanish throughout a plane while varying in the direction normal to that plane.