Potential Due to a Group of Point Charges — Question 9

Question 9

An antisymmetric pair makes the perpendicular bisector a zero-work track

Charges +Q at x=a and -Q at x=-a have Q,a>0. A probe q follows the y axis from y=b to y=c, constrained by an ideal guide.

1. Find V(y) and the electric work along the guide.

2. Find the electric field along the y axis and the transverse guide force needed to keep the probe on it.

Question 9 - Solution

An antisymmetric pair makes the perpendicular bisector a zero-work track

(a) Method and calculation

$$V(0,y)=\frac{kQ}{\sqrt{a^2+y^2}}-\frac{kQ}{\sqrt{a^2+y^2}}=0.$$$$W_e=q[V(0,b)-V(0,c)]=0.$$

(b) Interpretation and further calculation

$$\mathbf E(0,y)=-\frac{2kQa}{(a^2+y^2)^{3/2}}\hat{\mathbf x}.$$$$\mathbf F_{\mathrm{guide}}=+\frac{2kQqa}{(a^2+y^2)^{3/2}}\hat{\mathbf x}.$$

Physical check

The guide force is transverse to the allowed velocity and does no work, while cancelling a generally nonzero electric force.