Chapter 14.1, Problem 49E

Electric field due to a line of charge The electric field in the xy-plane due to an infinite line of charge along the z-axis is a gradient field with a potential function $V\left(x,y\right)=c\ln \left(\frac{r_0}{\sqrt{x^2+y^2}}\right).$ where c > 0 is a constant and r₀ is a reference distance at which the potential is assumed to be 0 (see figure).

a.    Find the components of the electric field in the x-and y directions, where $E\left(x,y\right)=-\nabla V\left(x,y\right)$.

b.    Show that the electric field at a point in the xy-plane is directed outward from the origin and has magnitude $\left|E\right|=c/r,$.where $r=\sqrt{x^2+y^2}.$

c.    Show that the vector field is orthogonal to the equipotential curves at all points in the domain of V.