Chapter 17.1, Problem 54E

Electric field due to a point charge The electric field in the xy-plane due to a point charge at (0,0) is a gradient field with a potential function $V\left(x,y\right)=\frac{k}{\sqrt{x^2+y^2}}$ where k > 0 is a physical constant.

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

b.    Show that the vectors of the electric field point in the radial direction (outward from the origin) and the radial component of E can be expressed as E_r = k/r², where $\sqrt{x^2+y^2}.$

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