Question

The electric potential in some region in space is given by V (r) = C/r where r = p x 2 + y 2 + z 2 and C is a positive constant. (a) calculate an expression for the electric field in this region. (b) note that ~r = xˆi + ˆj + z ˆk and ~r = rrˆ, express E~ in spherical coordinates. Here ˆr is the radial unit vector of spherical coordinates, sometimes denoted as ˆer. (c) Look and report the expression for the gradient operator in spherical coordinates. Re-derive the expression for E~ , now directly in spherical coordinates

Answer 1

$\vec r = x\vec i + y\vec j + z\vec k$

$\frac{\partial}{\partial x}(\frac 1 r) = -\frac{1}{r^2}\frac{x\vec i}{r}$

From symmetric nature of radius formula:

$\vec E = -\frac{C}{r^2}\frac{\vec r}{r}$