Find field yielding the potential ln(1/r)

[timn]

Dear Physics Forums,

Homework Statement

Given the potential

$$\Phi(\bar{r}) = \Phi_0\ln{\left(\frac{r_0}{r}\right)}$$

calculate the electric field strength.

Homework Equations

$$- \nabla \Phi = \bar{E}$$

The Attempt at a Solution

Due to the symmetry of the problem, consider the position vector $$\bar{r}$$ as a vector in spherical coordinates.

$$\bar{E}(\bar{r}) = -\nabla \Phi(\bar{r})$$

The potential depends only on the radial component, so the other terms of the gradient vanish.

$$\bar{E}(\bar{r}) = - \Phi_0 \frac{\partial}{\partial r} \ln{\frac{r_0}{r}} \hat{r} = \Phi_0 \frac{1}{r} \hat{r}$$

where $$\hat{r}$$ denotes the normalised radial base vector.

The answer is supposed to be $$\Phi_0 \frac{1}{r^2} \hat{r}$$, with a square in the denominator. What did I do wrong?

 

[ehild]

"rhat" in the answer means the position vector, independent of the coordinate system.

ehild

 

[timn]

Aha! Thank you, ehild.

Sorry about copying the answer wrong.