# Magnetic monopoles, electric field lines and equipotential surfaces

## Homework Statement

(i) Explain why it would not be possible to write the magnetic Field (B-field) in terms of a vector potential (A) IF magnetic monopoles existed.

(ii) For an electrostatic field (E-field), define the electrostatic potential (Fi), and explain CONCISELY what is meant by a field line and an equipotential surface.

(iii) At what angle do field lines and equipotential surfaces generally intersect? Briefly explain your answer.

Cheers guys!

## The Attempt at a Solution

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for (ii)

$\phi(\vec{r})=\frac{1}{4 \pi \epsilon_0} \int_{V'} \frac{\rho(\vec{r'})}{|\vec{r}-\vec{r'}|} dV'$

a field line indicates the direction in which a small positive test charge would move it it were placed in the field.
an equipotental surface is just as it sounds - a surface on which the potential is equal at all points. can you prove that?

consider a conductor surface with normal $\mathbf{\hat{n}}$, you know $\vec{E}=-\nabla \phi(\vec{r})$, so $\nabla \phi(\vec{r})$ points in what direction...., and $\vec{dr}$ on the equipotential surface points in what direction?
now consider $d \phi$ - why would it be 0 if you are on the surface?

actually i just realised that the above argument can be used to answer (iii) as well , just start by assuming $d \phi=0$ this time