Magnetic monopoles, electric field lines and equipotential surfaces

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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.


Please help me with the three questions at above.
Cheers guys!

Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
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for (ii)

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

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 [itex]\mathbf{\hat{n}}[/itex], you know [itex]\vec{E}=-\nabla \phi(\vec{r})[/itex], so [itex]\nabla \phi(\vec{r})[/itex] points in what direction...., and [itex]\vec{dr}[/itex] on the equipotential surface points in what direction?
now consider [itex]d \phi[/itex] - 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 [itex]d \phi=0[/itex] this time
 

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