Scalar and vector potentials and magnetic monopoles

[peterjaybee]

Homework Statement

Write down expressions oer E and B fields in terms of [tex]\varphi[/tex] and [tex]\bar{A}[/tex]. Demonstrate that this definition of B is consistent with the non-existence of magnetic monopoles.

Homework Equations

The Attempt at a Solution

The first part of the question is easy. i.e. bookwork
[tex]B = \nabla \times \bar{A}[/tex]
[tex]E = \nabla\varphi - \frac{\partial{A}}{\partial t}\[/tex]

I don't know how to approach the second part of the question though

 

[tiny-tim]

Write down expressions oer E and B fields in terms of [tex]\varphi[/tex] and [tex]\bar{A}[/tex]. Demonstrate that this definition of B is consistent with the non-existence of magnetic monopoles.

Hi peterjaybee!

Hint: there is electric charge, but (if there are no magnetic monopoles) there's no magnetic charge (ie, you can't have a particle or an object with overall magnetic charge) … so how does electric charge come out of the equations in a way that magnetic charge can't?

 

[Cyosis]

I believe you're missing a minus sign for the E-field. Grab your book and go to the page where the Maxwell equations are listed. Can you identify the equation that has electric charge as a source term?

 

[peterjaybee]

thanks, yes you are right the minus sign should be there.

With regards to the second part, i think I have it thanks to your hints. Div B = 0 is the maxwell eqn that shows there are no magnetic monopoles (as there are no source terms - unlike for div E). The definition of the vector potential is consistent with this because the divergence of a curl is always zero.

 

[Cyosis]

Yes that is correct.