Magnetic monopoles, electric field lines and equipotential surfaces

In summary, the first question addresses the relationship between the magnetic field and vector potential in the presence of magnetic monopoles. The second question defines the electrostatic potential and explains the concepts of field lines and equipotential surfaces in an electrostatic field. Finally, the third question asks about the general intersection angle between field lines and equipotential surfaces. This can be determined by considering the direction of \nabla \phi and \vec{dr} on the surface and assuming d \phi=0.
  • #1
wam_mi
81
1

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

 
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  • #2
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 realized that the above argument can be used to answer (iii) as well , just start by assuming [itex]d \phi=0[/itex] this time
 

1. What are magnetic monopoles?

Magnetic monopoles are hypothetical particles that have a single magnetic pole, either a north or south pole, unlike regular magnets that have both poles. They are predicted by some theories of physics, but have not yet been observed in experiments.

2. How do electric field lines work?

Electric field lines are used to represent the direction and strength of an electric field. They point in the direction that a positive test charge would move, and the closer the lines are together, the stronger the electric field is.

3. What is the difference between magnetic monopoles and electric field lines?

Magnetic monopoles are hypothetical particles, while electric field lines are a visual representation of an electric field. Magnetic monopoles have a single pole, while electric field lines have both positive and negative charges represented by the direction of the lines.

4. How do equipotential surfaces relate to electric fields?

Equipotential surfaces are imaginary surfaces that connect points with the same electric potential. In an electric field, they are always perpendicular to the electric field lines. The closer the equipotential surfaces are together, the stronger the electric field is in that area.

5. Can magnetic monopoles exist in nature?

While there is no concrete evidence for the existence of magnetic monopoles, some theories suggest that they may exist in certain conditions, such as in the early universe or in high-energy particle collisions. However, their existence has not been confirmed by experiments.

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