Short Aerial: Find the scalar potential (retarded sources)

In summary, the conversation discussed determining the vector potential and scalar potential due to a short wire carrying a current. The vector potential was found using the fact that the distance from the wire to the observation point is much larger than the length of the wire. The scalar potential was not explicitly given, but it was suggested that it could be derived from the vector potential using a gauge condition. The issue of determining ρ was also mentioned, but it was not relevant to the problem at hand.
  • #1
Poirot
94
2

Homework Statement


Determine the vector potential due to a short wire running from (−L/2, 0, 0) to (L/2, 0, 0) carrying a current I = I0cos(ωt) (consider only points at distance r >> L from the origin). (You may neglect the effect of the return circuit.) Now determine the corresponding scalar potential for large r.

Homework Equations


V(x, t) = 1/4πε0 ∫d3x' ρ(x' , t - |x' - x|/c) / |x' - x|

A
(x, t) = μ0/4π ∫d3x' j (x' , t - |x' - x|/c) / |x' - x|

Where A is the vector potential and V is the scalar potential.

The Attempt at a Solution


I managed to solve the first part for the vector potential by using the fact that r>>L and therefore |x' - x| ≈ r
and using that j = I/c x(hat) (in the x direction, and I let C= the cross sectional area of the wire)
this gave A(x, t) = (μ0L I0 cos(ω(t-r/c)))/4πr

When trying to determine the vector potential I ran into an issue, I'm not entirely sure what ρ is explicitly. I had a guess that is was ρ = Q/LC, and then using the fact that I=dQ/dt and integrating to find Q, but this was a stab in the dark and I have no idea if it's right.

Any help would be greatly appreciated, thank you!
 
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  • #2
The problem seems to suggest that you can derive V from A. There is a gauge condition for the potentials that must be satisfied in order for your relevant equations to be valid. This condition can be used to relate V to A.
 
  • #3
TSny said:
The problem seems to suggest that you can derive V from A. There is a gauge condition for the potentials that must be satisfied in order for your relevant equations to be valid. This condition can be used to relate V to A.
That seems to make much more sense since ρ was not specified. Thank you!
 
  • #4
Poirot said:
When trying to determine the vector potential I ran into an issue, I'm not entirely sure what ρ is explicitly. I had a guess that is was ρ = Q/LC, and then using the fact that I=dQ/dt and integrating to find Q, but this was a stab in the dark and I have no idea if it's right.
You didn't relate ρ to your equation for A so I don't know what you mean by it. It's usually used to mean charge density, not applicable here.

The vector potential A is
A = (1/4π)(j/r)dV
where j is current density (a vector),
dV is a differential of volume within the volume of current, and
r is distance from dV to the point of observation (not a vector).
(EDIT:
note I use A in the sense of H = ∇ x A).
In this case I assume r >> L also implies r >> R, the radius of the wire.

So I suggest orienting your short wire in say the z direction, middle at origin, then doing the simple integration over the length of the wire of j/r.

The problem suggests you compute A first, then V. Actually, the magnetic scalar potential is of academic interest only since, when current flows, the mag potential cannot be used to compute B or H since ∇ x H is not then zero (Maxwell: ∇ x H = j). So I don't know why you were called upon to compute it to begin with.
 

1. What is the concept of "retarded sources" in the context of finding the scalar potential in a short aerial?

In electromagnetism, "retarded sources" refer to the idea that the effects of a source (such as a charged particle) on the electric potential at a certain point in space are delayed by the time it takes for the electric field to propagate from the source to that point. This is important to consider when calculating the scalar potential in a short aerial, as the potential at a point is affected not only by the present position of the source, but also its previous positions.

2. How is the scalar potential calculated in a short aerial with retarded sources?

The scalar potential is calculated using the formula V(r) = (1/4πε0) ∫ρ(r')/|r-r'| dτ, where ρ(r') represents the charge distribution of the source and r and r' represent the position vectors of the point at which the potential is being calculated and the source, respectively. The integral takes into account the delayed effects of the source on the potential at the point.

3. Can the scalar potential be negative in a short aerial with retarded sources?

Yes, the scalar potential can be negative in this context. The potential at a point is dependent on the distance and position of the source, and can be positive or negative depending on these factors. It is important to note that the sign of the potential does not affect the physical behavior of the system, as it is the electric field that determines the motion of charged particles.

4. How does the presence of multiple sources affect the calculation of the scalar potential in a short aerial?

When there are multiple sources present, the scalar potential is calculated by summing the contributions from each individual source using the formula V(r) = (1/4πε0) ∑∫ρ(r')/|r-r'| dτ. This takes into account the delayed effects of each source on the potential at the point, and allows for the overall potential to be calculated.

5. How is the scalar potential related to the electric field in a short aerial with retarded sources?

The electric field can be calculated from the scalar potential using the formula E(r) = -∇V(r), where ∇ represents the gradient operator. This means that the electric field is directly proportional to the gradient of the scalar potential, and the direction of the field is opposite to the direction of the gradient. Therefore, the scalar potential can be used to determine the behavior of the electric field in a short aerial with retarded sources.

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