Radiation field of charge moving in a circular loop

[MCB277]

Homework Statement

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I need to calculate de radiation field produced by a charge moving in a circular loop of radius a, considering the motion is not relativistic and zone of radiation (far field) .

Homework Equations

The scalar potential is:

$$ V=\int \dfrac{\rho(x')}{|x-x'|}d^3x'$$

The potential vector is:

$$ V=\int \dfrac{J(x')}{|x-x'|}d^3x'$$

The Attempt at a Solution

For this case I express the density of charge that $$\rho(x')=\dfrac{q}{2\pi a} \delta(x'-a\cos (wt))\delta(y'-a\cos (wt))\delta(z)$$ and $$J(x')=q (-a\sin(wt)\hat{i}+a\cos(wt)\hat{j})\delta(r'-a)\delta(\theta-\pi/2)$$.

Where t is the retarded time.

This expression for density of current J and charge density $\rho$ are correct?.

Thanks

 

[mark726]

for your help.Hello,

Yes, your expressions for the density of current and charge are correct for a charge moving in a circular loop of radius a. The delta functions in the expressions account for the localized nature of the charge and current distribution.

To calculate the radiation field, you can use the equations you have provided for the scalar potential and potential vector. However, you will need to integrate over the entire loop to take into account the motion of the charge. You can use the velocity of the charge to determine the retarded time t in your expressions.

Additionally, you will need to consider the direction of the radiation field, which will depend on the orientation of the loop and the direction of motion of the charge. You can use the right-hand rule to determine the direction of the radiation field.

I hope this helps. Good luck with your calculations!