Vector potential and and Intensity of Radiation

[sid0123]

Homework Statement

Let a charge oscillate on a straight line between -a to +a with a frequency ω and according to the law:
κ (x.t) = κ° sin(πx/a) e^(-iωt)
I have to find the following:
1. Vector potential in the dipole approximation
2. Integral of the intensity of radiation

Homework Equations

The Attempt at a Solution

First of all, I tried to calculate the dipole using:
d = ∫ κ (x.t) x dx

After that I calculated vector potential using:
A = 1/(cR) × d(d)/dt
But, I worry that my formula is not correct and I may have used the incorrect formula.

Then I used the formula for calculating the intensity of radiation:

I = (2(second time derivative of dipole)^2) / ( 3c^2)

Please guide me! I have picked up this subject after a long time and things have just jumbled up inside my head.

 

[anathema]

Thank you for sharing your attempt at solving this problem. Here are a few comments and suggestions that may help guide you in the right direction:

1. The formula you used for calculating the dipole moment is correct, but make sure you are integrating over the entire oscillation range (-a to +a) and not just a single point.
2. The formula for the vector potential in the dipole approximation is slightly different from what you have used. It is given by A = (1/c^2R) x (d^2/dt^2) d, where d is the dipole moment. Make sure you are using the correct formula.
3. When calculating the intensity of radiation, make sure you are using the correct formula for the time derivative of the dipole moment. It should be d^2/dt^2, not d/dt.
4. Additionally, you may need to consider the direction of the dipole moment when calculating the intensity of radiation. This can be done by taking the dot product of the dipole moment with the unit vector in the direction of propagation.
5. Finally, make sure you are using the correct units in your calculations and that all the variables are consistent with each other.

I hope this helps. Good luck with your problem!