Lienard-Wiechert potential for 2 discrete Qs

[jmc8197]

I've tried a simple derivation of the Lienard-Wiecher potential for 2 discrete charges separated by dr, but end up with a result which isn't the same as the theoretically correct version:

Take two charges q1 @r1, q2 @r2 with r1 > r2, dr = r1 - r2, parallel
and both traveling at velocity v away from and along r1, r2. The
observation point is at r = 0. The potential of q1 that arrives at q2
is from a retarded time t' = dr/(v + c). This is when q1 was at r2 +
dr - vt' = r2 + dr - v dr/(v + c) = r2 + dr/(1 + B) where B = v/c. So
the potential at the observation point can be replaced by an
equivalent static q1 and q2 separated by dr/(1 + B), equivalent to
increasing the charge density by (1 + B) at r.

The total potential is then q2/r2 + q1/(r2 + dr/(1 + B)) which for
small dr and q1 = q2 = q can be written using the first two terms of a taylor expansion:


q/r2 + q/r2 (1 - dr/(1 + B)r2)


= 2q/r2 - q dr/(1 + B)r2^2 )


I'm not sure how to proceed further, since I was hoping to end up with
an expression 2q/(1 + B) r2. Can anyone see where I've gone wrong?

 

[Jhero]

Thank you for sharing your derivation of the Lienard-Wiechert potential. It seems like you are on the right track, but there are a few errors in your calculations.

Firstly, in your expression for the potential at the observation point, you have used the charge density of q1 to be (1 + B) at r, which is incorrect. The correct charge density at r would be q1, as q1 is the only charge present at r.

Secondly, in your expression for the equivalent static charge, you have used the distance between q1 and q2 to be dr/(1 + B). This is incorrect as the distance between q1 and q2 would be dr, as they are separated by dr in the direction of their velocities.

Finally, in your final expression, you have used the charge density of q1 to be q, which is incorrect. The charge density of q1 would be q at r2, but at r2 + dr, the charge density would be q(1 - dr/(1 + B)r2).

Correcting these errors, the final expression for the potential at the observation point would be:

q/r2 + q(1 - dr/(1 + B)r2)/(r2 + dr)

= q/r2 + q(1 - dr/(1 + B)r2)/(r2(1 + dr/r2))

= q/r2 + q(1 - dr/(1 + B)r2)/(r2 + dr)

= q/r2 + q/(r2 + dr) - q dr/(1 + B)r2(r2 + dr)

= q/r2 + q/(r2 + dr) - q dr/(1 + B)r2^2 - q dr/(1 + B)r2dr

= q/r2 + q/(r2 + dr) - q dr/(1 + B)r2^2 - q(1 - B)dr/r2

= q(1 + B)/r2 + q/(r2 + dr) - q dr/r2^2 - q(1 - B)dr/r2

= q(1 + B)/r2 + q/(r2 + dr) - q(1 + B)dr/r2^2

= q(1 + B)/r2 + q(1 + B)/(r2 + dr) - q(1 + B)dr/r2^