Power radiated by point charge Calculus

[HPRF]

It is found that Poyntings vector gives

P = ExH = (mu₀q²a²sin²(theta)/6pi²cr²)r

This apparently leads to

Total Power = (mu₀q²a²/6pi²c)$$\int$$(sin²(theta)/r²)(2pir²sin(theta)d$$\theta$$)

What I am unsure of is where the

(2pir²sin(theta)d$$\theta$$)

appears from. Can anyone help?

 

[caleb20k]

actually the true equation is that S=1/2Re{ExH*} that's the real part of ExH* and H* is the conjugate of H.

this leads to Pwr = Int[S.dA]. this is the eqn for power passing thru a surface of area dA (assumed to be spherical). in it we integrated over the surface area of the sphere because the poynting vector is defined as the power per area. So if we multiply S by the area we obtain the power

where dA is r-squared times the solid angle == r^2*d(Omega). Expressed another way, dA=r^2*sin(theta)d(theta)d(phi) the r^2 sin(theta) comes out of the jacobian because you are changing from cartesian coordinates to spherical coordinates.

so that makes the Power=Int[1/2*Re{ExH*} * (r^2*sin(theta)d(theta)d(phi)),

here theta is integrated over 0 to Pi, Phi is integrated from 0 to 2*Pi

upon integrating you get the 2*Pi from the Phi integral.
which leaves you with the d(theta)*r^2*sin(theta)

hope that the response isn't too late.

 

[caleb20k]

in all truth you don't need that 1/2 on the Poynting vector S. Just a bad habit of mine from dealing with equations using the Poynting vector in E&M. So omit that 1/2 from what I stated above

 

[caleb20k]

sorry, had a brain fart when I posted that. I meant keep the 1/2 and dismiss the Real part. that is only when calculating the power/solid angle.

S = 1/2 (ExH*)