# Magnetic field above rotating disk

• theneedtoknow
In summary, the question asks for the magnetic field at a distance z above a rotating disk with surface charge density inversely proportional to the square of the distance from the centre of the disk. The solution involves using the equations K = v*surface charge density and B = (constant)*integral of (K x rhat / r^2 )da, where K is the product of the angular velocity (w) and the surface charge density (c/s^2). To set up the integral, the horizontal components cancel and only the vertical components need to be integrated. If the surface charge density was proportional to the distance from the centre instead, the integral would be more complicated.

## Homework Statement

What is the magnetic field a distance z above a rotating disk (ang.velocity w) with surface charge density inversely proportional to the square of the distance from the centre of the disk.

## Homework Equations

K = v*surface charge density
B = (constant)*integral of (K x rhat / r^2 )da

## The Attempt at a Solution

I just want to know if I'm setting up this question right (basically, if I'm setting up the part in the integral correct, which is why i ommited the constant)

charge desntiy = c/s^2 where c is a constant (since it is inversely proportional to square of distance from centre of disk)
so K = v*charge density = ws*c/s^2 = wc/s
so K cross rhat = Ksin(alpha) where alpha is the angle between k and r
but in this case, the angle will always be 90 degrees since k and r are in perpendicular planes
so i need to inegrate (K/r^2)da
but all the horizontal components will cancel so i need to take only the vertical components, so i multiply by cos theta where theta is the angle r makes with the disk

so i have :
integral of (Kcostheta/r^2) da
costheta = s/(s^2+z^2)^0.5
K = wc/s
r^2 = s^2 + z^2

so i get
ingeral of wc/s * s/(s^2+z^2)^3/2 * da
da = sdsdtheta
so I have to integrate (wcs/(s^2+z^2)^3/2)dsdtheta from s=0 to s=R and theta= 0 to theta=2pi

is this correct?

Also, what about if the surface charge density was proportional to the distance from the centre instead?
then I would get an integral of the type s^4/(s^2+z^2)^3/2

How in the world would i do this integral? The one above is much easier but I can't figure out a way to do this one