18-4. (18.2.1) The magnetic flux density at the center of a sunspot
The Zeeman effect observed in the spectra of sunspots reveals the
existence of magnetic fields as large as 0.4 tesla. These fields are associated
with pancake-shaped current distributions in the plasma near the surface.
In effect, one has a disk of electrons, with a radius of, say, 107 meters,
rotating at an angular velocity of the order of 3 x 10-2 radian/second. The
thickness of the disk is small compared to its radius.
(a) Calculate the surface density of electrons required to achieve a B of
0.4 tesla at the center.
(b) Calculate the current.
I can calculate the surface density
I use a small area of a disk
dA = rdrdY
take dY as the derivative of angle Y
since its a surface this would be a surface charge
dQ = sigma * dA
since the disk is rotating this charge must leave the area in a small time
dQ/dt = sigma * rdrdY/dt
now since Y is an angle
dY/dt = w the rotational speed
I = sigma * w* r * dr
we use this in the formula for B calculating B at the center of the disk
for a loop radius r
dBz = (u/(4*pi))* I * dl x r1/r^2
dl will always be 90 degs to r1
by definition of cross product
dl x r1 = | dl | sin pi/2 = | dl |
I solve this to 2.12 coulombs surface charge
my only problem is finding the current I cant figure out how to do that ... Probably something stupid that I just am overlooking...
the current is supposed to be something around 3x10^12 amperes