- #1
psholtz
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I have three concentric metal, conducting spheres, of radii a < x < b.
The spheres a and b are fixed, while x can vary over the range from a to b.
Spheres a and b are also grounded, while sphere x is maintained at a constant potential V with respect to these two spheres.
The charge on sphere x will distribute itself partly on the "inner" surface of the sphere (that facing a) and partly on the "outer" surface of the sphere (that facing b). I would like to calculate what fraction of charge distributes itself on the inside and outside surfaces of sphere x as a function of x (i.e., as x varies from a to b).
Some charge will be induced on sphere a. Denote the surface density of this charge as [tex]\sigma_a[/tex]. We have:
[tex]Q_a = 4\pi a^2 \sigma_a[/tex]
and in the region between a and x, the electric field is given by:
[tex]E = \frac{4\pi a^2\sigma_a}{r^2}[/tex]
while the potential is given by:
[tex]\phi = -\frac{4\pi a^2 \sigma_a}{r}[/tex]
From which we get that the potential difference between the two spheres is:
[tex]\frac{4\pi a^2 \sigma_a (x-a)}{xa} = \frac{4\pi a \sigma_a (x-a)}{x} = V[/tex]
Since we're holding the potential difference V constant, and anticipating that the surface charge density changes as a function of time, we can express this (perhaps more clearly) as:
[tex]V = \frac{4\pi a (x-a)\sigma_a(x) }{x}[/tex]
Solving for the surface charge density on the inner sphere as a function of x and V, we have:
[tex]\sigma_a(x) = \frac{Vx}{4\pi a (x-a)}[/tex]
There are other components to this problem, and other reasons I have for wanting to solve this particular problem, but even here the problem that I'm running into raises its head: that is, if we look at the surface charge density when x=a, the answer diverges. In other words, [tex]\sigma_a(a) \rightarrow \infty[/tex].
This doesn't seem correct to me.. rather it seems that the surface charge density on the inner sphere should remain finite, in such a way [tex]4\pi a^2 \sigma_a(a) = Q[/tex], where Q is the total charge on sphere x.
Is there something wrong in my reasoning here?
The spheres a and b are fixed, while x can vary over the range from a to b.
Spheres a and b are also grounded, while sphere x is maintained at a constant potential V with respect to these two spheres.
The charge on sphere x will distribute itself partly on the "inner" surface of the sphere (that facing a) and partly on the "outer" surface of the sphere (that facing b). I would like to calculate what fraction of charge distributes itself on the inside and outside surfaces of sphere x as a function of x (i.e., as x varies from a to b).
Some charge will be induced on sphere a. Denote the surface density of this charge as [tex]\sigma_a[/tex]. We have:
[tex]Q_a = 4\pi a^2 \sigma_a[/tex]
and in the region between a and x, the electric field is given by:
[tex]E = \frac{4\pi a^2\sigma_a}{r^2}[/tex]
while the potential is given by:
[tex]\phi = -\frac{4\pi a^2 \sigma_a}{r}[/tex]
From which we get that the potential difference between the two spheres is:
[tex]\frac{4\pi a^2 \sigma_a (x-a)}{xa} = \frac{4\pi a \sigma_a (x-a)}{x} = V[/tex]
Since we're holding the potential difference V constant, and anticipating that the surface charge density changes as a function of time, we can express this (perhaps more clearly) as:
[tex]V = \frac{4\pi a (x-a)\sigma_a(x) }{x}[/tex]
Solving for the surface charge density on the inner sphere as a function of x and V, we have:
[tex]\sigma_a(x) = \frac{Vx}{4\pi a (x-a)}[/tex]
There are other components to this problem, and other reasons I have for wanting to solve this particular problem, but even here the problem that I'm running into raises its head: that is, if we look at the surface charge density when x=a, the answer diverges. In other words, [tex]\sigma_a(a) \rightarrow \infty[/tex].
This doesn't seem correct to me.. rather it seems that the surface charge density on the inner sphere should remain finite, in such a way [tex]4\pi a^2 \sigma_a(a) = Q[/tex], where Q is the total charge on sphere x.
Is there something wrong in my reasoning here?