Electrostatic Energy: Uniformly Charged Sphere
What is the energy of assembly of a uniformly charged sphere of radius R and total charge q?
QUESTIONS
(Score is number right minus number wrong.)
A good choice for setting V=0 in this case is at an infinite radius.
True
False
In order to calculate the energy of assembly W for this problem, we just need to know the charge distribution (given) and the potential V inside the sphere.
True
False
We can obtain the potential V from Equation (2) if we know the electric field E everywhere.
True
False
The magnitude of the electric field OUTSIDE the sphere can easily be obtained from the integral form of Gauss' Law (Equation (1)),
Equation (5a)
Equation (5b)
Equation (5c)
Equation (5d)
The magnitude of the electric field INSIDE the sphere can easily be obtained from the integral form of Gauss' Law (Equation (1)),
Equation (5a)
Equation (5b)
Equation (5c)
Equation (5d)
The electric field OUTSIDE the sphere is given correctly by,
Equation (6a)
Equation (6b)
Equation (6c)
Equation (6d)
The electric field INSIDE the sphere is given correctly by,
Equation (6a)
Equation (6b)
Equation (6c)
Equation (6d)
So, we can now use Equation (2) to get the potential V. You will probably need to do some calculation for that.
The potential V OUTSIDE the sphere is,
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
The potential V INSIDE the sphere is,
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (3) and Equation (4) are all equivalent expressions for calculating the assembly energy of the sphere of charge.
True
False
In Equation (4), the surface integral must be taken at the surface of the charge distribution.
True
False
If the surface is taken where r becomes infinite, the surface integral vanishes.
True
False
The integrals of Equation (3a) and Equation (3b) are taken over the volume actually occupied by charge.
True
False
If the surface integral of Equation (4) is taken at r=a, it will vanish for a spherically symmetric distribution of charge.
True
False
Now put some pieces together to see how everything fits together. That requires some more calculation.
The value of W given by Equation (3a) is
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
The value of Equation (8b) taken over the interior of the sphere only (r < R) is,
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
The value of Equation (8b) taken over the exterior of the sphere only (r > R) is,
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
The value of Equation (8b) taken over the interior of the sphere plus the exterior of the sphere is,
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
The value of Equation (8b) taken over the region from r = R to r = a only is,
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
The value of Equation (8c) taken over the surface at r = a sphere is,
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
The value of Equation (8b) taken over the interior of the sphere at r = a plus the surface integral at r = a is,
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
The radius of the surface at r = a does not appear in the final value of W calculated by Equation (4).
True
False
EQUATIONS
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