Electric Potential by Direct Integration: Uniformly Charge Sphere

Calculate the potential at an arbitrary point P inside a uniformly charged solid sphere of radius R and total charge q using direct integration of the Helmholtz integral (Equation (2)). Figure 1



QUESTIONS

(Score is number right minus number wrong.)

We lose no generality by putting the field point of interest on the z-axis at a position r < R.
True
False

Putting the pieces together...

The position of the FIELD point can be written as
Equation (6a)
Equation (6b)
Equation (6c)
Equation (6d)
The position of a SOURCE point can be written as
Equation (6a)
Equation (6b)
Equation (6c)
Equation (6d)
The vector "WAVY r" is given by
Equation (7a)
Equation (7b)
Equation (7c)

We could obtain the squared length of wavy r by taking the dot product of wavy r with itself. We could also simply use the Law of Cosines from trigonometry to get the distance between the source point and the field point. The LENGTH of wavy r is given by
Equation (8a)
Equation (8b)
Equation (8c)
Equation (8d)
A differential volume element for the integral is given by
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)

We now have the pieces to put together Equation (2).

The ANGULAR portion of the integration of Equation (2) can be put into the integral form of Equation (4) by making the substitution,
Equation (10a)
Equation (10b)
Equation (10c)
Equation (10d)
The right-hand-side of Equation (4) can be simplified to 2/z in this problem.
True
False
The right-hand-side of Equation (4) has two possible answers depending on whether z < r or z > r.
True
False
The essential problem in evaluating the right-hand-side of Equation (4) is that we must take the principal (positive) roots of the indicated radicals. That means,
Equation (11a)
Equation (11b)
Equation (11c)
Equation (11d)
In our case we get different answers depending on whether r' < r or r' > r. Hence, we must divide the radial integral (r') into two parts according to Equation (12).
True
False
The potential V inside the sphere is correctly given by Equation (13).
True
False
EQUATIONS

Equations


Equations


Equations









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