1. Applied Electricity and Magnetism
ECE 3318
Applied Electricity and Magnetism
Spring 2017
Prof. David R. Jackson
ECE Dept.
Notes 15

2. Potential From Charge
This is a method for calculating the potential function directly, without having to calculate the electric field first.
This is often the easiest way to find the potential function (especially when you don’t already have the electric field calculated). There are no vector calculations involved.
The method assumes that the potential is zero at infinity.
(If this is not so, you must remember to add a constant to the solution.)

3. Potential From Charge (cont.)
Point charge formula:
From the point charge formula:
Integrating, we obtain the following result:

4. Potential From Charge (cont.)
Summary for potential-charge formulas for
All possible types of charge densities:
Note that the potential is zero at infinity (R  )
in all cases.

5. Circular ring of line charge
Example
Circular ring of line charge
Find (0, 0, z)

6. Example (cont.)
Note: The upper limit must be larger than the lower limit, to keep dl positive.

7. Example (cont.)
Summary

8. Solid cube of uniform charge density
Example
Solid cube of uniform charge density
Find (0, 0, z)

9. Example (cont.)
The integral can be evaluated numerically.

10. Example (cont.)
 [V]
z [m]
Result from Mathcad

11. Example (cont.)  [V] z [m] Result from Mathcad 0.5 1.0 1.5 2.0
 [V]
Face of cube
Result from Mathcad

12. Limitation of Potential-Charge Formula
This method always works for a bounded charge density; that is, one that may be completely enclosed by a volume.
For a charge density that extends to infinity, the method might fail because it may not be possible to have zero volts at infinity.
The method will always fail for 2D problems.

13. Example of Limitation (cont.)
Here the potential integral formula fails.
The integral does not converge!
Infinite line charge

14. Example of Limitation (cont.)
The field-integration method still works:
(From Notes 14)
Note:
We can still use the potential integral method if we assume a finite length of line charge first, and then let the length tend to infinity after solving the problem.
(This will be a homework problem.) x
Infinite line charge