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

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.)

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

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.

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

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

Example (cont.) Summary

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

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

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

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

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.

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

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