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Prof. David R. Jackson ECE Dept.

ECE 2317 Applied Electricity and Magnetism. Spring 2014. Prof. David R. Jackson ECE Dept. Notes 15. Potential Integral Formula. This is a method for calculating the potential function directly, without having to calculate the electric field first.

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Prof. David R. Jackson ECE Dept.

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  1. ECE 2317 Applied Electricity and Magnetism Spring 2014 Prof. David R. Jackson ECE Dept. Notes 15

  2. Potential Integral Formula 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 Integral Formula (cont.) Point charge formula: r(x, y, z) z R From the point charge formula: y x Integrating, we obtain the following result:

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

  5. Example z r = (0, 0, z) R a y x l0[C/m] Find (0, 0, z) Circular ring of line charge Note: The upper limit must be larger than the lower limit, to keep dl positive.

  6. Example (cont.) For (This agrees with the point charge formula.)

  7. Example z Find (0, 0, z) r = (0, 0, z) R a y a a x v0[C/m] Solid cube of uniform charge density The integral can be evaluated numerically.

  8. Example (cont.) z (0, 0, z) [V] a y a x a h [m] Result from Mathcad

  9. Example (cont.) 2.0 Face of cube z 1.5 (0, 0, z) [V] 1.0 a y 0.5 a x 0 a 1.5 1.0 0.5 2.0 h [m] Result from Mathcad

  10. Limitation of Potential Integral Method 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. • This will happen when there is an infinite voltage drop going to infinity.

  11. Example of Limitation Can we put the reference point at infinity? z Assume that  () = 0 Try integrating the electric field: l0[C/m] y r  x  The integral does not exist! Infinite line charge

  12. Example of Limitation (cont.) z l0[C/m] z R y r  x  Since we cannot put the reference point at infinity, the potential integral method will fail. The integral does not converge! Infinite line charge

  13. Example of Limitation (cont.) z The field-integration method still works: l0[C/m] (From Notes 14) y r  Note: We can still use the potential integral method if we assume a finite length of line charge first, and then after solving the problem let the length tend to infinity. (This will be a homework problem.) b x R( =b) Infinite line charge

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