10/26/2004 Electric Potential Function for Charge Densities.doc 1/3 Jim Stiles The Univ. of Kansas Dept. of EECS Electric Potential Function for Charge Densities Recall the total static electric field produced by 2 different charges (or charge densities) is just the vector sum of the fields produced by each: ()()()12rrr=+EEE Since the fields are conservative, we can write this as: ()()()() () ()() () ()()121212rrrrrrrrrVVVVVV=+−∇ = −∇ − ∇−∇ = −∇ +EEE Therefore, we find, ()()()12rrrVVV=+ In other words, superposition also holds for the electric potential function! The total electric potential field produced by a collection of charges is simply the sum of the electric potential produced by each. Consider now some distribution of charge, ()rvρ. The amount of charge dQ, contained within small volume dv, located at position r′, is: ()rvdQ dvρ′′=10/26/2004 Electric Potential Function for Charge Densities.doc 2/3 Jim Stiles The Univ. of Kansas Dept. of EECS The electric potential function produced by this charge is therefore: ()()00r4r-rr4r-rvdQdVdvπερπε=′′′=′ Therefore, integrating across all the charge in some volume V, we get: ()()0rr4r-rvVVdvρπε′′=′∫∫∫ Likewise, for surface or line charge density: ()()()()00rr4r-rrr4r-rsSCVdsVdρπερπε′′=′′′=′∫∫∫AA Note that these integrations are scalar integrations—typically they are easier to evaluate than the integrations resulting from Coulomb’s Law.10/26/2004 Electric Potential Function for Charge Densities.doc 3/3 Jim Stiles The Univ. of Kansas Dept. of EECS Once we find the electric potential function ()Vr, we can then determine the total electric field by taking the gradient: ()()rVr=−∇E Thus, we now have three (!) potential methods for determining the electric field produced by some charge distribution ()vrρ: 1. Determine ()rE from Coulomb’s Law. 2. If ()vrρ is symmetric, determine ()rE from Gauss’s Law. 3. Determine the electric potential function ()Vr, and then determine the electric field as ()()rVr=−∇E . Q: Yikes! Which of the three should we use?? A: To a certain extent, it does not matter! All three will provide the same result (although ()vrρ must be symmetric to use method 2!). However, if the charge density is symmetric, we will find that using Gauss’s Law (method 2) will typically result in much less work! Otherwise (i.e., for non-symmetric ()vrρ), we find that sometimes method 1 is easiest, but in other cases method 3 is a bit less stressful (i.e., you
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