10/26/2004 Electric Potential for a Point Charge.doc 1/7 Jim Stiles The Univ. of Kansas Dept. of EECS Electric Potential for Point Charge Recall that a point charge Q, located at the origin (r=0′), produces a static electric field: ()ˆ20r4rQarπε=E Now, we know that this field is the gradient of some scalar field: ()()rrV=−∇E Q: What is the electric potential function ()rV generated by a point charge Q, located at the origin? A: We find that it is: ()0r4QVrπε= Q: Where did this come from ? How do we know that this is the correct solution? A: We can show it is the correct solution by direct substitution!10/26/2004 Electric Potential for a Point Charge.doc 2/7 Jim Stiles The Univ. of Kansas Dept. of EECS ()()ˆˆ0020rr4044rrVQrQarrQarπεπεπε=−∇⎛⎞=−∇⎜⎟⎝⎠⎛⎞∂=−+⎜⎟∂⎝⎠=E The correct result! Q: What if the charge is not located at the origin ? A: Substitute r with r-r′, and we get: ()0r4r-rQVπε=′ Where, as before, the position vector r′ denotes the location of the charge Q, and the position vector rdenotes the location in space where the electric potential function is evaluated.10/26/2004 Electric Potential for a Point Charge.doc 3/7 Jim Stiles The Univ. of Kansas Dept. of EECS The scalar function ()rV for a point charge can be shown graphically as a contour plot: Or, in three dimensions as: −4−2 0 2 4x−4−2024y−4−2024x−4−2024y0.20.4volts−4−2024x−4−2024y0.20.4volts10/26/2004 Electric Potential for a Point Charge.doc 4/7 Jim Stiles The Univ. of Kansas Dept. of EECS Note the electric potential increases as we get closer to the point charge (located at the origin). It appears that we have “mountain” of electric potential; an appropriate analogy, considering that the potential energy of a mass in the Earth’s gravitational field increases with altitude (i.e., height)! Recall the electric field produced by a point charge is a vector field that looks like: −4−2 0 2 4x−4−2024y10/26/2004 Electric Potential for a Point Charge.doc 5/7 Jim Stiles The Univ. of Kansas Dept. of EECS Combining the electric field plot with the electric potential plot, we get: Given our understanding of the gradient, the above plot makes perfect sense! Do you see why ? Now lets examine another example, where three point charges (one of them negative!) are present. −4−2 0 2 4x−4−2024y10/26/2004 Electric Potential for a Point Charge.doc 6/7 Jim Stiles The Univ. of Kansas Dept. of EECS −4−2 0 2 4x−4−2024y−4−2024x−4−2024y−0.4−0.200.20.4volts−4−2024x−4−2024y−0.4−0.200.20.4volts10/26/2004 Electric Potential for a Point Charge.doc 7/7 Jim Stiles The Univ. of Kansas Dept. of EECS −4−2 0 2
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