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ps02sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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PS02 Solution - 1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2008 Problem Set 2 Solutions Problem 1: Charges on a Square Three identical charges + Q are placed on the corners of a square of side a , as shown in the figure. (a) What is the electric field at the fourth corner (the one missing a charge) due to the first three charges? We’ll just use superposition: ( ) ( ) ( ) 3 2 3 3 3 0 0 ˆ ˆ ˆ ˆ ˆ ˆ 1 2 4 4 2 Q a a a a Q a a a πε πε + = + + = + + i i j j E i j G (b) What is the electric potential at that corner? A common mistake in doing this kind of problem is to try to integrate the E field we just found to obtain the potential. Of course, we can’t do that we only found the E field at a single point, not as a function of position. Instead, just sum the point charge potentials from the 3 points: 0 0 0 1 1 1 2 4 4 4 2 2 i i j ij q Q Q Q Q V r a a a a πε πε πε = = + + = + (c) How much work does it take to bring another charge, + Q , from infinity and place it at that corner? The work required to bring a charge + Q from infinity (where the potential is 0) to the corner is: 2 0 1 2 4 2 Q W Q V a πε = = + (d) How much energy did it take to assemble the pictured configuration of three charges? The work done to assemble three charges as pictured is the same as the potential energy of the three charges already in such an arrangement. Now, there are two pairs of charges situated at a distance of a , and one pair of charges situated at a distance of 2 a , thus we have
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