Problem Set 3 Solutions

Problem Set 3 Solutions - MASSACHUSETTS INSTITUTE OF...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2007 Problem Set 3 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 33 2 00 ˆˆ 12 44 2 Qa a a a Q aa a a πε ⎛⎞ + ⎜⎟ =+ + + ⎝⎠ ii jj Ei G j (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 11 2 4 22 i ij ij q QQQ Q V ra a a a == + + = + 1 (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 WQV a =Δ= + (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 PS03 Solution - 1
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Problem 1: Charges on a Square continued… 22 2 00 0 11 1 21 2 44 4 2 QQ Q W aa a πε ⎛⎞ =+ = + ⎜⎟ ⎝⎠ 2 2 Alternatively we could have started with empty space, brought in the first charge for free, the second charge in the potential of the first and so forth. We’ll get the same answer. Problem 2: Electric Potential Suppose an electrostatic potential has a maximum at point P and a minimum at point M. (a) Are either (or both) of these points equilibrium points for a negative charge? If so are they stable? The electric field is the gradient of the potential, which is zero at both potential minima and maxima. So a negative charge is in equilibrium (feels no net force) at both P & M. However, only the maximum (P) is stable. If displaced slightly from P, a negative charge will roll back “up” hill, back to P. If displaced from M a negative charge will roll away from the potential minimum. (b) Are either (or both) of these points equilibrium points for a positive charge? If so are they stable? Similarly, both P & M are equilibria for positive charges, but only M is a stable equilibrium because positive charges seek low potential (this is probably the case that seems more logical since it is like balls on mountains). Problem 3: A potential for finding charge The electric potential V(r) for a distribution of charge with spherical symmetry is: 2 0 0 2 Region I: ( )= for 0 2 Region II: ( ) for 2 2 R egion III: ( ) 0 for 2 Vr V r R R VR V R r R r r R −+ < < =− < < => Where V 0 is a constant of appropriate units. This function is plotted above.
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Problem Set 3 Solutions - MASSACHUSETTS INSTITUTE OF...

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