webct12 - Triangle of charges 1. Find the energy stored in...

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Triangle of charges 1. Find the energy stored in the arrangement of charges below, in J. Q 1 = 5 μC, Q 2 = 5 μC, Q 3 = 4 μC, a = 4 cm. The energy stored is equal to sum of the potential energy for each pair: () 92 12 23 31 12 2 91 0 Nm/C 25 20 20 10 C 14.6 J 41 0 m × =++= + + ×= × QQ Uk k k aaa This is also equal to the external work that has to be done to build the system, ie to bring each of the charges from infinity. For the first charge ( Q 1 ), no work must be done. For the second charge ( Q 2 ), the external work (which must be equal to minus the work done by the electric field) is. ext 0 =∆ = − = WU k k aa For the third charge ( Q 3 ), the external work is: ext 0 + + W U kk Thus the total work to build the system is:
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23 31 12 ext total =++ QQ Wk k k aaa
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Gauss cylinder 1. A very long (i.e. infinite) cylindrical rod of radius a has a positive charge uniformly distributed along its length. The linear charge density is λ . We would like to determine the electric field at a distance r ( r > a ) from the rod. This can be done in two ways: 1. Cutting the rod down to infinitely small pieces of length dx , finding the contribution from each piece, and adding (integrating) them all. 2. Using Gauss’s law, taking advantage of the symmetry of the problem. We’ll use the second way. You can try the first method as well, to keep your calculus skills in good shape. We first need to analyze the symmetry of the problem. Which one of the figures below seems an appropriate representation of what the electric field lines will look like in this case? a. A b. B c. C *d. D
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e. E The symmetry of the charge distribution indicates that the field will everywhere point radially away from the line. The appropriate figure is (d). If we maintain a fixed distance r from the line, the magnitude of the field will be constant. 2. We must choose a Gaussian surface that satisfies these two conditions: 1. It respects the symmetry of the problem. 2. It contains the point at which we are trying to find the electric field. An appropriate Gaussian surface for the present problem is: a. A sphere of radius r . *b. A cylinder of radius r and (arbitrary) height l . c. A cube of side r . d. A cylinder of radius a and (arbitrary) height l . e. A sphere of radius a . The symmetry of the problem indiactes that an appropriate Gaussian surface for this problem is a cylinder of radius r , (concentric with line charge) and some height l . 3. Ok, draw the surface. We’ll now compute the flux through this surface in two different ways: 1. Using the definition of flux. Before you start calculating, think if the flux through any part of your Gaussian surface is zero. Think also what is the angle between the area vector and the electric field.
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webct12 - Triangle of charges 1. Find the energy stored in...

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