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Unformatted text preview: Ph 214, Spring 2011 Exam # 1 Question #1 SOLUTIONS 1. [25 pts] Avalanche breakdown. a. (15 pts) Find an expression for vector E rad at this position for all times. Be sure to completely specify the direction (in cartesian coordinates) as well as its dependence on the given parameters and t . Electromagnetic radiation is produced by accelerating point charges. We know that the radiated electric field from a single accelerated point charge is given by vector E rad ( vector r, t ) = vectora ( t ) q 4 c 2 r so our first order of business will be to determine the acceleration. We are given the position as a function of time so we can easily compute vectora ( t ) = d 2 dt 2 vector z e ( t ). We obtain d 2 dt 2 ( z ) z = t t vectora ( t ) = d 2 dt 2 bracketleftbiggparenleftbigg 1 6 t 3 1 2 t 2 t parenrightbigg z bracketrightbigg = t z t < t < t d 2 dt 2 ( z ) z = t t [3 pts for correct expressions. Answer must include domain and correct direction.] x x z z R 2 3 2 R R 30 observer 30 vectora vectora Figure 1: Geometry of observer relative to accelerating charges. Now that we have determined vectora , we can compute the projection of vectora perpendicular to the direction of propagation. The observer is located along a radius vector lying in the x z plane and making an angle of 30 with respect to the z axis (refer to Figure 1). For acceleration in + z , | vectora | = | vectora | sin 30 = 1 2 | vectora | . The corresponding direction is given by cos 30 x + sin 30 z = 3 2 x + 1 2 z . Therefore, 1 Ph 214, Spring 2011 Exam # 1 Question #1 SOLUTIONS t t vectora ( t ) = parenleftbigg 1 2 t parenrightbigg parenleftBigg 3 2 x + 1 2 z parenrightBigg t < t < t t t [5 pts for correct magnitude and direction. Breakdown is 1 + 3 + 1 pts for the three cases.] The distance from the charge to the observer is R and therefore the time it takes for the electromagnetic radiation to reach the observer is R/c . This implies that the retarded time is given by t = t R/c . Treating all N electrons as a single charged object of total charge Ne , we can put all the pieces together to obtain t t + R c vector E rad ( R, t ) = N e 8 c 2 R parenleftbigg t R c parenrightbigg parenleftBigg 3 2 x + 1 2 z parenrightBigg t + R c < t < t + R c t t + R c [7 pts broken down as 1 + 5 + 1 pts for the three cases. To receive full credit: the direction must be correct, the retarded time must be specifically included & the time limits for each of the three cases must be correct.] b. (5 pts) For this observer, make a sketch of vector E rad as a function of time....
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This note was uploaded on 02/27/2012 for the course CHEMISTRY/ CH/ECE/PH/ taught by Professor Faculty during the Spring '08 term at Cooper Union.
- Spring '08