SampleFC

# SampleFC - 8.02 Sample Final C (Modified from Spring 1993)...

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8.02 Sample Final C (Modified from Spring 1993) p. 1/4 1. Displacement Current A parallel-plate capacitor consists of two circular plates, each with radius R , separated by a distance d . The electric field E between the plates is directed upwards (see sketch). R r d P E (a) What is the total energy stored in the electric field o the capacitor, in terms of E , R , d , and appropriate physical constants? Assume that the electric field is uniform between the plates and zero outside of the plates (i.e., neglect fringing fields). f (b) Now, suppose that the electric field is increasing with time ( dE/dt > 0 ). The point P is located between the plates at radius r < R (see sketch) . Derive an expression for the magnitude of the magnetic field B at point P in terms of dE/dt , r , and appropriate physical constants. Indicate the direction of B at P on the sketch. (c) What is the Poynting flux at point P in terms of the quantities given? Give both direction and magnitude. (d) Derive an expression for the total electromagnetic energy flowing into the capacitor per unit time across r = R, in terms of E , dE/dt , and given constants. Write down an equation relating this quantity to the electric energy contained in the capacitor (see part (a)). 2. Waves For the case of electromagnetic fields in a vacuum with the form E = E(x,t) ; B = B(x,t) where are unit vectors in the x- , y- , and z- directions, respectively, Faraday's and Ampere's Laws may be written as y ˆ z ˆ z y x ˆ and , ˆ , ˆ t B x E = (Faraday’s Law); t E x B = 0 0 ε µ (Ampere’s Law) (a) Using these equations, derive a second-order differential equation for E(x,t) alone. (b) Consider the function E(x,t) = E 0 (ax-bt) 2 , where a and b are constants. What condition must hold for this function to be a solution to the differential equation you obtained in part (a)? (c) Again consider the function

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## This note was uploaded on 04/27/2009 for the course 8 8.02 taught by Professor Hudson during the Spring '07 term at MIT.

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SampleFC - 8.02 Sample Final C (Modified from Spring 1993)...

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