SampleFC_Solutions

SampleFC_Solutions - 8.02 Sample Final C Solutions...

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8.02 Sample Final C Solutions (Modified from Spring 1993) p. 1/5 1. Displacement Current (a) The energy stored is in the electric field. Since E is nearly constant we can just multiply the energy density by the volume inside the capacitor: R 2 2 0 2 2 0 2 2 E d R d R E V u U E E π ε = = = (b) With the electric field increasing, we have an upwards displacement current: ( ) P at page of out 2 1 2 0 0 2 0 0 0 2 0 2 0 0 dt dE r B dt dE r I r B dl B dt dE r dt r E d dt d I nt displaceme E nt displaceme µ = = = = = = Φ = (c) dt dE rE dt dE r E B E S 0 0 0 0 0 2 1 2 1 1 = = × = G G G (to the right/inwards!) (d) To find the total energy flowing in consider that the band at r = R has an area A = 2 π Rd , so () dt dE dE R Rd dt dE RE A R r S dt dU 2 0 0 2 2 1 = = = = G G . Notice that this is indeed the time derivative of U E that we calculated in part (a). 2. Waves E = E(x,t) ; B = B(x,t) , Faraday: y ˆ z ˆ t B x E = ; Ampere: t E x B = 0 0 (a) Take the partial of Faraday’s law w.r.t. x and the time derivative of Ampere’s Law and substitute: 2 2 0 0 2 2 2 2 0 0 2 2 2 t E x E t E t x B x E = = = (b) To see if E(x,t) = E 0 (ax-bt) 2 holds, plug it into the differential eqn. of (a): 0 0 2 0 2 0 0 0 2 0 2 0 0 0 0 0 2 2 0 0 ? 0 2 0 2 2 2 2 : Condition 2 2 2 2 = = = = = = = b a E b E a E b bt ax bE x t E E a bt ax aE x x E (c) Faraday: 2 0 0 2 bt ax b aE B t B bt ax aE x E = = = . Check in Ampere: bt ax bE b a t E bt ax b E a x B = = = 0 2 0 0 ? 0 2 2 2 Yes! Using the condition we derived in (b) we find this works. (d) Putting this wave in a dielectric medium with dielectric constant κ e changes Ampere’s law (since ε 0 becomes κ e ε 0 ). The speed of propagation ( b / a ) decreases by e / 1 r d P E
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8.02 Sample Final C Solutions (Modified from Spring 1993) p. 2/5 3. Wave B = 10 -7 sin [ x - 3 ·10 8 t] Tesla y ˆ (a) What is the wavelength λ of the wave? m k π λ 2 1 2 2 = = = (b) What is the amplitude E 0 of the associated magnetic field? E
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SampleFC_Solutions - 8.02 Sample Final C Solutions...

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