February 23, 2017

Homework Help: Physics

Posted by Anonymous on Saturday, May 11, 2013 at 11:22am.

We discussed in lectures that traveling Electromagnetic waves in vacuum of the form
E⃗ =E0xˆcos(kz−ωt),B⃗ =B0yˆcos(kz−ωt)
satisfy all 4 Maxwell's equations. In lectures, I showed that an application of the generalized Ampere's Law (closed loop surrounding area A2, see below), leads to: B0=ε0μ0cE0, and I mentioned that independently it follows from an application of Faraday's Law that B0=E0/c. Combining these two results then leads to the fantastic result that the “speed of light" in vacuum c=1/ε0μ0‾‾‾‾‾√. I want you to show that Faraday's Law indeed leads to the result B0=E0/c. You can show this by choosing a similar special area as we did in lectures:

We define the normal of the surface A1 in the figure above to point in the +yˆ direction. In the following , assume that ℓ=1 m, E0=1 V/m, f=610×1012 Hz, where f is the frequency of oscillation.

(a) We define f1(t)=∮E⃗ ⋅dℓ⃗ , where ∮E⃗ ⋅dℓ⃗ is the closed loop integral taken along the contour of the area A1 in the figure above. Evaluate the function f1(t) in Volts for the following value of t
t=2e-16 sec

(b) Consider the function f2(t)=ϕB(t). Following the method used in Lecture 27, calculate the function f2(t), and evaluate it (in Volts seconds) for the following time t
t=2e-16 sec

(c) Consider again the function f2(t) and evaluate −df2(t)dt in Volts for the following time t
t=2e-16 sec

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