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mPhys390-hw1-sol - Physics 390 Homework set#1 Solutions...

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Unformatted text preview: January 17, 2007 Physics 390: Homework set #1 Solutions Reading: Tipler & Llewellyn, Chapter 3 Questions: 1. Show that the classical wave equation ∂ 2 f ∂t 2- c 2 ∂ 2 f ∂x 2 = 0 is satisfied by any function f that depends on x and t in the combination u = x ± ct : f ( x, t ) = f ( u ) = f ( x ± ct ). Solution: To see this, plug into the equation above: ∂ ∂t f = ∂f ∂u ∂u ∂t = ∂f ∂u ( ± c ) ∂ 2 ∂t 2 f = ∂ ∂t parenleftbigg ∂f ∂t parenrightbigg = ∂ ∂t bracketleftbigg ∂f ∂u ( ± c ) bracketrightbigg = ( ± c ) ∂ ∂t ∂f ∂u = ( ± c ) ∂ ∂u ∂f ∂u ∂u ∂t = ( ± c ) ∂ 2 f ∂u 2 ( ± c ) = c 2 ∂ 2 f ∂u 2 similarly, ∂ 2 f ∂x 2 = ∂ 2 f ∂u 2 parenleftbigg ∂u ∂x parenrightbigg 2 = ∂ 2 f ∂u 2 (1) = ∂ 2 f ∂u 2 and, ∂ 2 f ∂t 2- c 2 ∂ 2 f ∂x 2 = c 2 ∂ 2 f ∂u 2- c 2 ∂ 2 f ∂u 2 = 0 identically! 2. Planck’s constant is h = 6 . 626 × 10 − 34 J · s. What familiar physical quantity from classical me- chanics also has dimensions of J · s? Solution: Angular momentum also has dimensions of J · s. We will see that Planck’s constant is closely related to quantization of angular momentum. 1 3. In what region of the electromagnetic spectrum does the blackbody radiation from a roomtem- perature object peak? What sorts of problems would we have if our eyes were sensitive in this region? Solution: Room temperature is about 290 K. Using the Wien displacement law, we have λ max = 2 . 898 × 10 − 3 m · K 290 K = 1 × 10 − 5 m ....
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mPhys390-hw1-sol - Physics 390 Homework set#1 Solutions...

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