ch37-p058 - 58. (a) The binomial theorem tells us that, for...

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58. (a) The binomial theorem tells us that, for x small, (1 + x ) ν 1 + ν x + ½ ν(ν − 1) x ² if we ignore terms involving x 3 and higher powers (this is reasonable since if x is small, say x = 0.1, then x 3 is much smaller: x 3 = 0.001). The relativistic kinetic energy formula, when the speed v is much smaller than c , has a term that we can apply the binomial theorem to; identifying – β ² as x and –1/2 as ν , we have γ = () 1 − β 2 − 1/2 1 + (–½)(– β ²) + ½ (–½) ( (–½) − 1 ) (– β ²) 2 . Substituting this into Eq. 37-52 leads to K = mc ²( γ – 1) mc ² ( (–½)(– β ²) + ½ (–½) ( (–½) − 1 ) (– β ²) 2 ) which simplifies to K 1 2 mc ² β 2 + 3 8 mc ² β 4 = 1 2 mv ² + 3 8 mv 4 / c ² . (b) If we use the mc ² value for the electron found in Table 37-3, then for β = 1/20, the classical expression for kinetic energy gives K classical = 1 2 mv ² = 1 2 mc ² β 2 = 1 2 (8.19 × 10 14 J) (1/20) 2 = 1.0 × 10 16 J . (c) The first-order correction becomes
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This note was uploaded on 06/03/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.

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ch37-p058 - 58. (a) The binomial theorem tells us that, for...

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