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(a) What is the sum free time tau between collisions for the conduction electrons in copper? (b) The mean free path lambda of the conduction electrons in a particular conductor is the average distance traveled by an electron between collisions. What is lambda for the conduction electrons in copper, assuming that their effective speed v_("eff")" is "1.6 xx 10^(6) m//s? |
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Answer» Solution :(a) The mean freetime `tau` of copper is approximately constant, and in particular does not depend on any electric field that might be APPLIED to a sample of the copper. THUS, we need not consider any particular value of applied, electric field. HOWEVER, because the resistivity p displaced by copper under an electric field depends on `tau`, we can find the mean free time `tau` from `(p=m//z^(2) n tau)`. Calculations: That equation gives us `tau=m/(ne^(2) p)` The number of conduction electrons per unit volume in copper is `8.49 xx 10^(28) m^(-3)`. We take the value of p fro Table 26.1. The denominator then BECOMES. `(8.49 xx 10^(28) m^(-3)) (1.6 xx 10^(-19) C)^(2) (1.69 xx 10^(-8)Omega.m)` `=3.67 xx 10^(-17)C. Omega//m^(2)=3.67 xx 10^(-17) kg//s`. where we converted units as `(C^2.Omega)/(m^(2))=(C^(2))/(m^(2)).V/A=(C^(2))/(m^(2)). (J//C)/(C//s) =(kg. m^(2)//s^(2))/(m^(2)//s)=(kg)/(s)`. Using these results and substitutingfor the electron mass m, we then have `tau=(9.1 xx 10^(-31)kg)/(3.67 xx 1^(-17) kg//s) =2.5 xx 10^(-14)s`. (b) The distance d any particular travels in a certain time t at a constant speed v is d=vt. Calculation : For the electrons in copper, this gives us `lambda =v_("eff") tau` `=(1.6 xx 10^(6) m//s) (2.5 xx 10^(-14)s)` `=4.0 xx 10^(-8)m=40nm`. This is about 150 times the distance between nearest-neighbor atoms in a copper lattice. Thus, on the average, each conduction electron passes many copper atoms before finally hitting one. |
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