2. Calculate the thermal velocity of electrons in a conductor, taking that their kinetic energy equals the thermal energy kkBBTT with kB bei

Question

2. Calculate the thermal velocity of electrons in a conductor, taking that their kinetic energy equals the thermal energy kkBBTT with kB being the Boltzmann constant, at the following temperatures: a. 100 K b. 300 K c. 1000 K Compare the velocities you obtained in each answer to the speed of light.

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Mít Mít 4 years 2021-08-12T15:57:29+00:00 1 Answers 32 views 0

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  1. kimcuc
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    2021-08-12T15:59:22+00:00
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    Answer:

    (a).v_{avg} = 6.8*10^{4}m/s, is 0.02% the speed of light.

    (b).v_{avg} = 1.2*10^5m/s, is 0.04%  the speed of light.

    (c). v_{avg} =2.1*10^5m/s is 0.07% the speed of light.

    Explanation:

    The average kinetic energy is related to the thermal energy of the electrons in a conductor by the the relation

    K.E_{avg} = \dfrac{1}{2}mv_{avg}^2 = \dfrac{3}{2}kT,

    where m = 9.1*10^{-31}kg is the mass of the electrons, v_{avg} is their average velocity, T is the temperature of the conductor, and k = 1.38*10^{-23}m^2kg \:s^{-2}\:K^{-1} is the Boltzmann constant.

    The equation, when solved for v_{avg}, gives

    v_{avg} = \sqrt{\dfrac{3kT}{m} }

    (a),

    For T = 100K, the thermal (average) velocity v_{avg} is

    v_{avg} = \sqrt{\dfrac{3(1.38*10^{-23})(100K)}{9.1*10^{-31}kg} }

    \boxed{v_{avg} = 6.8*10^{4}m/s}

    which when compared to the speed of light is

    \dfrac{6.8*10^4}{3*10^8} *100\% = (0.02\%)c

    0.02% the speed of light.

    (b).

    Similarly, for  T =300K

    v_{avg} = \sqrt{\dfrac{3(1.38*10^{-23})(300K)}{9.1*10^{-31}kg} }

    \boxed{v_{avg} = 1.2*10^5m/s}

    which is

    \dfrac{1,2*10^5m/s}{3*10^8m/s} *100\%= (0.04\%)c

    0.04% the speed of light.

    (c).

    Finally, for T =1000K

    v_{avg} = \sqrt{\dfrac{3(1.38*10^{-23})(1000k)}{9.1*10^{-31}kg} }

    \boxed{v_{avg} =2.1*10^5m/s}

    which is

    \dfrac{2.1*10^5m/s}{3*10^8m/s} *100\% = (0.07\%)c

    0.07% the speed of light.

    We see that the average electron velocities we obtain are always less than 1% the speed of light, which means relativistic effects are negligible, for they are apparent at about 25% the speed of light.

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