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15 December, 03:57

What is the escape speed for an electron initially at rest on the surface of a sphere with a radius of 1.3 cm and a uniformly distributed charge of 2.3 ✕ 10-15 C. That is, what initial speed must the electron have to reach an infinite distance from the sphere and have zero kinetic energy when it gets there?

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  1. 15 December, 04:19
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    2.37 * 10^4 m/s

    Explanation:

    Constants:

    Mass of electron = 9.11 * 10^ (-31) kg

    Electric charge of an electron = 1.602 * 10^ (-19) C

    Parameters given:

    Radius of sphere = 1.3cm = 0.013m

    Charge of sphere = 2.3 * 10^ (-15) C

    Using the law of conservation of energy, we have that:

    K. E. (initial) + P. E. (initial) = K. E. (final) + P. E. (final)

    K. E. (final) = 0, since final velocity is zero and P. E. (final) = 0 since the electron reaches a final distance of infinity.

    Hence,

    K. E. (initial) = P. E. (initial)

    0.5mv^2 = (kqQ) / r

    Where k = Coulumbs constant

    Q = charge of the sphere.

    r = radius of the sphere.

    => 0.5*m*v^2 = (kqQ) / r

    0.5 * 9.11 * 10^ (-31) * v^2 = (9 * 10^9 * 1.602 * 10^ (-19) * 2.3 * 10^ (-15)) / 0.013

    4.555 * 10^ (-31) * v^2 = 2550.88 * 10^ (-25)

    => v^2 = 2550.88 * 10^ (-25) / 4.555 * 10^ (-31)

    v^2 = 560 * 10^6 = 5.60 * 10^8

    => v = 2.37 * 10^4 m/s
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