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Question: An electron gun on a space probe starts emitting an electron beam into the space. Electrons in the b...

An electron gun on a space probe starts emitting an electron beam into the space. Electrons in the beam have energy W=9.0×104W = 9.0 \times 10^4 eV and the beam is equivalent to electric current I=1.0I = 1.0 mA. If the space probe is a metal sphere of radius r=1.0r = 1.0 m, find electric field strength at the surface of the space probe 1.0 min after starting the electron gun. Charge on an electron is e=−1.6×10−19e = -1.6 \times 10^{-19} C.

Answer

5.4×108 N/C5.4 \times 10^8 \text{ N/C}

Explanation

Solution

The problem asks to find the electric field strength at the surface of a metal sphere after a certain time, given the current of an electron beam emitted from it.

Here's the step-by-step solution:

  1. Calculate the total charge accumulated on the sphere:

    The electron gun emits electrons from the space probe. This means the space probe loses negative charge, effectively becoming positively charged. The rate of charge flow is given by the current II.

    The total charge QQ accumulated on the sphere after time tt is given by:

    Q=I×tQ = I \times t

    Given:

    Current I=1.0 mA=1.0×10−3 AI = 1.0 \text{ mA} = 1.0 \times 10^{-3} \text{ A}

    Time t=1.0 min=60 st = 1.0 \text{ min} = 60 \text{ s}

    Q=(1.0×10−3 A)×(60 s)Q = (1.0 \times 10^{-3} \text{ A}) \times (60 \text{ s})

    Q=0.06 CQ = 0.06 \text{ C}

  2. Calculate the electric field strength at the surface of the space probe:

    For a conducting sphere of radius rr with charge QQ uniformly distributed on its surface, the electric field strength EE at its surface is given by Coulomb's law for a point charge located at the center of the sphere:

    E=kQr2E = \frac{kQ}{r^2}

    where k=14πϵ0k = \frac{1}{4\pi\epsilon_0} is Coulomb's constant, approximately 9.0×109 N m2/C29.0 \times 10^9 \text{ N m}^2/\text{C}^2.

    Given:

    Radius r=1.0 mr = 1.0 \text{ m}

    Charge Q=0.06 CQ = 0.06 \text{ C}

    Constant k=9.0×109 N m2/C2k = 9.0 \times 10^9 \text{ N m}^2/\text{C}^2

    E=(9.0×109 N m2/C2)×(0.06 C)(1.0 m)2E = \frac{(9.0 \times 10^9 \text{ N m}^2/\text{C}^2) \times (0.06 \text{ C})}{(1.0 \text{ m})^2}

    E=9.0×109×0.06 N/CE = 9.0 \times 10^9 \times 0.06 \text{ N/C}

    E=0.54×109 N/CE = 0.54 \times 10^9 \text{ N/C}

    E=5.4×108 N/CE = 5.4 \times 10^8 \text{ N/C}

    The information about the energy of the electrons (W=9.0×104W = 9.0 \times 10^4 eV) and the charge on an electron (e=−1.6×10−19e = -1.6 \times 10^{-19} C) is not required for this calculation, as the current directly provides the rate of charge flow.