In vacuum, is held at rest at the origin and is held at rest at on the -axis. No…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential · Electrostatic fields and potential

Electricity, Magnetism and Electromagnetism, Ch Electrostatic potential, potential energy and the energy of the field

Problem

In vacuum,

q_1 = +8.0\,\mathrm{nC}

is held at rest at the origin and

q_2 = -8.0\,\mathrm{nC}

is held at rest at

x = d = 0.40\,\mathrm{m}

on the

x

-axis. No other charges are present. Take

V = 0

at infinity, and let

M

be the midpoint

x = 0.20\,\mathrm{m}

. Use

1/4\pi\varepsilon_0 = 8.99\times10^{9}\,\mathrm{N\,m^2\,C^{-2}}

. ((a)) Find the potential at

M

by adding the potentials of the two charges. ((b)) For

0 < x < d

the potential on the axis is

V(x) = \frac{1}{4\pi\varepsilon_0}\left(\frac{q_1}{x} + \frac{q_2}{d-x}\right).

Differentiate this to obtain the

x

-component of the field, evaluate it at

M

and state its direction. ((c)) A third charge

q_3 = +5.0\,\mathrm{nC}

is brought in slowly from infinity and held at

M

. Find the total electrostatic potential energy of the three charges.

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