ELECTROMAGNETISM & RELATIVITY - Chapter 1, Exercise 2 Solution
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Finding Charge Magnitude from a Known Force

PROBLEM
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Two identical point charges, each of magnitude q, are 0.2 m apart and
repel each other with a force of 9 N. Using Coulomb's law, find the
magnitude of q.

SOLUTION
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Starting from Coulomb's law, with both charges equal to q:

    F = k q^2 / r^2

Rearranged to solve for q:

    q^2 = F r^2 / k
    q = sqrt(F r^2 / k)

Substitute the given values:

    F = 9 N
    r = 0.2 m
    k = 8.99 x 10^9 N.m^2.C^-2

    q^2 = (9 x 0.2^2) / (8.99 x 10^9)
    q^2 = (9 x 0.04) / (8.99 x 10^9)
    q^2 = 0.36 / (8.99 x 10^9)
    q^2 = 4.00 x 10^-11

    q = sqrt(4.00 x 10^-11)
    q = 6.33 x 10^-6 C (approximately 6.33 uC)

ANSWER: Each charge has a magnitude of approximately 6.33 uC. Since
the problem states the charges REPEL each other, both charges must
have the SAME sign (both positive or both negative) - Coulomb's law
alone cannot determine which.

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WHY THIS WORKS AS AN ANSWER

Because both charges are stated to be equal in magnitude, Coulomb's
law simplifies from F = k q1 q2 / r^2 to F = k q^2 / r^2, letting the
single unknown q be isolated by algebra and then found by taking a
square root. The repulsive nature of the force (stated directly in the
problem) confirms the two charges share the same sign, even though
Coulomb's law's magnitude calculation alone is sign-blind - exactly
the same two-part reasoning (magnitude from the formula, direction
from the signs) used in Exercise 1.