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Three charges lie along the x-axis. One positive charge, q1 = 4.80*10^-18 C, is at x = 3.72 m, and another positive charge, q2 = 1.60*10^-19 C, is at the origin.

At what point on the x-axis must a negative charge, q3, be placed so that the resultant force on it is zero?

1 Answer

4 votes

Answer:

The third charge needs to be placed at
x \approx 0.57\; \rm m.

Step-by-step explanation:

Both
q_1 and
q_2 would attract
q_3.

These two electrostatic attractions need to balance one another. Hence, they need to be opposite to one another. Therefore,
q_1 and
q_2 need to be on opposite sides of
q_3. That is possible only if
q_3 \! is on the line segment between
q_1 \! and
q_2 \!.

Assume that
q_3 is at
x\; \rm m, where
0 < x < 3.72 (in other words,
q_3 \! is on the line segment between
q_1 and
q_2, and is
x\; \rm m \! away from
q_2 \!.)

Let
k denote Coulomb's constant.

The magnitude of the electrostatic attraction between
q_1 and
q_3 would be:


\displaystyle (k\cdot q_1 \cdot q_3)/((3.72 - x)^(2)).

Similarly, the magnitude of the electrostatic attraction between
q_2 and
q_3 would be:


\displaystyle (k\cdot q_2 \cdot q_3)/(x^(2)).

The magnitudes of these two electrostatic attractions need to be equal to one another for the resultant electrostatic force on
q_3 to be
0. Equate these two expressions and solve for
x:


\displaystyle (k\cdot q_1 \cdot q_3)/((3.72 - x)^(2)) = (k\cdot q_2 \cdot q_3)/(x^(2)).


\displaystyle (q_1)/((3.72 - x)^(2)) = (q_2)/(x^(2)).


\displaystyle (x^2)/((3.72 - x)^(2)) = (q_2)/(q_1).


\displaystyle (x^2)/((3.72 - x)^(2)) = (q_2)/(q_1) = (1)/(30).

By the assumption that
(0 < x < 3.72), it should be true that
(x > 0) and
(3.72 - x > 0). Therefore,
\displaystyle (x)/((3.72 - x)) > 0.

Take the square root of both sides of the equation
\displaystyle (x^2)/((3.72 - x)^(2)) = (1)/(30).


\displaystyle \sqrt{(x^2)/((3.72 - x)^(2))} = \sqrt{(1)/(30)}.


\displaystyle (x)/(3.72 - x) = (1)/(√(30)).


√(30)\, x = 3.72 - x.

Therefore:


\left(1 + √(30)\right)\, x = 3.72.


\displaystyle x = (3.72)/(1 + √(30)) \approx 0.57.

Hence,
q_3 should be placed at
x \approx 0.57\; \rm m.

User Jonathan Rich
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