Question

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?

Answer 1

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`.