Question

Three particles are fixed on an x-axis. Particle 1 of charge q₁ is at x=-a, and particle 2 of charge q₂is at x=+a. If their net electrostatic force on particle 3 of charge -Qis to be zero, what must be the ratio q₁ /q₂when particle 3 is at (a)x=+0.500a and (b) x=+1.50a?

Short answer

1. The ratio when particle 3 is at$x=+0.500a$is 9.0 .
2. The ratio when particle 3 is at $x=+1.50a$is -25.

Step-by-step solution

The given data

Charge $q_{1}isatx=-a$

Charge $q_{2}isatx=+a$

The net force on the third particle of charge -Q is to be zero.

Understanding the concept of Coulomb’s law

Using the given data of separation, the ratio of the charges can be found easily by substituting these values in the formula of force obtained from Coulomb's law.

Formula:

The magnitude of the electrostatic force between any two particles,

$F_1=\frac{1}{\left(4\pi {\epsilon }_0\right)}\frac{\left|q_1\right|\left|q_2\right|}{r^2}$ (1)

a) Calculation of ratio q1 /q2 when particle 3 is at x=+0.500a

The net force from particles 1 and 2 on third particle 3 (acting on -Q) is zero. Thus, the relation for x=+0.500a is given using equation (1) as:

$$\begin{gathered} \frac{1}{\left(4\pi {\epsilon }_0\right)}\frac{\left|q_1\right|Q}{(-a-{\displaystyle \frac{a}{2}}{)}^2}=\frac{1}{\left(4\pi {\epsilon }_0\right)}\frac{\left|q_2\right|Q}{(a-{\displaystyle \frac{a}{2}}{)}^2} \\ \left|q_1\right|=9.0\left|q_2\right| \\ \frac{\left|q_1\right|}{\left|q_2\right|}=9.0 \end{gathered}$$

Since Q is located between q₁ and q₂, we conclude q₁and q₂ are like-sign.

Hence, the value of the ratio is 9.0.

b) Calculation of ratio q1 /q2 when particle 3 is at x=+1.50a.

The net force from particles 1 and 2 on third particle 3 (acting on -Q) is zero. Thus, the relation for x=+1.50a is given using equation (1) as:

$$\begin{gathered} \frac{1}{\left(4\pi {\epsilon }_0\right)}\frac{\left|q_1\right|Q}{(-a-{\displaystyle \frac{3a}{2}}{)}^2}=\frac{1}{\left(4\pi {\epsilon }_0\right)}\frac{\left|q_2\right|Q}{(a-{\displaystyle \frac{3a}{2}}{)}^2} \\ \left|q_1\right|=25\left|q_2\right| \\ \frac{\left|q_1\right|}{\left|q_2\right|}=25 \end{gathered}$$

Now, Q is not located between q₁ and q₂ ; one of them must push and the other must pull. Thus, they are unlike-sign, the value of the ratio is -25