Question

Three charges \(+Q\), \(q\), \(+Q\) are placed respectively, at distances \(0\), \(\frac{d}{2}\), and \(d\) from the origin on the x-axis. If the net force experienced by \(+Q\) placed at \(x = 0\) is zero, then the value of \(q\) is

• A. \(-\frac{Q}{4}\)
• B. \(+\frac{Q}{2}\)
• C. \(+\frac{Q}{4}\)
• D. \(-\frac{Q}{2}\)

Hint:

Use Coulomb’s Law and set net force to zero. Pay attention to direction and sign of charges.

Answer 1

The Correct Option is A

Let’s analyze the net force on the \(+Q\) charge at \(x = 0\): - The force due to the charge \(+Q\) at distance \(d\) (right side) is repulsive: \[ F_1 = \frac{kQ^2}{d^2} \] - The force due to charge \(q\) at distance \(\frac{d}{2}\) is: \[ F_2 = \frac{kQq}{\left(\frac{d}{2}\right)^2} = \frac{4kQq}{d^2} \] Net force on the charge at \(x = 0\) is zero: \[ F_1 = F_2 \Rightarrow \frac{kQ^2}{d^2} = \frac{4kQq}{d^2} \Rightarrow Q = 4q \Rightarrow q = \frac{Q}{4} \] Since the force from \(q\) must be attractive to balance repulsion from the +Q on the right, \(q\) must be negative: \[ q = -\frac{Q}{4} \]

Answer 2

Step 1: Understand the setup
- Three charges are placed on the x-axis:
  At \(x = 0\): charge \(+Q\)
  At \(x = \frac{d}{2}\): charge \(q\)
  At \(x = d\): charge \(+Q\)

Step 2: Forces on charge at \(x=0\)
- Force due to charge at \(x = \frac{d}{2}\):
\[ F_1 = k \frac{Q |q|}{\left(\frac{d}{2}\right)^2} = k \frac{4Q |q|}{d^2} \]
Direction depends on sign of \(q\).

- Force due to charge at \(x = d\):
\[ F_2 = k \frac{Q \cdot Q}{d^2} = k \frac{Q^2}{d^2} \]
Direction is repulsive (away from \(+Q\) at \(d\)).

Step 3: Set net force on charge at \(x=0\) to zero
For equilibrium, forces must be equal and opposite:
\[ F_1 = F_2 \]
If \(q\) is negative, force \(F_1\) is attractive, towards \(q\), i.e., towards \(x = \frac{d}{2}\).
Force \(F_2\) is repulsive, pushing \(+Q\) at 0 away from \(+Q\) at \(d\).

So,
\[ k \frac{4 Q |q|}{d^2} = k \frac{Q^2}{d^2} \implies 4 Q |q| = Q^2 \implies |q| = \frac{Q}{4} \]
Since \(q\) is negative for the directions to oppose,
\[ q = -\frac{Q}{4} \]

Step 4: Final answer
The value of \(q\) is \(-\frac{Q}{4}\).