A charge q is placed at the centre of the line joining two equal charges Q. The system of the three charges will be in equilibrium if q is equal to:

Options

A. `-(Q)/(2)`
B. `-(Q)/(4)`
C. `+(Q)/(4)`
D. `+(Q)/(2)`

Answer

To solve the problem of finding the charge \( q \) that will keep the system of three charges in equilibrium, we can follow these steps:

### Step 1: Understand the Configuration
We have two equal charges \( Q \) placed at a distance \( 2d \) apart. A charge \( q \) is placed at the center of the line joining these two charges, which means it is at a distance \( d \) from each charge \( Q \).

**Hint:** Visualize the arrangement of the charges and their positions.

### Step 2: Analyze Forces Acting on Charge \( q \)
For the charge \( q \) to be in equilibrium, the net force acting on it must be zero. The forces acting on \( q \) are due to the two charges \( Q \).

**Hint:** Remember that like charges repel and opposite charges attract.

### Step 3: Calculate the Force on Charge \( q \)
The force exerted on \( q \) by one of the charges \( Q \) (let's say the left one) is given by Coulomb's law:
\[ F_{Q \to q} = \frac{k \cdot Q \cdot |q|}{d^2} \]
where \( k \) is Coulomb's constant.

Since there are two charges \( Q \), the force due to the other charge \( Q \) will be the same in magnitude but directed towards the right.

**Hint:** Write down the expression for the force due to both charges acting on \( q \).

### Step 4: Set Up the Equilibrium Condition
For \( q \) to be in equilibrium, the force exerted by charge \( q \) on one of the charges \( Q \) must balance the force exerted by the other charge \( Q \) on \( q \). The force exerted by \( q \) on one of the charges \( Q \) is:
\[ F_{q \to Q} = \frac{k \cdot |q| \cdot Q}{d^2} \]
Setting the magnitudes of the forces equal gives:
\[ \frac{k \cdot |q| \cdot Q}{d^2} = \frac{k \cdot Q^2}{4d^2} \]

**Hint:** Simplify the equation by canceling common terms.

### Step 5: Solve for \( q \)
From the equilibrium condition, we can simplify:
\[ |q| \cdot Q = \frac{Q^2}{4} \]
This leads to:
\[ |q| = \frac{Q}{4} \]
Since \( q \) must be of opposite sign to \( Q \) for the forces to balance (as one is repulsive and the other attractive), we conclude:
\[ q = -\frac{Q}{4} \]

**Hint:** Consider the sign of the charge based on the nature of the forces.

### Final Answer
Thus, the charge \( q \) that will keep the system in equilibrium is:
\[ q = -\frac{Q}{4} \]