Three point charges `Q_(1), Q_(2) and Q_(3)` in that order are placed equally spaced along a striaght line. `Q_(2) and Q_(3)` are equal in magnitude but opposite is sign. If the net force on `Q_(3)` is zero, the value of `Q_(1)` is

Options

A. `Q_(1) = |Q_(3)|`
B. `Q_(1)= sqrt2 |Q_(3)|`
C. `Q_(1) = 2|Q_(3)|`
D. `Q_(1)= 4 |Q_(3)|`

Answer

To solve the problem, we need to find the value of charge \( Q_1 \) such that the net force acting on charge \( Q_3 \) is zero. Given that \( Q_2 \) and \( Q_3 \) are equal in magnitude but opposite in sign, we can denote:

- \( Q_2 = -q \)
- \( Q_3 = +q \)

Let’s denote the distance between each charge as \( d \). Therefore, the distances between the charges are:

- Distance between \( Q_1 \) and \( Q_2 \) = \( d \)
- Distance between \( Q_2 \) and \( Q_3 \) = \( d \)
- Distance between \( Q_1 \) and \( Q_3 \) = \( 2d \)

### Step 1: Calculate the forces acting on \( Q_3 \)

The force acting on \( Q_3 \) due to \( Q_2 \) (denoted as \( F_{23} \)) and the force acting on \( Q_3 \) due to \( Q_1 \) (denoted as \( F_{13} \)) can be calculated using Coulomb's Law:

\[ F_{23} = k \frac{|Q_2 Q_3|}{d^2} = k \frac{|-q \cdot q|}{d^2} = k \frac{q^2}{d^2} \]

The direction of \( F_{23} \) is towards \( Q_2 \) (to the left).

The force acting on \( Q_3 \) due to \( Q_1 \) is:

\[ F_{13} = k \frac{|Q_1 Q_3|}{(2d)^2} = k \frac{|Q_1 \cdot q|}{4d^2} \]

The direction of \( F_{13} \) is away from \( Q_1 \) (to the right).

### Step 2: Set the forces equal to each other for equilibrium

For the net force on \( Q_3 \) to be zero, the magnitudes of these forces must be equal:

\[ F_{13} = F_{23} \]

Substituting the expressions we obtained:

\[ k \frac{|Q_1 \cdot q|}{4d^2} = k \frac{q^2}{d^2} \]

### Step 3: Simplify the equation

We can cancel \( k \) and \( d^2 \) from both sides:

\[ \frac{|Q_1 \cdot q|}{4} = q \]

### Step 4: Solve for \( Q_1 \)

Now, we can solve for \( Q_1 \):

\[ |Q_1| = 4 \]

Since \( Q_1 \) must be positive (as it is not specified to be negative), we conclude:

\[ Q_1 = 4q \]

### Final Answer

Thus, the value of \( Q_1 \) is \( 4 \times Q_3 \).

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