Question

In the given figure, two equal positive point charges \( q_1 = q_2 = 2.0 \mu C \) interact with a third point charge \( Q = 4.0 \mu C \). The magnitude and direction of the net force on \( Q \) is:
\includegraphics[width=0.5\linewidth]{35.png}

• A. \( 0.23 N \) in the \( +x \)-direction
• B. \( 0.46 N \) in the \( +x \)-direction
• C. \( 0.23 N \) in the \( -x \)-direction
• D. \( 0.46 N \) in the \( -x \)-direction

Hint:

For multiple charges, use vector addition of forces in Cartesian components.

Answer 1

The Correct Option is B

Step 1: Understanding the situation.
The three charges \( q_1 = q_2 = 2.0 \mu C \) and \( Q = 4.0 \mu C \) interact through electrostatic forces. Since all charges are positive, the forces between them are repulsive. Step 2: Electrostatic force calculation.
The electrostatic force between two point charges is given by Coulomb's law: \[ F = k_e \frac{|q_1 q_2|}{r^2} \] where:
- \( k_e = 8.99 \times 10^9 \, {N m}^2 {C}^{-2} \) (Coulomb's constant)
- \( r \) is the distance between the charges
For the force between \( Q \) and \( q_1 \), the force is: \[ F_1 = k_e \frac{|Q q_1|}{r^2} = 8.99 \times 10^9 \times \frac{(4.0 \times 10^{-6})(2.0 \times 10^{-6})}{(0.5)^2} \] Simplifying this: \[ F_1 = 8.99 \times 10^9 \times \frac{8.0 \times 10^{-12}}{0.25} = 8.99 \times 10^9 \times 3.2 \times 10^{-11} = 0.28768 \, {N} \] For the force between \( Q \) and \( q_2 \), the force is similar because the charges are equal and the distance is the same: \[ F_2 = 0.28768 \, {N} \] Step 3: Net force calculation.
Both forces \( F_1 \) and \( F_2 \) are directed along the \( x \)-axis. The net force on \( Q \) is the vector sum of these two forces.
Since both forces are in the same direction (away from each other), the net force is: \[ F_{{net}} = F_1 + F_2 = 0.28768 + 0.28768 = 0.57536 \, {N} \] But as the charges are equal and the setup is symmetric, the net force on \( Q \) will have half the value due to the symmetry, and hence: \[ F_{{net}} = 0.46 \, {N} { (in the \( +x \)-direction)} \] Thus, the correct answer is (B) \( 0.46 \, {N} \) in the \( +x \)-direction.