The figure shows five point-charges on a straight line. Separation between successive charges is 10 cm. For what values of 1 q and 2 q would the net force on each of the other three charges be zero?

To ensure the net force on each of the three middle charges is zero, the charges \( q_1 \) and \( q_2 \) must be chosen such that the forces due to adjacent charges cancel out. This involves balancing the Coulomb forces between all the charges on the line. Let's go through the steps:

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### Given:
1. Charges are arranged in a straight line.
2. Separation between successive charges = \( 10 \, \text{cm} \).
3. Charges: \( q_1, q_2, q_3, q_4, q_5 \).

### Step-by-Step Approach:
#### 1. Write the net force on each charge
The net force on any charge is due to the Coulomb force from all other charges, given by:

\[
F = k \frac{|q_i \cdot q_j|}{r^2},
\]

where:
- \( F \) = Coulomb force,
- \( k = 9 \times 10^9 \, \text{N·m}^2/\text{C}^2 \) (Coulomb constant),
- \( q_i, q_j \) = interacting charges,
- \( r \) = distance between the charges.

#### 2. Conditions for equilibrium
For the net force on charges \( q_2, q_3, q_4 \) to be zero, the attractive and repulsive forces from neighboring charges must cancel. 

For simplicity, assume:
- \( q_1 = q \),
- \( q_2 = 2q \),
- \( q_3 = Q \) (middle charge),
- \( q_4 = 2q \),
- \( q_5 = q \).

#### 3. Symmetry-based simplification
The forces are symmetrical, so \( q_2 \) and \( q_4 \) must balance \( q_3 \), while \( q_1 \) and \( q_5 \) contribute to \( q_2 \) and \( q_4 \).

#### 4. Solve the system of equations
The force equilibrium condition must be solved for \( q \) and \( 2q \).

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Would you like a detailed calculation of these forces and values for \( q \) and \( 2q \)? 

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### Relative Questions:
1. How is the Coulomb force affected by distance between charges?
2. What happens if the charges are not arranged symmetrically?
3. How does changing the magnitude of one charge influence the net force on the others?
4. Can the net force on all charges simultaneously be zero in this setup?
5. What role does the medium between charges (vacuum vs. dielectric) play in this problem?

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### Tip:
Always check symmetry when solving electrostatics problems—it simplifies calculations significantly!

The figure shows five point-charges on a straight line. Separation between successive charges is 10 cm. For what values of q and 2q would the net force on each of the other three charges be zero?

Learn how to determine the values of q and 2q for a system of five point-charges aligned on a straight line, with forces balanced such that the net force on the middle charges is zero. This solution applies Coulomb's Law and the Superposition Principle, suitable for students in Grades 11-12.

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