Four point charge of $+{10}^{-7}\mathrm{C},-{10}^{-7}\mathrm{C}\text{, }$ $-2\times {10}^{-7}\mathrm{C}$ and $+2\times {10}^{-7}\mathrm{C}$ are placed respectively at the corners A, B, C, D of a 0.05m square. Find the magnitude of the resultant force on the charge at D.

To find the magnitude of the resultant force on the charge at point D due to the other charges at points A, B, and C, we proceed as follows:

Step 1: Identify the Charges and Their Positions

The charges are located at the corners of a square with side length 0.05 m:

• Charge at A: q_A = +10^-7 C
• Charge at B: q_B = -10^-7 C
• Charge at C: q_C = -2 × 10^-7 C
• Charge at D: q_D = +2 × 10^-7 C

Step 2: Calculate the Force on Charge D

The force between two charges is given by Coulomb's Law:

F = k × |q₁ × q₂| / r²

Where:

• k = 9 × 10⁹ N·m²/C² (Coulomb's constant)
• r is the distance between the charges

Force Due to Charge at A

The distance between A and D is 0.05 m:

F_AD = (9 × 10⁹ × |10^-7 × 2 × 10^-7|) / (0.05)² 
F_AD = 0.072 N

Direction: Away from A (opposite charges attract).

Force Due to Charge at B

The distance between B and D is also 0.05 m:

F_BD = (9 × 10⁹ × |(-10^-7) × (2 × 10^-7)|) / (0.05)² 
F_BD = 0.072 N

Direction: Towards B (opposite charges attract).

Force Due to Charge at C

The distance between C and D is 0.05 m:

F_CD = (9 × 10⁹ × |(-2 × 10^-7) × (2 × 10^-7)|) / (0.05)² 
F_CD = 0.144 N

Direction: Towards C (opposite charges attract).

Step 3: Combine the Forces

Net Force Between A and C

The forces between A and D, and C and D, act along the same line but in opposite directions:

F_net,AC = F_AD - F_CD 
F_net,AC = 0.072 - 0.144 
F_net,AC = -0.072 N (towards C)

Resultant Force

The force due to B is perpendicular to the net force between A and C. Using the Pythagorean theorem:

F_resultant = √(F_net,AC² + F_BD²) 
F_resultant = √((-0.072)² + (0.072)²) 
F_resultant = √(2 × (0.072)²) 
F_resultant = 0.072 √2 ≈ 0.102 N

Final Answer

The magnitude of the resultant force on the charge at D is approximately 0.102 N.