Question

Two charges \(q_1\) and \(q_2\) are separated by a distance of 30cm. A third charge \(q_3\) initially at point C is shown in figure, is moved along the circular path of radius 40 cm from C to D. If the difference in potential energy due to movement of \(q_3\) from C to D is \(q_3k/4πε_0\), the value of K is (\(ε_0\) = permittivity of free space)

• A. \(8q_2\)
• B. \(8q_1\)
• C. \(6q_2\)
• D. \(6q_1\)

Hint:

Only q2 changes its distance to q3; q1 stays at a fixed 40 cm because the path is a circle about q1.

Answer 1

The Correct Option is A

Step 1: Read the figure
Charge \(q_1\) is at \(A\) and the path of \(q_3\) is a circular arc of radius \(40\ \text{cm}\) about \(A\). So the distance to \(q_1\) stays \(0.4\ \text{m}\), and the energy due to \(q_1\) does not change.

Step 2: Distances from q2
\(q_2\) is at \(B\), \(30\ \text{cm}\) from \(A\) along \(AD\). At \(C\) the distance \(BC = \sqrt{0.3^2+0.4^2} = 0.5\ \text{m}\). At \(D\) the distance \(BD = 0.4 - 0.3 = 0.1\ \text{m}\).

Step 3: Energy difference
\[ U_D - U_C = \frac{q_3q_2}{4\pi\varepsilon_0}\left(\frac1{0.1} - \frac1{0.5}\right) = \frac{q_3}{4\pi\varepsilon_0}(10-2)q_2 \]

Step 4: Result
Comparing with \(\frac{q_3K}{4\pi\varepsilon_0}\), \(K = 8q_2\) (with distances in metres). Option (A).

Final Answer:
The value of K is 8 q2. \[ \boxed{\text{(A)}\ K=8q_2} \]