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A semi-circular arc of radius 'a' is charged uniformly and the charge per unit length is λ. The electric field at the centre of this arc is

1. \( \frac{\lambda}{2 \pi \epsilon_0 a} \)
2. \( \frac{\lambda}{2 \pi \epsilon_0 a^2} \)
3. \( \frac{\lambda}{4 \pi \epsilon_0 a} \)
4. \( \frac{\lambda}{2 \pi \epsilon_0 a} \)

Correct answer: \( \frac{\lambda}{2 \pi \epsilon_0 a} \)

Solution

The electric field at the center of a uniformly charged semi-circular arc is calculated by integrating the contributions of all charge elements. The vertical components cancel due to symmetry, leaving only the horizontal components. The result is \( \frac{\lambda}{2 \pi \epsilon_0 a} \).

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