Question

Two infinite planes each with uniform surface charge density to are kept in such a way that the angle between them is 30^o. The electric field in the region shown between them is given by :

• A. \( {\sigma \over {{ \in _0}}}\left[ {\left( {1 + {{\sqrt 3 } \over 2}} \right)\widehat y + {{\widehat x} \over 2}} \right] \)
• B. \( {\sigma \over {2{ \in _0}}}\left[ {\left( {1 + \sqrt 3 } \right)\widehat y + {{\widehat x} \over 2}} \right] \)
• C. \( {\sigma \over {2{ \in _0}}}\left[ {\left( {1 + \sqrt 3 } \right)\widehat y - {{\widehat x} \over 2}} \right] \)
• D. \( {\sigma \over {2{ \in _0}}}\left[ {\left( {1 - {{\sqrt 3 } \over 2}} \right)\widehat y - {{\widehat x} \over 2}} \right] \)

Hint:

Each infinite sheet gives a field of magnitude \(\sigma/2\epsilon_0\) pointing away from it; add the two vectors in the wedge region.

Answer 1

The Correct Option is D

The horizontal plane lies along the x-axis; in the wedge region (above it) its field is \(\dfrac{\sigma}{2\epsilon_0}\hat y\).
The inclined plane makes 30° with the x-axis, its direction pointing up-left at 150° from +x. Its unit normal pointing into the wedge (below the plane, away from it) is \(\hat n_2=(-\sin30^\circ,-\cos30^\circ)=\left(-\tfrac12,-\tfrac{\sqrt3}{2}\right)\). Field \(=\dfrac{\sigma}{2\epsilon_0}\left(-\tfrac12\hat x-\tfrac{\sqrt3}{2}\hat y\right)\).
Sum: \(\vec E=\dfrac{\sigma}{2\epsilon_0}\left[\left(1-\tfrac{\sqrt3}{2}\right)\hat y-\tfrac{\hat x}{2}\right]\).
Answer: (D)