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

(A)
(B)
(C)
(D)

Option 1 - <math ><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>σ</mi><mrow><mn>2</mn><msub><mi>ϵ</mi><mn>0</mn></msub></mrow></mfrac></mstyle><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msqrt><mn>3</mn></msqrt><mo stretchy="false">)</mo><mover accent="true"><mi>y</mi><mo>^</mo></mover><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mover accent="true"><mi>x</mi><mo>^</mo></mover><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\dfrac{\sigma}{2\epsilon_0} \left[(1 - \sqrt{3}) \hat{y} - \tfrac{1}{2} \hat{x}\right]</annotation></semantics></math>2ϵ0σ[(1−3)y^−21x^]

Option 2 - <math ><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>σ</mi><mrow><mn>2</mn><msub><mi>ϵ</mi><mn>0</mn></msub></mrow></mfrac></mstyle><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msqrt><mn>3</mn></msqrt><mo stretchy="false">)</mo><mover accent="true"><mi>y</mi><mo>^</mo></mover><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mover accent="true"><mi>x</mi><mo>^</mo></mover><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\dfrac{\sigma}{2\epsilon_0} \left[(1 + \sqrt{3}) \hat{y} - \tfrac{1}{2} \hat{x}\right]</annotation></semantics></math>2ϵ0σ[(1+3)y^−21x^]

Option 3 - <math ><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>σ</mi><mrow><mn>2</mn><msub><mi>ϵ</mi><mn>0</mn></msub></mrow></mfrac></mstyle><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msqrt><mn>3</mn></msqrt><mo stretchy="false">)</mo><mover accent="true"><mi>y</mi><mo>^</mo></mover><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mover accent="true"><mi>x</mi><mo>^</mo></mover><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\dfrac{\sigma}{2\epsilon_0} \left[(1 + \sqrt{3}) \hat{y} + \tfrac{1}{2} \hat{x}\right]</annotation></semantics></math>2ϵ0σ[(1+3)y^+21x^]

Option 4 - <math ><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>σ</mi><mrow><mn>2</mn><msub><mi>ϵ</mi><mn>0</mn></msub></mrow></mfrac></mstyle><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msqrt><mn>3</mn></msqrt><mo stretchy="false">)</mo><mover accent="true"><mi>y</mi><mo>^</mo></mover><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mover accent="true"><mi>x</mi><mo>^</mo></mover><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\dfrac{\sigma}{2\epsilon_0} \left[(1 + \sqrt{3}) \hat{y} + \tfrac{1}{2} \hat{x}\right]</annotation></semantics></math>2ϵ0σ[(1+3)y^+21x^]

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Answered by

Vishal Baghel | Contributor-Level 10

7 months ago

Correct Option - 3

Detailed Solution:

$\frac {I_0} {10} = I_0 \cos^2 \theta$ $\cos \theta = \frac {1} {\sqrt {10} = 0.31 < \frac {1} {\sqrt {2} \quad \text {which is 0.707}$