  Two infinite grounded parallel conducting planes are separated by a distance d. A point charge q is placed between the planes. UseGreen's reciprocation theorem to prove that the total induced charge on one of the planes is equal to -q times the fractional perpendiculardistance of the point charge from the other plane. (Hint: As your comparison electrostatic problem with the same surfaces choose one|whose charge densities and potential are known and simple.)

Question

3 help_outlineImage TranscriptioncloseTwo infinite grounded parallel conducting planes are separated by a distance d. A point charge q is placed between the planes. Use Green's reciprocation theorem to prove that the total induced charge on one of the planes is equal to -q times the fractional perpendicular distance of the point charge from the other plane. (Hint: As your comparison electrostatic problem with the same surfaces choose one |whose charge densities and potential are known and simple.) fullscreen
Step 1

Write the Green’s reciprocation theorem as follows: help_outlineImage Transcriptionclosedx'da= , p«Dd°x + [ o{Dda (1) Here p is the volume charge density, V is the volume bounded, S is the conducting surface area Dis the potential due to volume charge density, o is the surface charge density, Dis the potential due to another charge distribution, and p', o' are another charge distribution parameters fullscreen
Step 2 help_outlineImage Transcriptioncloseand the point charge at 2 Consider top plane at z = bottom plane at z = -- 2 (-0y-02-) d x 0, y 0, z a Here, a is the distance between the top plane and the point charge. Consider that, there is no charge density inside the volume. Therefore, p' = 0 Consider that, there is a uniform charge +o on the lower plane and a uniform charge on the upper plane. 0 fullscreen
Step 3

Use the symmetrical property, physical considerations, and write the electr...

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