Explanation Formula used: The required diagram for the infinite parallel grounded plates is shown in Figure 1 Figure 1 The case when the average charge is on the left side of the plate the at a distance of ( a − x ) then the image charge − q will be at a distance − ( a − x ) and by the method of induction all the charges are distributed on the infinite plane. The expression to determine the value of the force is given by, F = k q 2 r 2 Here, q is the charge and, k is the proportionality constant and r is the distance. Calculation: The positive charge image charge force cancels in pairs and the net force is evaluated as, F = q 2 4 π ε 0 { 1 [ 2 ( a − x ) ] 2 + 1 [ 2 a + 2 ( a − x ) ] 2 + 1 [ 4 a + 2 ( a − x ) ] 2 + .... − [ 1 ( 2 x ) 2 − 1 ( 2 a + 2 x ) 2 − 1 ( 4 a + 2 x ) 2 − ............ ] } = q 2 4 π ε 0 { 1 4 [ ( a − x ) ] 2 + 1 4 [ a + ( a − x ) ] 2 + 1 4 [ 2 a + ( a − x ) ] 2 + .... − [ 1 4 x 2 − 1 4 ( a + x ) 2 − 1 4 ( 2 a + x ) 2 − ............ ] } = q 2 16 π ε 0 { 1 [ ( a − x ) ] 2 + 1 [ a + ( a − x ) ] 2 + 1 [ 2 a + ( a − x ) ] 2 + .... − [ 1 x 2 − 1 ( a + x ) 2 − 1 ( 2 a + x ) 2 − ............ ] } The above equation for the condition when a ≫ x is evaluated as, F = q 2 16 π ε 0 { 1 a 2 + 1 4 a 2 + 1 9 a 2 + .... − [ 1 x 2 + 1 a 2 + 1 2 a 2 + ....... ] } The above equation when a → ∞ is derived as, F = q 2 16 π ε 0 { 1 [ ( a − x ) ] 2 + 1 [ a + ( a − x ) ] 2 + 1 [ 2 a + ( a − x ) ] 2 +

4th Edition

ISBN: 9781108420419

Author: David J. Griffiths

Publisher: Cambridge University Press

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Electric Charges and Fields

One of the fundamental properties is the electromagnetic property. Charge cannot be destroyed by any process and this contributes formally to the law of charge conservation.

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Chapter 3.4, Problem 3.39P

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The force on q , also check the answer for the special cases when a→∞ and x=a2 .

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Chapter 3.4, Problem 3.38P

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The force on q , also check the answer for the special cases when a → ∞ and x = a 2 .

Solution Summary: The author explains how the force between the infinite parallel conductors is 0.

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The force on q , also check the answer for the special cases when a→∞ and x=a2 .

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