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Solved Problem-6

Problem-6
(a) A point charge ‘Q’ is located at a distance ‘h’ from a perfect conducting infinite plane. Obtain the image charge and show that the induced charge on the conducting plane is ‘-Q’.
 
(b) An infinitely long line charge of uniform line charge density ‘λ’ is situated parallel to and at a distance ‘x’ from the grounded infinite plane conductor. Obtain the image charge and show that the induced surface charge on the conductor per unit length is ‘-λ’.
 
Solution
(a) The image charge -Q is placed at a distance h in the conducting region and the conducting grounded plane is replaced by an equipotential surface of zero potential.
 
For the charge Q at (0, 0, h) and its image Q’ at (0, 0, -h), the potential at any point
P(x, y, z) is,
Description: 168340.png
 
At z = 0, i.e., on the conducting plane, V = 0.
Description: 168333.png
 
Thus, we see that the magnitude of the image charge is -Q.
 
Hence, the potential at any point P(x, y, z) is,
 
Description: 168325.png
 
The electric field at any point P(x, y, z) can be found out by using the relation Description: 168318.png or may also be written as,
 
Description: 168309.png
 
The electric field component Ez on the xy plane, i.e., on the conductor surface is,
Description: 168302.png
 
The parallel field components Ex and Ey vanish as they should.
 
The induced surface charge density is given by σ = εEz. The total induced surface charge is given by,
Description: 168293.png
Description: 214327.png
 
(b) The finite line charge λ is assumed at x = 0, z = h and the image λ’ is assumed at x = 0, z = -h, so that the two are parallel to y-axis.
 
The field at point P (x, y, z) is,
Description: 168286.png
Here,
Description: 168275.png
 
Description: 168267.png
 
Potential at P is,
Description: 168257.png
 
At the conducting plane, V = 0 ⇒ λ’ = -λ (with z = 0)
 
Description: 168250.png
 
Description: 168243.png
 
The surface charge induced on the conducting plane is,
 
Description: 168235.png
 
The induced charge per unit length on the conducting plane is,
Description: 168228.png
 
Description: 168219.png
 





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