A point charge q is located at the origin. It lies between (but not necessarily

Question

A point charge q is located at the origin. It lies between (but not necessarily midway between) two grounded conducting planes that are separated by a distance d and are infinite in extent. The grounded plane on the left of the point charge is at x = -b and is perpendicular to the x axis. The grounded plate on the right of the point charge is at x = +a and is also perpendicular to the x axis, so that d = a + b. a) Find the locations of the infinite number of image charges that are needed. Give separate expressions for the locations of all the positive charges and for the locations of all the negative charges, and identify the pattern that the charges form. b) Express the force F_x on the charge q as an infinite series. Write this series for F_x as a function of the dimensionless ratio u = a/d.

Explanation / Answer

part a:

image charges will be needed as below:

-q at x=-2*b,x=2*a,x=-2*a-4*b,x=4*a+2*b and so on

q at x=-2*b-2*a, x=2*a+2*b ,x=4*a+4*b, x=-4*a-4*b and so on

pattern is as follows:

negative charges required at x=-2*(k*b+(k-1)*a), 2*((k-1)*b+k*a), where k=1,2,3...

positive charges required at x=-2*k*(a+b),2*k*(a+b) where k=1,2,3....

part b:

positive charges which are placed at equal distance on either side will cancel each other's force on charge q.

force due to negative charges being placed on -ve x axis will be along -ve x axis and force due to charges on +ve x axis will be along +ve x axis

total force=(q/(4*pi*espilon))*( ((1/(2*b)^2) + (1/(2*a+4*b)^2) + ...) - ((1/(2*a)^2)+(1/(2*b+4*a)^2) + ...) )

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