Positive charge +Q is distributed uniformly along the +x-axis from x = 0 to x =

Question

Positive charge +Q is distributed uniformly along the +x-axis from x = 0 to x = a. Negative charge - Q is distributed uniformly along the -x-axis from x = 0 to x = - a.A positive point charge q lies on the positive y-axis, a distance y from the origin. Find the force (magnitude and direction) that the positive and negative charge distributions together exert on q. Show that this force is proportional to y^{ - 3} for y gg a. Essay answers are limited to about 500 words (3800 characters maximum, including spaces). 3800 Character(s) remaining

Explanation / Answer

From Give, after dividing into components,

Fx = 2F*Cosm, where m is the angle made by the Force with respect to the x-Axis.

Fy = 0, No force along the y-direction.

Cosm = x/Sqrt(x^2+y^2)

So, F = KqPdx/(x^2 + y^2)

Now, dFx = 2KqPdx/[(x^2 + y^2) Sqrt(x^2 + y^2)]

Now, integrate it from limits 0 to r ( later substitute 'a' for 'r')

we, get,

Fx = 2KqQ Integral( 0 to r) [dx/(x^2 + y^2 )^3/2

=> Fx = [(x^2 + y^2)^-1/2]/-1 from o to r

=>Fx = 2KqQ [1/ry - 1/rSqrt(y^2 + r^2]

Now, we take the a is ver smaller compareed to y,

=>Fx = 2KqQ/a * [1/y^3]

Hence Proved.

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