Red Rod

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two thin rods of equal length L carry equal charges Q?

two thin rods of equal length L carry equal charges Q. The linear charge density for blue rod on the left is uniformly distributed over its length; however, the linear charge density is non-uniform on the red rod on the right and is given by expresion (landa)(x)=C x (coulumbs/meter) where C is a constant. The two rods are aligned and their nearest ends are separated by a distance D.

a) determine the constant C in the charge distribution for the red rod on the right.

b) find the net force the blue rod on the left exerts on the red rod on the right.

Simple calculus:

a) total charge is the integral of charge per unit length over the length of the rod.

Q = integral, from x = 0 to x = L, of Cx dx

b) total force on either rod is the same (Newton's third law).

For the force exerted by a small length dx of one rod on the other rod, you have to integrate:

dF(x) = dQ(x) (integral from y = 0 to y = L, of dQ(y)/r(x,y)^2)

where r is the distance from point x on one rod to point y on the other.

The total force is the integral of dF(x) over x, which is the double integral over dx dy.

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