A thin rod of length and uniform charge per unit length ? lies along the x axis

Question

A thin rod of length and uniform charge per unit length ? lies along the x axis as shown in the figure below.

http://www.webassign.net/serpop/p19-22.gif

(a) Determine the magnitude of the components of the electric field at P, a distance y from the rod along its perpendicular bisector. Suggestion: First, calculate the field at P due to an element of length dx, which has a charge ?dx. Then, change variables from x to ?, using the relationships x = y tan (?) and dx = y sec2 ? d ?, and integrate over ?. (Use k_e for ke, lambda for ?, y for y, and theta_0 for ?0 as necessary.)

x-component
y-component

(b) Using your result to part (a), determine the components of the electric field of a rod of infinite length.
x-component
y-component

Explanation / Answer

(A) Since P is on the perpendicular bisector, there will be no x-component; all of the Ex component coming from one side of P will be offset by an opposite component on the other side of P.

So we only need the y component. If we calculate dE from dx as suggested, the y component will be dE cos , and we will need to integrate from -0 to 0.

dE = k dq/r2 = k dx / r2 = ky sec2 d / y2 sec2   where r = y sec

= k d/y

which we must multiply by cos to get the y component, then integrate. The integral of cos is -sin , so the resulting y-component is:

(k/y)(-sin 0 + sin(-0)) = 2k sin 0/y

and if you want to express in terms of l and y instead of 0 and y, you can see that

sin 0 = (l/2)/(y2 + l2/4)½ and substitute that in.

(B) for an infinite line the angle 0 goes to 90 degrees, its sine is 1, so the x-component is still 0 and the y component is just 2k/y.

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