A uniform line charge of length L = 3.4 cm and total charge Q= 2.1 ?C lies on th

Question

A uniform line charge of length L = 3.4 cm and total charge Q= 2.1 ?C lies on the y-axis, centered on the x-axis (so half the line lies above the x-axis and half below). A small spherical charge, of equal charge Q, lies on the x-axis at x = L.

What is the electric force on the spherical charge due to the line charge?

Give your answer in Newtons, to three significant figures. Do not include units in your answer.

Remember: 1 ?C = 10-6 C and 1 cm = 10-2 m

A uniform line charge of length L = 3. 4 cm and total charge Q= 2. 1 ?C lies on the y-axis, centered on the x-axis (so half the line lies above the x-axis and half below). A small spherical charge, of equal charge Q, lies on the x-axis at x = L. What is the electric force on the spherical charge due to the line charge? Give your answer in Newtons, to three significant figures. Do not include units in your answer. Remember: 1 ?C = 10-6 C and 1 cm = 10-2 m

Explanation / Answer

When setting up the integral, take advantage of the symmetry. Consider two pieces of the line, one on the left and one on the right. When these pieces are the same distance from the center of the line the fields they set up are equal in magnitude. The vertical components cancel, and the horizontal components add. The net field has no horizontal component, so in the integral we just need to sum all the horizontal components.

The charge Q is spread uniformly over the line, which has length L. There is therefore a constant charge per unit length l which is:

l = Q/L

If a small piece of the line has a width dy, the charge on it is:

dq = l dy

The field this piece sets up at the point is:

dE = k dq / r2, where r2 = d2 + y2

Therefore dE = k l dy / (d2 + y2)

We just need the horizontal component, so we multiply by
cos(q) = d/r = d/(d2 + y2)1/2

dEx = k l d dy / (d2 + y2)3/2

Summing over the whole line gives:

Ex =

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