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How to set up an integral to calculate electric field:
For a given charge distribution, the total electric field at a point is the vector sum of the
fields at this point due to each point charge in the charge distribution. If the charges are
continuously distributed along a line, over a surface, or through a volume, i.e. the charges
cannot be considered as discrete point charges, it requires to integrate over the charge
distribution to calculate the total electric field. Here is the general guideline how hot set up
an integral for this type of problems.
1. Make a drawing showing the distribution of the charges and your choice of coordinate
system.
2. Divide the charge distribution into infinitesimal segments. Try to use symmetry and do
it in a way such that one of the electric field components can be cancelled.
3. Calculate the electric field due to an infinitesimal segment and express the electric
field in terms of variables so that the expression can be generalize to all segments.
4. Construct an integral based on this expression and integrate over the whole charge
distribution.
Calculate the integral to find the total electric field.
Positive charge Q is distributed uniformly along the x-axis from 𝑥 = 0 to 𝑥 = 𝑎. A
positive point charge q is located on the positive x-axis at 𝑥 = 𝑎 + 𝑟, a distance r to the
right of the end of Q.
(a) Calculate the electric field produced by the charge distribution Q at the point 𝑥 =
𝑎 + 𝑟.
(b) Calculate the force (magnitude and direction) that the charge distribution Q exerts
on q.
(c) If 𝑟 ≫ 𝑎, what is approximately the magnitude of the force?