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Homework Statement
This is regarding setting up an integral to calculate the electric field on a point charge that is at a distance "a" from a uniformly charged rod of length "L". I have attached a picture of my work, which includes a diagram of the problem, and wanted to know if my thought process is correct.
Homework Equations
The Attempt at a Solution
Finding a general expression for the electric field produced by an infinitely small "piece" of charge and then adding them all up. The charged rod is parallel to the yaxis and it's center is at the origin.
We know that the linear charge density is
λ=(Q/L).
For an infinitely small piece of charge, we say λ=(dQ/dL), so dQ=λ*dL
The formula for the Electric field on a point charge is E=q/((4∏ε)*r^2)
To find an expression for the Electric field produced by the small piece of charge, we can replace q with dQ
dE=dQ/((4∏ε)*r^2)
(substitue λ*dL for dQ)
dE=λ*dL / ((4∏ε)*r^2)
The part that I am unsure about is finding an expression for the "r". Would it be reasonable to say that "r = (a+y)", where y is the variable that I am integrating, and my limits of integration would be [(L/2),(L/2)]? The reason I am saying (a+y) is because if I plug "L/2" in for y, then (a(L/2)) is the distance from the point charge "a" to the end of the rod that is above the origin. If I plug in "L/2" for y, then (a+(L/2)) takes care of the distance from the point charge "a" to the end of the rod below the origin. If I plug in 0 for y, then I simply get "a" which makes sense, since that is the distance from the origin to the point charge.
I hope my post was formatted correctly, please let me know if It is not, and I will be sure to make changes in the future
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