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Homework Statement
(Taken from Serway and Jewett Chapter 23, Q44, 9th Edition)
A thin rod of length ## l ## and uniform charge per unit length ##λ## lies along the x axis as shown in the image attached.
(a) Show that the electric field at P, a distance d from the rod along its perpendicular bisector, has no x component and is given by ## E=2k_eλ sinθ_0/d.##
b) What If? Using your result to part (a), show that the field of a rod of infinite length is ##E=2k_eλ/d.##
Homework Equations
##dq=λ dl##
##\vec E=k_e∫ \frac {dq}{r^2} \hat {\mathbf r} ##
Pythagoras i.e. ## r^2 = d^2 + (l/2)^2.##
The Attempt at a Solution
Starting out with replacing the infinitesimal ##dq## in integral with ## λ dl## to give an expression of ##\vec E=k_e λ ∫ \frac {dl} {r^2}\hat {\mathbf r}## (I took the ##λ## outside the integral since it is uniform and thus I assumed it was a constant). From there I could only find one way to continue.
That way involved using Pythagoras saying that ##r^2 = {d^2 + \frac{l^2} {4}} ## which I then put back into the integral but found that I could not make any more progress with this. (Maybe some standard integral would work here but I do not know it).
Where I stopped:
##\vec E= k_e λ ∫ \frac {dl}{ {d^2} +\frac {l^2}{4}}\hat {\mathbf r}##.
What I would like to know is how to get rid of the vector by turning it into a scalar and ultimately make progress with this question.
For part (b) I noticed that ##sinθ## disappears and becomes 1. Thus ##θ## must become ##90°## to allow this to happen. However, I struggle to deduce the other reasoning necessary for reaching the answer. Thus I would like help with this.
I thank you for any help offered.
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