Can you please help me with this problem. I need a full explanation and steps of

Question

Can you please help me with this problem.

I need a full explanation and steps of how you did it

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Thanks.

2. Two protons and an electron are aligned on the x axis as shown in the right figure. Distance between two protons is -O 2d, and the electron is located at the distance, r, from the middle point between those two protons. (10 points) a. Find the net electrostatic force on the electron as a function ofx b. For an electron far from those protons (r >»> d), show the net electrostatic force on the electron is

Explanation / Answer

given two protons (p) and electron (e) lined on the x axis
distance between protons = 2d
distance of electron from the middle point of protons = r

a) Net force on electrons F = ke^2/(r - d)^2 + ke^2/(r + d)^2
   here e is the charge on a proton
   F = ke^2[1/(r - d)^2 + 1/(r + d)^2]
   F = ke^2[1/(r/d - 1)^2 + 1/(r/d + 1)^2]/d^2

b) now, if r >> d
   then r/d > > 1
   r^4/d^4 > > 1

   F = ke^2[1/(r/d - 1)^2 + 1/(r/d + 1)^2]/d^2 = ke^2[(r/d + 1)^2 + (r/d - 1)^2]/d^2*(r^2/d^2 - 1)^2
   F = ke^2[r^2/d^2 + 1 + 2r/d + r^2/d^2 + 1 - 2r/d]/d^2*(r^2/d^2 - 1)^2
   F = 2ke^2[r^2/d^2 + 1]/d^2*(r^4/d^4 + 1 - 2r^2/d^2)
   F = 2ke^2/(r^2 - 2d^2) = e^2/2*pi*epsilon( r^2 - 2d^2)

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