# Find the electric potential at point P due to the dipole shown. a) (10 pts) Give your answer in terms of q, k,?

Find the electric potential at point P due to the dipole shown.

a) (10 pts) Give your answer in terms of q, k, a, and b. (d, r, and θ don’t matter here)

b) Prove that if r >> d, (is much greater than), the electric potential at P can be accurately approximated by: V = kq(dcosθ/r^2).

Hint: in the second situation a ≈ b ≈ r ; therefore a and b can also be considered parallel.

Relevance

a) This part is easy, just write down the potential...

V = k q [1/a-1/b]

b) Notice that the extra distance from the negative charge to point P can be written as d cosθ when compared to the distance from the positive charge to point P, so:

V = k q [1/a-1/b] ≈ k q [1/r-1/(r+d cosθ)]

This last step uses a ≈ r and since a and b are now approximately parallel, we only needed to add the extra distance dcosθ to write b. Now I will write this in a more suggestive way:

V ≈ k q [1/r-1/r 1/(1+d/r cosθ)]

Since d << r we can Taylor expand about x = 0, where x = d/r cosθ. Recall:

1/(1+x) = 1 - x + ... for |x| < 1.

V ≈ k q [1/r-1/r (1 - d/r cosθ)] = k q [d cosθ/r²]

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