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PH213 Written Homework 1 1 A Strange Dipole Consider the distribution of charges pictured below. In this problem we will calculate the electric field of this distribution on the positive z-axis and approximate its behavior far from the origin. -24 (a) Calculate [10pts] Use superposition to find an exact expression for the electric field at an arbitrary point on the z-axis with z > 2. (b) Simplify [5pts] Pull out a factor from your expression from (a). Your answer should now look like: kq (...) E = There should be three terms in the parenthesis. 1

(c) Approximate [10pts] We will now approximate our answer in the limit that z>> or equivalently<< 1. Consider the binomial approximation, that for || << 1: ≈ (1-nx) 1 (1+x)" Apply this approximation to your answer from (b) and simplify. (d) Unit Check [5pts] Make sense of your answer by checking that the units are correct. (e) Dipole Moment [5pts] In class we found that the electric field (on the z-axis) of a dipole has the form: Ē= 2kp 23 What would you need to plug in for the dipole moment, p, to get this to match your answer from (c)? (f) Dipole Calculation [5pts] In general, the dipole moment of a distribution of point charges can be calculated as: p= Calculate the dipole moment of this charge distribution and make sure it matches your answer from (e). 2