Question 4.8: Show that the interaction energy of two dipoles separated by...

Introduction to Electrodynamics – Solution Manuals [EXP-2863]

Show that the interaction energy of two dipoles separated by a dis placement r is

$U=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}\left[ p _{1} \cdot p _{2}-3\left( p _{1} \cdot \hat{ r }\right)\left( p _{2} \cdot \hat{ r }\right)\right]$ (4.7)

[Hint: Use Prob. 4.7 and Eq. 3.104.]

$E _{\text {dip }}( r )=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}[3( p \cdot \hat{ r }) \hat{ r }- p ]$ (3.104)

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U=- p _{ 1 } \cdot E _{ 2 }, \text { but } E _{ 2 }=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}\left[3\left( p _{ 2 } \cdot \hat{ r }\right) \hat{ r }- p _{2}\right] . \text { So } U=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}\left[ p _{ 1 } \cdot p _{ 2 }-3\left( p _{ 1 } \cdot \hat { \mathbf { r } } \right)\left( p _{ 2 } \cdot \hat{ r }\right)\right]

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