Exercise 3: Approximating Electric Field of Electric Dipoles using Binomial Series

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In this video, I use the binomial series approximation to demonstrate that the electric field (E) of an electric dipole (two equal but opposite charges separated by a small distance) is proportional to 1 over the distance cubed when measured from a far distance. Normally, the electric field (E) of a single charge is proportional to the charge divided by the distance squared. However, for a dipole, we must subtract the electric field of one charge from the electric field of the other. By applying the binomial series to rewrite E and considering the limit at large distances, higher-order terms (second-order and beyond) approach zero. This simplification yields the approximate result: E ≈ 2dq/D³, where D is the distance from the dipole, d is the separation between the charges, and q is the magnitude of the charge.

Timestamps:

  • Exercise 3: Electric Dipoles: 0:00
  • Solution: Rewrite Electric Field equation to get d/D term: 1:38
  • Binomial series approximation for d/D term: 4:24
  • Plug in binomial series into electric field equation: 8:25
  • Electric field of a dipole is proportional to 1/D^3 when D is much greater than d: 10:58
    • E ≈ 2dq/D^3: 12:31

Full video, notes, and playlists:


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