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

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:

- Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0F76sIU8xm09oqBTq1mlry3
- HIVE notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-applications-of-taylor-polynomials
- Infinite Sequences and Series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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