September 19, 2017

Infinite Number of Charges Problem

Let's discuss a problem on Infinite Number of Charges from Physics Olympiad. Try the problem first, and then read the solution here.

The Problem: Infinite Number of Charges

An infinite number of charges, each equal to (q), are placed along the x-axis at (x=1),(x=2),(x=4),(x=8) etc. Find the potential and electric field at the point x=0 due to the set of the charges.

Discussion:

An infinite number of charges, each equal to (q), are placed along the x-axis at (x=1),(x=2),(x=4),(x=8) etc.

Electric potential $$ V=\frac{1}{4\pi\epsilon_0}(\frac{q}{1}+\frac{q}{2}+\frac{q}{4}+\frac{q}{8}+...)$$ $$=\frac{1}{4\pi\epsilon_0}(1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....)$$
The terms in brackets form a geometric progression of infinite terms whose sum is $$ S= \frac{a}{1-r}=\frac{1}{1-1/2}=2$$

Hence the potential $$ V=q/2\pi\epsilon_0$$

Some useful links:

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com