Question Detail
Question
An infinite number of charges, each equal to q, are placed along the x-axis at x = 1, x = 2, x = 4, x = 8, and so on. Find the potential and the electric field at the point x = 0 due to this set of charges.
Correct Answer
Option A is the correct answer.
Detailed Explanation
The charges are placed at distances r = 1, 2, 4, 8, ... which are powers of 2 (20, 21, 22, ...).1. Potential (V) at x=0: V = Σ (kq/ri) = kq Σ (1/2n) for n=0 to ∞. This is a geometric series with a=1 and r=1/2. The sum is a/(1-r) = 1/(1-1/2) = 2. So, V = 2kq = 2q/(4πε₀).2. Electric Field (E) at x=0: E = Σ (kq/ri2) = kq Σ (1/(2n)2) = kq Σ (1/4n) for n=0 to ∞. This is a geometric series with a=1 and r=1/4. The sum is a/(1-r) = 1/(1-1/4) = 4/3. So, E = kq(4/3) = (4/3)q/(4πε₀) = 4q/(12πε₀). Both results match option A.
Hint
Recognize the pattern of charge placement as a geometric series. Calculate potential using 1/r and electric field using 1/r2, then sum the series.