Question

A 1C charge is placed at the origin. Other infinite numbers of unit charges are placed at $\sqrt{2}$, $\sqrt{4}$, $\sqrt{8}$, $\sqrt{16}$... (up to infinite) distances from the origin in a straight line. What will be the total force acting on the 1st charge?

• A. $9*10^9 N$
• B. 1 N
• C. $18*10^9 N$
• D. Infinite

Answer 1

Answer: (A)

9*10^9 N

Answer 2

The problem asks for the total force acting on a 1C charge placed at the origin due to an infinite number of unit charges (1C each) placed at specific distances along a straight line.

1. Identify the charges and constant:

   • Charge at the origin, $q_0 = 1 \text{ C}$.
   • Other charges, $q = 1 \text{ C}$.
   • Coulomb's constant, $k = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ N m}^2/\text{C}^2$.

2. Identify the distances: The charges are placed at distances $r_n$ from the origin given by: $r_1 = \sqrt{2}$ $r_2 = \sqrt{4} = 2$ $r_3 = \sqrt{8} = 2\sqrt{2}$ $r_4 = \sqrt{16} = 4$ ... and so on. This sequence of distances can be written as $r_n = (\sqrt{2})^n$.

3. Apply Coulomb's Law and sum the forces: Since all charges are positive and are placed along a straight line (implying they are all on one side of the origin, so forces add up), the total force ($F_{total}$) on the charge at the origin will be the sum of the individual forces exerted by each charge. The force exerted by the $n^{th}$ charge on the charge at the origin is given by Coulomb's Law: $F_n = k \frac{q_0 q}{r_n^2}$

   Substitute $q_0 = 1 \text{ C}$ and $q = 1 \text{ C}$: $F_n = k \frac{(1)(1)}{r_n^2} = \frac{k}{r_n^2}$

   Now, substitute $r_n = (\sqrt{2})^n$: $r_n^2 = ((\sqrt{2})^n)^2 = (2^{1/2})^n)^2 = 2^n$

   So, $F_n = \frac{k}{2^n}$.

   The total force is the sum of these individual forces: $F_{total} = \sum_{n=1}^{\infty} F_n = \sum_{n=1}^{\infty} \frac{k}{2^n}$ $F_{total} = k \left( \frac{1}{2^1} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \dots \right)$ $F_{total} = k \left( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots \right)$

4. Sum the infinite geometric series: The expression in the parenthesis is an infinite geometric series with: First term ($a$) = $\frac{1}{2}$ Common ratio ($r$) = $\frac{1/4}{1/2} = \frac{1}{2}$

   Since $|r| < 1$ (i.e., $|1/2| < 1$), the series converges to a sum $S = \frac{a}{1-r}$. $S = \frac{1/2}{1 - 1/2} = \frac{1/2}{1/2} = 1$

5. Calculate the total force: $F_{total} = k \times S = (9 \times 10^9 \text{ N m}^2/\text{C}^2) \times 1$ $F_{total} = 9 \times 10^9 \text{ N}$