An infinite number of charges, each of charge, 1µC, are placed on the x-axis with co-ordinates x = 1, 2, 4, 8, ... ∞. If a charge of 1c is kept at the origin, then what is the net force acting on 1 C charge?a)9000 Nb)12000 Nc)24000 Nd)36000 NCorrect answer is option 'B'. Can you explain this answer?

Question

An infinite number of charges, each of charge, 1&#181;C, are placed on the x-axis with co-ordinates x = 1, 2, 4, 8, ... &#8734;. If a charge of 1c is kept at the origin, then what is the net force acting on 1 C charge?a)9000 Nb)12000 Nc)24000 Nd)36000 NCorrect answer is option 'B'. Can you explain this answer?

Answer

Answer

From Coulomb's law, the force between two points charges q1,q2 separated by distance r is

The schematic diagram of distribution of charges on xaxis is shown.

The total force acting on 1 C charge is given by

Using the fact that sum of GP series is

where a is first term of series and r thecommon ratio.

Answer

Understanding the problem:
An infinite number of charges with a charge of 1C are placed at x = 1, 2, 4, 8, ... on the x-axis. We need to find the net force acting on a 1C charge kept at the origin.

Calculating the force:
- The force between two charges q1 and q2 separated by a distance r is given by Coulomb's law: F = k * (q1 * q2) / r^2
- Since all charges are positive, they will repel each other, causing a force in the positive x-direction.
- The net force between the charge at the origin and the other charges can be calculated by summing up the individual forces.
- The distance between the origin (x=0) and the charges at x = 1, 2, 4, 8, ... follows a geometric progression: 1, 2, 4, 8, ... which can be represented as 2^(n-1) where n is the position of the charge.
- Substituting the values in the formula, the net force can be calculated as F = k * (1 * 1) / 1^2 + k * (1 * 1) / 2^2 + k * (1 * 1) / 4^2 + k * (1 * 1) / 8^2 + ...
- This simplifies to F = k * (1 + 1/4 + 1/16 + 1/64 + ...) = k * (4/3)
- Substituting the value of k (9 * 10^9 N m^2/C^2) gives F = 4/3 * 9 * 10^9 = 12 * 10^9 N = 12000 N
Therefore, the net force acting on the 1C charge at the origin is 12000 N.

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