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&#956;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

Total Force= K(1C)(10^(-6))[(1/1)+(1/4)+(1/16)....] This is sum of infinite GP... K=9x10^9... Answer is 9000(1/(1-4/3))=12000.

Answer

Calculating total force on 1C charge due infinite 1 micro coulomb charges

https://edurev.in/studytube/Calculating-total-force-on-1C-charge-due-infinite-1-micro-coulomb-charges/16a86480-4820-4b14-b422-f5dd7d001212_t

Answer

Understanding the Problem
An infinite series of charges, each with a charge of 1 µC, is placed at coordinates x = 1, 2, 4, 8, ... on the x-axis. We need to find the net force acting on a charge of 1 C placed at the origin.
Calculating the Force from Each Charge
- The electrostatic force (F) between two point charges is given by Coulomb's law:
F = k * |q1 * q2| / r²
where:
- k = 9 x 10^9 N m²/C² (Coulomb's constant)
- q1 = 1 C (the charge at the origin)
- q2 = 1 µC = 1 x 10^-6 C (charge of each point on the x-axis)
- r = distance from the origin to the charge.
- The distances to the charges are powers of 2: 1, 2, 4, 8, ...
Calculating Individual Forces
- For each charge located at x = 2^n:
r = 2^n
F_n = k * (1 * 1 x 10^-6) / (2^n)²
Thus, F_n = 9 x 10^9 * 1 x 10^-6 / (4^n) = 9 x 10^3 / (4^n)
Summing the Forces
- The net force is the sum of the forces from all charges:
F_net = ∑ (9 x 10^3 / 4^n) for n = 0 to ∞
- This series is a geometric series with:
- First term (a) = 9 x 10^3
- Common ratio (r) = 1/4
- The sum of an infinite geometric series is given by S = a / (1 - r):
S = (9 x 10^3) / (1 - 1/4) = (9 x 10^3) / (3/4) = 12,000 N
Conclusion
The net force acting on the 1 C charge at the origin is 12,000 N. Thus, the correct answer is option 'B'.

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