Problem 53 - Master Resource Book in JEE Main Physics

Question

An infinite number of charges, each \(1 \mu \mathrm{C}\) are placed on the \(x\)-axis with coordinates \(x=1,2,4,8 \ldots \infty\). If a charge of \(1 \mathrm{C}\) is kept at the origin, then what is the net force acting on \(1 \mathrm{C}\) charge (a) \(10000 \mathrm{~N}\) (b) \(32000 \mathrm{~N}\) (c) \(12000 \mathrm{~N}\) (d) \(18000 \mathrm{~N}\)

Step-by-Step Solution

Answer

The net force acting on the 1 C charge is approximately 12000 N, so the answer is (c) 12000 N.

Step 1: Understanding the Setup

We have an infinite series of charges, each of magnitude \(1 \mu \mathrm{C}\), placed at positions with coordinates \(x = 2^n\) where \(n = 0, 1, 2, \, \ldots \,\infty\) on the x-axis. A charge of \(1 \mathrm{C}\) is at the origin.

Step 2: Using Coulomb's Law

The force between two charges is calculated using Coulomb's Law: \( F = \dfrac{k |q_1 q_2|}{r^2} \). Here, \(k = 8.99 \times 10^9 \mathrm{Nm^2/C^2}\), \(q_1 = 1 \mathrm{C}\), \(q_2 = 1 \mu \mathrm{C} = 10^{-6} \mathrm{C}\), and \(r = 2^n\) for the \(n\)th charge.

Step 3: Summing Forces Exerted by Each Charge

The total force on the charge at the origin is the sum of the forces due to each of the charges at positions \(x = 2^n\). This is a geometric series: \[ F = k q Q \sum_{n=0}^{\infty} \dfrac{1}{(2^n)^2} = 8.99 \times 10^9 \times 1 \times 10^{-6} \sum_{n=0}^{\infty} \dfrac{1}{4^n}. \]

Step 4: Solving the Geometric Series

The series \( \sum_{n=0}^{\infty} \dfrac{1}{4^n} \) is a geometric series with the first term \(a = 1\) and common ratio \(r = \dfrac{1}{4}\). The sum of an infinite geometric series is given by \( S = \dfrac{a}{1-r} \), so \[ S = \dfrac{1}{1 - \frac{1}{4}} = \dfrac{1}{\frac{3}{4}} = \dfrac{4}{3}. \]

Step 5: Calculating the Net Force

Substitute the sum \(S = \frac{4}{3}\) into the expression for force:\[ F = 8.99 \times 10^9 \times 10^{-6} \times \dfrac{4}{3}, \]resulting in the force \( F \approx 11,986.67 \mathrm{~N} \approx 12000 \mathrm{~N} \).

Key Concepts

Geometric Series

An infinite geometric series is a special type of series where the ratio between successive terms is constant. In this exercise, each charge is placed at positions that are powers of 2, like 1, 2, 4, 8, and so forth. This means we can express these positions as a series: \( x = 2^n \) for \( n = 0, 1, 2, \ldots, \infty \).
To evaluate the force acting on a charge at the origin due to these infinitely placed charges, we used a geometric series. This is essential because it simplifies the sum of infinite forces into a finite number.
The geometric series formula, \( S = \frac{a}{1-r} \), allows us to find the total effect of these infinite series of forces. Here, the first term \( a \) is 1, and the common ratio \( r \) is \( \frac{1}{4} \). By applying this to the problem, we find the series' sum is \( \frac{4}{3} \), which simplifies our calculations greatly.

Electrostatics

Electrostatics is the study of forces between charged bodies at rest. It's a core concept required to solve problems involving charges and electric fields, like this exercise where charges are placed along the x-axis.
The fundamental law governing electrostatics is Coulomb's Law, which represents the force between two point charges. The formula is given by \( F = \frac{k |q_1 q_2|}{r^2} \), where \( k \) is the electrostatic constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.
In this problem, we apply Coulomb's Law repeatedly for each charge in the infinite sequence to calculate the force on the charge at the origin. The innovation here is how we break this down into an understandable series, using geometric series to sum the forces efficiently. The interplay of forces, charges, and distances exemplifies electrostatics' elegance.

Infinite Charge Distribution

Infinite charge distributions often appear complex, but understanding them becomes manageable with the right approach. In this exercise, charges are arranged at positions that increase exponentially along the x-axis, specifically at \( x = 2^n \).
Handling an infinite charge distribution emphasizes comprehending how each charge contributes to the net force. While it sounds daunting, the geometric series helps collate these effects into something quantifiable and manageable using finite mathematics.
By recognizing the repetitive and predictable placement (\( 2^n \)), we apply mathematical tools to analyze their impacts systematically. Although the charges seem endless, their effect becomes clear when recalculated through series summation, showcasing how infinity in mathematics doesn't mean computational impossibility.