Problem 62 - Master Resource Book in JEE Main Physics

Question

Two infinitely long parallel wires having linear charge densities \(\lambda_{1}\) and \(\lambda_{2}\) respectively are placed at a distance of \(R\) metres. The force per unit length on either wire will be \(\left(K=\frac{1}{4 \pi \varepsilon_{0}}\right)\) (a) \(K \frac{2 \lambda_{1} \lambda_{2}}{R^{2}}\) (b) \(K \frac{2 \lambda_{1} \lambda_{2}}{R}\) (c) \(K \frac{\lambda_{1} \lambda_{2}}{R^{2}}\) (d) \(K \frac{\lambda_{1} \lambda_{2}}{R}\)

Step-by-Step Solution

Answer

The force per unit length is \(K \frac{2 \lambda_1 \lambda_2}{R}\).

Step 1: Understanding the Setup

We have two parallel wires, each with a linear charge density, \(\lambda_1\) and \(\lambda_2\), separated by a distance \(R\). The task is to find the force per unit length between these two wires.

Step 2: Coulomb's Law for Line Charges

For a linear charge density, the force per unit length between two wires is given by the expression for the electric force between line charges. Using Coulomb's Law adjusted for line charges: \[ F/L = \frac{K \cdot 2 \lambda_1 \cdot \lambda_2}{R} \]

Step 3: Applying the Formula

Plug the given linear charge densities into the expression. The force per unit length \( (F/L) \) becomes: \[ F/L = K \cdot \frac{2 \lambda_1 \lambda_2}{R} \]

Step 4: Compare with Options

Reviewing the provided options, we notice that option (b) \(K \frac{2 \lambda_{1} \lambda_{2}}{R}\) matches the derived expression for the force per unit length.

Key Concepts

Linear Charge Density

Linear charge density is an important concept when dealing with long, charged objects like rods or wires. It tells us how much charge is distributed per unit length along the object.

In mathematical terms, it's represented by the Greek letter lambda (\( \lambda \)), and if we have a wire with a total charge \( Q \) spread evenly over its length \( L \), the linear charge density would be \( \lambda = \frac{Q}{L} \). This concept is crucial because it allows us to simplify problems by focusing on how charge density interacts over lengths rather than dealing with the total charge all at once.

When two wires each have their own linear charge densities, their interactions can be calculated using these densities instead of the total charge, making the math more manageable.

Electric Force

Electric force is the force between charged objects due to their electric fields. This force can be attractive or repulsive, depending on the nature of the charges involved.

With Coulomb's Law, we can calculate this electric force between the charged linear objects, like wires, by using their linear charge densities and the distance between them. The modified version of Coulomb's Law for line charges calculates the force per unit length between two wires based on their charge densities \( \lambda_1 \) and \( \lambda_2 \), and the separation \( R \).

The formula is:\[ F/L = K \cdot \frac{2 \lambda_1 \lambda_2}{R} \]where \( K \) is Coulomb's constant adjusted for permittivity of space. Here, the factor of 2 comes in because each unit length interacts with the entire length of the other wire.

• This formula shows us the importance of both the linear charge densities and the distance between the wires.
• A smaller distance \( R \) results in a stronger force.
• This gives insight into how electric fields interact over distances.

Parallel Wires

Parallel wires carrying charges illustrate a fascinating phenomenon in electromagnetism. These wires interact with each other because of the electric fields they produce.

When wires are placed parallel to each other, each charged wire creates an electric field that influences the other. The force experienced by each wire is influenced by their linear charge densities and the distance \( R \) between them.

In our context, the presence of two parallel wires with linear charge densities \( \lambda_1 \) and \( \lambda_2 \) means that we look for the force per unit length that arises due to these linear charges. Understanding this interaction is vital in many practical applications like electrical cables and transmission lines.

• This setup shows how wires in parallel configurations exert forces on each other.
• Learning about this interaction helps in designing better electrical systems.
• Problems like these help us explore the principles of electrostatic forces in the real world.