Problem 7 - Master Resource Book in JEE Main Physics

Question

Two long and parallel straight wires \(A\) and \(B\) carrying currents of \(8.0 \mathrm{~A}\) and \(5.0 \mathrm{~A}\) in the same direction are separated by a distance of \(4.0 \mathrm{~cm}\). Estimate the force on a \(10 \mathrm{~cm}\) section of wire \(A ?\) (a) \(1.5 \times 10^{-5} \mathrm{~N}\) (b) \(2 \times 10^{-5} \mathrm{~N}\) (c) \(4 \times 10^{-5} \mathrm{~N}\) (d) \(3.2 \times 10^{-5} \mathrm{~N}\)

Step-by-Step Solution

Answer

The force on a 10 cm section of wire A is \(2 \times 10^{-5}\,\text{N}\), option (b).

Step 1: Understanding the formula for force

The magnetic force per unit length between two parallel wires carrying currents can be calculated using the formula: \( F/L = \frac{\mu_0 I_1 I_2}{2 \pi d} \), where \( F \) is the force, \( L \) is the length of the wire, \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \; \text{T}\cdot\text{m/A}) \), \( I_1 \) and \( I_2 \) are the currents through wire A and B, and \( d \) is the distance between the two wires.

Step 2: Substituting known values

Substitute the given values into the formula: \( I_1 = 8.0 \; \text{A} \), \( I_2 = 5.0 \; \text{A} \), and \( d = 4.0 \; \text{cm} = 0.04 \; \text{m} \). The formula becomes: \( F/L = \frac{4\pi \times 10^{-7} \times 8.0 \times 5.0}{2 \pi \times 0.04} \).

Step 3: Simplifying the equation

Cancel out \( \pi \) from the numerator and denominator: \( F/L = \frac{4 \times 10^{-7} \times 8.0 \times 5.0}{2 \times 0.04} \). Next, calculate the expression within the numerator \( 4 \times 10^{-7} \times 8.0 \times 5.0 = 1.6 \times 10^{-5} \).

Step 4: Solving for force per unit length

Divide the result by the remaining terms in the denominator: \( F/L = \frac{1.6 \times 10^{-5}}{0.08} = 2 \times 10^{-4} \; \text{N/m} \).

Step 5: Calculating force over given length

Multiply the force per unit length by the length of wire: \( F = (2 \times 10^{-4}) \times 0.10 \; \text{m} = 2 \times 10^{-5} \; \text{N} \).

Step 6: Choosing the correct answer

Comparing the calculated force with the options, match \( 2 \times 10^{-5} \; \text{N} \) with option (b).

Key Concepts

Ampere's Force Law

Ampere's force law plays a crucial role in understanding how currents in parallel conductors interact. It describes the magnetic force experienced between two parallel current-carrying wires. The fundamental principle of this law is that current flowing through a wire generates a magnetic field around it, which can influence other nearby currents.
To determine the force between two parallel wires, we use the formula: \[ F/L = \frac{\mu_0 I_1 I_2}{2 \pi d} \] Where:

• \( F \) is the magnetic force between the wires.
• \( L \) is the length of the wire over which the force is measured.
• \( \mu_0 \) is the permeability of free space \((4\pi \times 10^{-7} \; \text{T}\cdot\text{m/A})\).
• \( I_1 \) and \( I_2 \) are the currents in the wires.
• \( d \) is the distance between the wires.

This law helps us calculate the force in various real-world scenarios where wires carry different magnitudes of current. Understanding Ampère's force law is essential for efficiently designing and analyzing electrical systems.

Current-Carrying Conductors

Current-carrying conductors are simply wires or any other elements that allow electric current to flow through them. These conductors bear an electrical charge due to the movement of electrons, creating a magnetic field around them.
When such conductors are placed parallel to each other, as in the given exercise, they exert magnetic forces on one another. The strength and direction of these forces depend on the direction and magnitude of the currents as well as the distance between the wires.
Parallel wires carrying current in the same direction attract each other, while those with currents in opposite directions repel each other. This principle aids in understanding the interactions between electrical conductors in circuits and is vital in applications such as power distribution.

Magnetic Fields

Magnetic fields are unseen but powerful fields around magnetic objects or flowing electrical currents. These fields exert forces on other magnets and current-carrying conductors.
For a straight wire carrying current, the magnetic field can be understood by using the right-hand rule. If you point your thumb in the direction of current flow, your fingers curl in the direction of the magnetic field.
The magnetic field strength around a long, straight conductor depends on several factors:

• The magnitude of current through the conductor; a higher current means a stronger magnetic field.
• The distance away from the wire; the magnetic field strength decreases with increased distance.

These principles help describe how the magnetic field of one current-carrying wire affects another nearby wire, as seen in the parallel wires from the exercise. Understanding magnetic fields is key in technology involving motors, generators, and electromagnets.