Problem 5 - Master Resource Book in JEE Main Physics

Question

A current \((i)\) carrying circular wire of radius \(R\) is placed in a magnetic field \(B\) perpendicular to its plane. The tension \(T\) along the circumference of wire is (a) \(B \bar{R}\) (b) \(2 \pi \mathrm{BiR}\) (c) \(\pi B i R\) (d) \(2 B i R\)

Step-by-Step Solution

Answer

The tension along the circumference of the wire is \( 2 \pi BiR \).

Step 1: Understanding the Problem

A circular wire carrying a current is placed in a uniform magnetic field. The magnetic field is perpendicular to the plane of the wire. We are asked to find the tension in the wire due to the interaction between the magnetic field and the current.

Step 2: Recall the Magnetic Force on a Current Loop

For a current-carrying loop in a magnetic field, the magnetic force \( F \) can be determined using the formula: \ \[ F = B i L \] where \( B \) is the magnetic field, \( i \) is the current, and \( L \) is the length of the wire in the field (in this case, the circumference).

Step 3: Calculate the Circumference of the Wire

Since the wire is a circle of radius \( R \), its circumference \( L \) is: \ \[ L = 2 \pi R \]

Step 4: Find the Magnetic Force on the Wire

Substituting \( L = 2 \pi R \) into the magnetic force formula, we get: \ \[ F = B i (2 \pi R) = 2 \pi B i R \] This force acts inward along the circumference of the circle.

Step 5: Determine the Tension in the Wire

For a wire under uniform magnetic force, this force is balanced by the tension \( T \) along the circumference of the wire. Therefore, \ \( T = 2 \pi B i R \).

Key Concepts

Current Carrying Wire

In electrical physics, a wire carrying an electric current is described as a current-carrying wire. The current is the flow of electric charge, typically measured in amperes (A). In a circular wire, this current travels along the path of the circle.

When the wire is bent into a loop, such as a circle, it forms a current loop. This loop can create its own magnetic field. The strength and direction of this field depend on factors like the size of the current and the shape of the loop.

• The current creates its own magnetic field inside the loop.
• The direction of the field depends on the direction of the current.
• A loop influences external magnetic fields through interaction.

Understanding the behavior of a current-carrying wire in different scenarios is essential for exploring complex topics like magnetic field interactions and forces.

Magnetic Field Interaction

A magnetic field is a region around a magnetic material or moving electric charge within which the force of magnetism acts. In our scenario, this field is perpendicular to the plane of the current-carrying loop. This perpendicular arrangement influences how the magnetic force acts on the wire.

The interaction between the magnetic field and the current-carrying wire can be broken down into these steps:

• The magnetic field exerts a force on the wire due to its current.
• This force can be calculated using the formula \( F = B i L \), where \( B \) is the field strength, \( i \) is the current, and \( L \) is the length of the wire within the field.
• For a circular wire, \( L \) equals its circumference.

This force is responsible for pushing the wire, which in a circular setup, acts inward around the loop, altering the wire's tension as a physical interaction.

Tension in Wire

Tension in a wire refers to the force that pulls along its length. For a circular wire in a magnetic field, this tension arises from the magnetic forces acting on the wire. As the magnetic field applies a force outward, the wire must generate an inward tension to balance this force.

In our case, the tension can be computed by acknowledging the magnetic force balance:

• The tension acts uniformly around the loop.
• Since the magnetic force \( F = 2 \pi B i R \), so too is the tension \( T = 2 \pi B i R \).
• This means the tension is the wire's reaction to the magnetic force, ensuring stability and equilibrium.

Understanding tension is crucial as it demonstrates how forces within a current-carrying wire can maintain balance under magnetic fields, a fundamental principle in electromagnetism.