The magnetic field at the centre of a circular loop of area A is b. the magnetic moment of the loop will be

To determine the magnetic moment of a circular loop given the magnetic field at its center, we can use a relationship that connects these two quantities. The magnetic field (B) at the center of a circular loop is directly related to the magnetic moment (μ) of that loop. Let's break this down step by step.

Understanding the Relationship

The magnetic moment of a current-carrying loop is defined as the product of the current (I) flowing through the loop and the area (A) of the loop. Mathematically, it is expressed as:

μ = I × A

Where:

• μ is the magnetic moment.
• I is the current flowing through the loop.
• A is the area of the loop.

Magnetic Field at the Center

The magnetic field at the center of a circular loop can be expressed in terms of the current and radius of the loop. The formula for the magnetic field (B) at the center of a circular loop is given by:

B = (μ₀ / 4π) × (2πI / R²)

Where:

• B is the magnetic field strength at the center.
• μ₀ is the permeability of free space.
• R is the radius of the loop.

Connecting the Dots

To find the magnetic moment in terms of the magnetic field, we can rearrange the equation. Given that the area of the loop is A = πR², we can express the magnetic moment in terms of the magnetic field.

From the formula for B, we can also derive:

μ = (B × 4π) × (A / μ₀)

Since we know the area A, we can relate this back to the magnetic moment:

μ = (B × A) / (μ₀ / 2)

Final Expression

When we simplify this, we find that the magnetic moment of the loop is directly proportional to the magnetic field at its center and the area of the loop. Therefore, if you know the magnetic field (B) and the area (A), you can calculate the magnetic moment as:

μ = (B × A) / (μ₀ / 2)

Example Calculation

Let’s say you have a circular loop with an area of 0.01 m² and the magnetic field at its center is 0.1 T. Assuming μ₀ is approximately 4π × 10⁻⁷ T m/A, you can plug these values into the formula:

μ = (0.1 T × 0.01 m²) / (4π × 10⁻⁷ T m/A / 2)

This calculation will give you the magnetic moment of the loop in ampere-square meters (A·m²).

In summary, the magnetic moment of a circular loop can be easily derived from its area and the magnetic field at its center, allowing you to understand the relationship between these fundamental electromagnetic properties.