Magnetic field due to a circular loop

Explore the fundamentals of the magnetic field due to a circular current loop, with a clear breakdown of related laws and an illustrative example.

Magnetic Field Due to a Circular Loop

The concept of a magnetic field due to a circular current loop is a fundamental concept in the field of electromagnetism. It is governed by Ampère’s circuital law and the Biot-Savart law. When current flows through a circular loop, it generates a magnetic field around it. The strength of this field depends on the amount of current and the radius of the loop.

1. Ampère’s Circuital Law: According to this law, the integral of the magnetic field around a closed loop is proportional to the current enclosed by the loop.
2. Biot-Savart Law: This law describes how currents produce magnetic fields. It states that the magnetic field at any point due to a current carrying conductor is directly proportional to the current in the conductor, the length of the conductor, and inversely proportional to the square of the distance between the point and the conductor.

The magnetic field at the center of a circular loop due to a current I is given by the formula:

B = μ₀I / 2r

• Here, B is the magnetic field.
• μ₀ is the permeability of free space.
• I is the current flowing through the loop.
• r is the radius of the circular loop.

This formula indicates that the magnetic field at the center of the loop is directly proportional to the current and inversely proportional to the radius of the loop. It’s worth noting that the magnetic field is stronger inside the loop and decreases as we move away from it.

In summary, the understanding of the magnetic field due to a circular loop has wide-ranging applications in many areas of physics and engineering, from the design of electric motors to the study of electromagnetic waves.

Example of Calculation

Consider a circular loop of radius r = 0.5m, carrying a current I = 5A. Let’s calculate the magnetic field at the center of the loop.

1. The formula for the magnetic field at the center of a circular loop is: B = μ₀I / 2r.
2. We know that the value of μ₀, the permeability of free space, is approximately 4π × 10^-7 T m/A.
3. Substituting the values into the formula:

B = (4π × 10^-7 T m/A * 5A) / (2 * 0.5m)

After simplification, we find the magnetic field strength B at the center of the loop to be approximately 2 × 10^-6 T.

Therefore, for a current of 5A flowing through a circular loop of radius 0.5m, the magnetic field at the center is approximately 2 × 10^-6 T.