Moving Charges and Magnetism | Class 12 Physics Notes

4.5 MAGNETIC FIELD ON THE AXIS OF A CIRCULAR CURRENT LOOP

This section calculates the magnetic field produced by a circular current loop of radius R carrying current I at a point P located on the axis of the loop at distance x from its center. Using the Biot-Savart law, the magnetic field contribution dB due to an infinitesimal current element dl is perpendicular to the plane containing dl and the vector r from the element to point P.

Since all such dB vectors have components perpendicular to the axis that cancel out due to symmetry, only the components along the axis add up. The axial component dBx is given by dB cos θ, where cos θ = R / √(x² + R²).

Integrating over the entire loop circumference 2πR, the magnetic field at point P is:

B = (μ₀ I R²) / [2 (x² + R²)^(3/2)]

At the center of the loop (x=0), this reduces to:

B₀ = (μ₀ I) / (2 R)

The magnetic field lines form closed loops around the current loop, and the direction of the magnetic field is given by the right-hand thumb rule: curling the fingers in the direction of current, the thumb points in the direction of B.

Example 4.5 discusses the magnetic field at the center of a semicircular arc and straight segments; Example 4.6 calculates the field at the center of a tightly wound coil with multiple turns.

📊 Diagram: Figure 4.9 shows magnetic field dB due to element dl and its components along and perpendicular to the axis; Figure 4.10 shows magnetic field lines for a current loop with direction given by right-hand thumb rule; Figure 4.11 shows semicircular arc and straight segments.

🧪 Activity: Calculation of magnetic field on axis of circular loop and understanding field line patterns.

🔗 Connection: Leads to Ampere's circuital law as an alternative method to relate magnetic fields and currents.