The magnetic field B at a distance x from the center of a circular coil of radius R carrying a current I can be calculated using Ampere's law and Biot-Savart law.

Step 1: Biot-Savart law states that the magnetic field dB at a point due to a small current element Idl is given by:

dB = μ0/4π * (Idl x r)/r^3

where μ0 is the permeability of free space, r is the position vector from the current element to the point, and x denotes the cross product.

Step 2: For a circular coil, the current element Idl and the position vector r are always perpendicular, so the cross product simplifies to the product of their magnitudes. The magnitude of r is the distance from the current element to the point, which is sqrt(R^2 + x^2) for points on the axis of the coil.

Step 3: Integrate dB over the coil to find the total magnetic field B. The result is:

B = μ0/4π * I * ∫dl/sqrt(R^2 + x^2)^3

Step 4: The integral ∫dl over the coil is just the circumference of the coil, 2πR. So:

B = μ0/4π * I * 2πR/sqrt(R^2 + x^2)^3

Step 5: Simplify to get the final result:

B = μ0IR^2/2(x^2 + R^2)^(3/2)

This is the intensity of the magnetic field at a distance x from the center of the coil. Note that this formula is valid only for x R, i.e., for points far from the coil compared to its radius.

Question

The magnetic field B at a distance x from the center of a circular coil of radius R carrying a current I can be calculated using Ampere's law and Biot-Savart law.

Step 1: Biot-Savart law states that the magnetic field dB at a point due to a small current element Idl is given by:

dB = μ0/4π * (Idl x r)/r^3

where μ0 is the permeability of free space, r is the position vector from the current element to the point, and x denotes the cross product.

Step 2: For a circular coil, the current element Idl and the position vector r are always perpendicular, so the cross product simplifies to the product of their magnitudes. The magnitude of r is the distance from the current element to the point, which is sqrt(R^2 + x^2) for points on the axis of the coil.

Step 3: Integrate dB over the coil to find the total magnetic field B. The result is:

B = μ0/4π * I * ∫dl/sqrt(R^2 + x^2)^3

Step 4: The integral ∫dl over the coil is just the circumference of the coil, 2πR. So:

B = μ0/4π * I * 2πR/sqrt(R^2 + x^2)^3

Step 5: Simplify to get the final result:

B = μ0IR^2/2(x^2 + R^2)^(3/2)

This is the intensity of the magnetic field at a distance x from the center of the coil. Note that this formula is valid only for x >> R, i.e., for points far from the coil compared to its radius.

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