Magnetic field on the axis of a circular loop of radius R carrying current I at a distance z along the axis is given by which expression?

Magnetic field on the axis of a circular loop of radius R carrying current I at a distance z along the axis is given by which expression?

• A. μ0 I R^2 / [2 (R^2 + z^2)^{3/2}]
• B. μ0 I / [2 √(R^2 + z^2)]
• C. μ0 I R / (R^2 + z^2)^{3/2}
• D. μ0 I R^2 / (R^2 + z^2)^{2}

Answer: (A)

The field on the axis of a circular loop points along the axis, and its magnitude comes from adding up the contributions of all current elements using Biot-Savart, with symmetry eliminating any transverse components. For a point on the axis at distance z from the loop, every current element dl lies at radius R in the loop, and the distance to the field point is r = sqrt(R^2 + z^2), the same for all elements. The axial component of the magnetic field from an element is proportional to dl times the sine of the angle between dl and the line to the field point, which for this geometry gives sinα = R/r. The contribution to the axis from that element is dB_z = (μ0 I / 4π) (dl sinα) / r^2 = (μ0 I R dl) / (4π r^3). Integrating around the loop, ∮ dl = 2πR, so the total axial field is B_z = (μ0 I R / 4π r^3) (2πR) = μ0 I R^2 / [2 (R^2 + z^2)^{3/2}]. Thus the magnetic field on the axis is along the axis with magnitude μ0 I R^2 / [2 (R^2 + z^2)^{3/2}], which matches the given expression. The direction is set by the right-hand rule.

The field on the axis of a circular loop points along the axis, and its magnitude comes from adding up the contributions of all current elements using Biot-Savart, with symmetry eliminating any transverse components.

For a point on the axis at distance z from the loop, every current element dl lies at radius R in the loop, and the distance to the field point is r = sqrt(R^2 + z^2), the same for all elements. The axial component of the magnetic field from an element is proportional to dl times the sine of the angle between dl and the line to the field point, which for this geometry gives sinα = R/r. The contribution to the axis from that element is dB_z = (μ0 I / 4π) (dl sinα) / r^2 = (μ0 I R dl) / (4π r^3).

Integrating around the loop, ∮ dl = 2πR, so the total axial field is

B_z = (μ0 I R / 4π r^3) (2πR) = μ0 I R^2 / [2 (R^2 + z^2)^{3/2}].

Thus the magnetic field on the axis is along the axis with magnitude μ0 I R^2 / [2 (R^2 + z^2)^{3/2}], which matches the given expression. The direction is set by the right-hand rule.