Apply Biot-Savart’s law to identify the magnetic field strength, H on the Z- axis at a point P(0,0,h) due to a current carrying conductor loop, x2 +y2 =a2 in Z =0 plane

Biot-Savart's law states that the magnetic field at any point in space due to a current element is directly proportional to the current, the length of the current element, and the sine of the angle between the current element and the line joining the point and the current element, and inversely proportional to the square of the distance between the current element and the point.

The magnetic field due to a current element is given by the formula:

dB = μ0/4π * (Idl x r')/r'^2

where:

• dB is the magnetic field,
• μ0 is the permeability of free space,
• I is the current,
• dl is the length of the current element,
• r' is the vector from the current element to the point in space, and
• x denotes the cross product.

The magnetic field at a point P(0,0,h) due to a current carrying conductor loop x^2 + y^2 = a^2 in the Z = 0 plane can be found by integrating the above formula over the loop.

The loop is a circle of radius a in the xy-plane. We can parameterize the loop as r(t) = a cos(t) i + a sin(t) j for t in [0, 2π], where i and j are the unit vectors in the x and y directions, respectively. The current element is then dl = -a sin(t) i + a cos(t) j dt.

The vector from the current element to the point P is r' = -a cos(t) i - a sin(t) j + h k, where k is the unit vector in the z direction. The magnitude of this vector is r' = sqrt(a^2 + h^2).

The angle between dl and r' is 90 degrees, so their cross product is simply dl x r' = a h dt k.

Substituting these into the formula for dB and integrating over the loop gives:

B = ∫ dB = μ0/4π * ∫ (I a h dt)/(a^2 + h^2) = μ0 I h/4π * ∫ dt/(a^2 + h^2) for t in [0, 2π] = μ0 I h/2 * 1/(a^2 + h^2)

This is the magnetic field at the point P due to the current carrying conductor loop.