A circular loop located on x2 +y2 =9, z=0 carries a direct current of 10A along loop. Apply Biot-Savart’s law to identify the magnetic field strength, H on the Z- axis at a point P(0,0,4) and (0,0,-4) and at (0,0,0)

To solve this problem, we will use the Biot-Savart Law, which states that the magnetic field dH at a point due to a small current element Idl carrying current I is given by:

dH = (Idl x r') / (4π|r'|^3)

where r' is the position vector from the element to the point in question.

Step 1: Define the problem

We have a circular loop of radius 3 (from the equation x^2 + y^2 = 9) in the xy-plane (z=0) carrying a current of 10A. We want to find the magnetic field strength at points P(0,0,4), P(0,0,-4), and P(0,0,0).

Step 2: Apply Biot-Savart Law

For a circular loop, the magnetic field at a point on the axis is given by:

H = (μ0IR^2) / (2*(R^2 + z^2)^(3/2))

where μ0 is the permeability of free space, I is the current, R is the radius of the loop, and z is the distance from the center of the loop.

Step 3: Calculate H at P(0,0,4)

Substitute I=10A, R=3, z=4, and μ0=4π10^-7 Tm/A into the equation:

H = (4π10^-7 Tm/A * 10A * 3^2) / (2*(3^2 + 4^2)^(3/2)) = 1.35*10^-6 T

Step 4: Calculate H at P(0,0,-4)

The magnetic field is symmetric about the xy-plane, so H at P(0,0,-4) is the same as at P(0,0,4):

H = 1.35*10^-6 T

Step 5: Calculate H at P(0,0,0)

Substitute z=0 into the equation:

H = (4π10^-7 Tm/A * 10A * 3^2) / (23^3) = 210^-6 T

So, the magnetic field strength at points P(0,0,4) and P(0,0,-4) is 1.3510^-6 T, and at P(0,0,0) is 210^-6 T.