Magnetic Potential and Field at a Point on the Axis of a Flat Circular Magnetic Shell

Consider a flat circular magnetic shell ABC of uniform strength Φ. Let O be its centre and a, its radius. P is a point on the axis at a distance x from O. Let ∟APO = θ.



If the shell subtends a solid angle ω at P, the potential at P is
        V = μ₀/4π Φω
ω is determined as follows:
Let ADEC be the side view of the shell having width ds.
Talking P as centre, draw a sphere of radius PA = r. The circumference of the shell then lies on the surface of the sphere. The solid angle subtended by the shell at P is given by
        ω = Area of the cap ABC/r²
The area of the rim of the shell = (2πa)ds
The area of cap ABC can be calculated by integrating this area between the limits 0 and θ.

Area of cap ABC = ∫^θ₀ 2π a ds
    a = r sin θ and ds = r dθ
. :    area of the cap = ∫^θ₀ 2π r² sin θ dθ = 2πr² (1-cosθ)
Solid angle subtended by the cap ABC at P is
    ω = 2πr² (1-cos θ)/r² = 2π (1 – cos θ)
    = 2π (1 – x/r) = 2π (1 – x/ (a² + x²)^1/2)
The potential at P is given by
    V = μ₀ /4π Φω
    V = μ₀ /4π [2πΦ (1 – x/(a² + x²)]
If x = 0, i.e., if the point is very close the shell, V = μ0Φ/2
Magnetic Induction field (B)
Magnetic induction B = -dV/dx
. :    B = -d/dx [μ₀Φ/2{1 – x/ (a2 + x2)^1/2}] = μ0Φa² /2(a² + x²)^3/2
If x = 0, then B = μ0Φ/2a