## Potential At Any Point

**Potential At Any Point** Assignment Help | **Potential At Any Point** Homework Help

# Potential at any Point due to a Magnetic Shell

Consider a uniform magnetic shell of strength Φ. Let P be a point at a distance r from the centre O of a shell. Let OP make an angle θ with the normal to the shell, i.e., with the direction of magnetization OL. To find the potential at the point P, the sell, may be imagined to be divided into a large number of elementary area. Consider a small element of area dA of the shell around O.The magnetic moment of the element, m = Φ dA, directed along OL.

. : Magnetic moment of the element along OP = Φ dA cos θ

Potential at P due to the element of area dA is

dV = μ

_{0}/4π ΦdA cos θ/r

^{2}

But dA cos θ/r

^{2}= dω, the solid angle subtended by the element at P.

. : dV = μ

_{0}/4π Φ dω

The potential due to the whole shell is the sum of the potential due to all such elements.

The total potential at P due to the entire shell is given by

V = μ

_{0}/4π ∫ Φ dω

For a uniform magnetic shell, Φ is the same at all points.

. : V = μ

_{0}/4π Φ ∫dω

Or V = μ

_{0}/4π Φω (In Vacuum)

Where ω is the solid angle subtended at P by the entire shell.

Thus the magnetic potential at a point due to a uniform magnetic shell is μ

_{0}/4π times the produce of the strength of the shell and the solid angle subtended by the shell at the point.

(i) If the point P lies on the side of the face having north polarity the potential at P is considered positive.

(ii) If the point P lies on the side of the face having south polarity the potential at P is negative.

## Special Cases

### 1. Potential inside and outside a closed spherical shell

(i) If the point is outside the shell, ω = 0 . : v = 0(ii) If the point is inside the shell, ω = 4π . : v = μ

_{0}Φ

The strength of the magnetic shell Φ is the same at all points of the shell. Hence V is uniform inside the shell. Therefore, the magnetic field inside the shell is zero.

### 2. Infinitely long plane shell

Consider an infinitely long plane magnetic shell. If the point P1 lies on the side of N-pole, then potential at P1 isV = μ

_{0}/4πΦω = μ

_{0}/4πΦ .2π

Similarly, if the point P2 is on the face having south polarity.

V = - μ

_{0}Φ/2

. : Magnetic potential difference between two points exactly opposite to each other, once on each face of the shell

= μ

_{0}Φ/2 – (–μ

_{0}Φ/2) = μ

_{0}Φ

For more help in

**Potential at any Point due to a Magnetic Shell**please click the button below to submit your homework assignment.