PPT No. 19

* Magnetic Scalar Potential * Magnetic Vector Potential Magnetic Potentials

The Magnetic Potential is a method of representing the Magnetic field by using a quantity called Potential instead of the actual B vector field. Magnetic Potentials

Magnetic field can be related to a potential by two methods which give rise to two possible types of magnetic potentials used in different situations:

1. Magnetic Scalar Potential

2. Magnetic Vector Potential A) Magnetic Scalar Potential

In Electrostatics, electric field E is derivable from the electric potential V.

V is a scalar quantity and easier to handle than E which is a vector quantity.

In Magnetostatics, the quantity Magnetic scalar potential can be obtained using analogues relation A) Magnetic Scalar Potential

In regions of space in the absence of currents, the current density j =0

= 0 B is derivable from the gradient of a potential

Therefore B can be expressed as the gradient of a scalar quantity φm B = - φm φm is called as the Magnetic scalar potential. ∇ A) Magnetic Scalar Potential

The presence of a magnetic moment m creates a magnetic field B which is the gradient of some scalar field φm.

The divergence of the magnetic field B is zero, .B = 0

∇By definition, the divergence of the gradient of the scalar field is also zero, - . φm = 0 or 2 φm = 0. ∇ ∇ ∇The operator 2 is called the Laplacian and 2 φm = 0 is the Laplace’s equation. ∇ ∇ A) Magnetic Scalar Potential

2 φm = 0

∇Laplace’s equation is valid only outside the magnetic sources and away from currents.

Magnetic field can be calculated from the magnetic scalar potential using solutions of Laplace’s equation. B) Magnetic Vector Potential

The magnetic scalar potential is useful only in the region of space away from free currents.

If J=0, then only magnetic flux density can be computed from the magnetic scalar potential

The potential function which overcomes this limitation and is useful to compute B in region where J is present is . Magnetic Vector Potential B) Magnetic Vector Potential

Magnetic fields are generated by steady (time-independent) currents & satisfy Gauss’ Law

Since the divergence of a curl is zero, B can be written as the curl of a vector A as B) Magnetic Vector Potential

Any solenoidal vector field (e.g. B) in physics can always be written as the curl of some other vector field (A).

The quantity A is known as the Magnetic Vector Potential. B) Magnetic Vector Potential

{However, magnetic vector potential is not directly associated with work the way that scalar potential (e.g. Electric potential V) is associated with work}

Work done against the electric field E is stored as electric potential energy U given in terms of electric dipole moment p and E as B) Magnetic Vector Potential

The vector potential is defined to be consistent with Ampere’s Circuital Law and

It can be expressed in terms of either current i or current density j (i.e. the sources of magnetic field) as follows B) Magnetic Vector Potential

However, A is Not uniquely defined by the above equation.

Any function whose curl is zero, can be added to A, then the result would still be the same field B. e.g.

If ψ, the Gradient of a scalar ψ is added to A x (A + ψ )= x A + x ψ = x A = B ∇ ∇ ∇ ∇ ∇ ∇ ∇ B) Magnetic Vector Potential

To make A more specific/ unique, additional condition needs to be imposed on A.

In Magnetostatics a convenient condition which makes calculations easier can be specified as . A = 0 (In Electrodynamics, this condition cannot be imposed) ∇ B) Magnetic Vector Potential

The set of equations which uniquely define the vector potential A and also satisfy the fundamental equation of Gauss’ Law . B = 0 {the magnetic field is divergence-free}, ∇are as follows B) Magnetic Vector Potential

From Ampere’s law

Therefore the equation

can be written as

This equation is similar to Poisson's equation, the only difference is that A is a vector. B) Magnetic Vector Potential

Each component (e.g. along x, y, z axes) of A must satisfy the differential equation of the type

A unique solution to the above Poisson's equation can be found (By combining the solutions for components on x, y, z).

It specifies the magnetic vector potential A generated by steady currents. B) Magnetic Vector Potential

First A is determined using Poisson's equation then it is substituted in the equation

Thus the field B produced by a steady current can be computed. Gauge Transformation

According to Helmholtz's theorem a vector field is fully specified by its divergence and its curl.

The curl of the vector potential A gives the magnetic field B via Eq.

However, the divergence of A has no physical significance

can be chosen freely as desired B) Magnetic Vector Potential

According to the equation the magnetic field is invariant under the transformation

In other words, the vector potential is undetermined to the gradient of a scalar field

can be chosen as desired B) Magnetic Vector Potential

The electric scalar potential is undetermined to an arbitrary additive constant, since the transformation

leaves the electric field invariant in Equation

The transformations

and are examples of gauge transformations in Mathematics. B) Magnetic Vector Potential

In electromagnetic theory, several "gauges" have been used to advantage depending on the specific types of calculations

The choice of a particular function ψ or a particular constant c is referred to as a choice of the gauge. B) Magnetic Vector Potential

The gauge can be fixed as desired. Usually it is chosen to make equations simplest possible. It is convenient to choose gauge for the scalar potential Ф such that Ф → 0 at infinity.

The gauge for A is chosen such that

This particular choice is known as the Coulomb gauge