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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 .
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