The same derivation1 may be followed as for the electrostatic. This concept is useful, since it is sometimes more convenient to compute the potential and then compute the electric. This lecture introduces the concept of the magnetic vector potential, which is analogous to the electric potential. Classical electromagnetism university of texas at austin. Now suppose the particle is bound in a strong central potential and b is relatively weak. V jr0 jr r0j d3r0 1 provided that all currents are contained within a. For a stationary point charge p q6 x r, where r is the charge position, 4 ul for brevity lets write r x r. The magnetic vector potential contributed by a length. The vector potential can then be written as a 1 2 b. Recall that a vector field is uniquely determined by specifying its divergence and curl. Calculating the magnetic vector and scalar potentials gri ths.
Electromagnetic fields and energy mit opencourseware. Conception of the scalarvector potential in contemporary. Magnetic vector potential, a, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field. Although we cannot express the magnetic field as the gradient of a scalar potential function, we shall define a vector quantity. Dalarsson, in tensors, relativity, and cosmology second edition, 2015.
What is the direction of the magnetic vector potential in. First visualize a simple concept from basic physics. We consider a uniform sheet of current in the xyplane, carrying surface current density kx using the same argument as in the case of a slab of current, the magnetic. Simple analytic expressions for the magnetic field of a circular current loop james simpson, john lane, christopher immer, and robert youngquist abstract analytic expressions for the magnetic induction and its spatial derivatives for a circular loop carrying a static current are presented in cartesian, spherical and cylindrical coordinates. We define the action integral for the motion of a charged particle in the electromagnetic field. The magnetic field b due to a single current loop of radius a, as shown in figure 1, can be computed by evaluating the curl of the magnetic vector potential a. This is similar to how the electric field e can be conveniently reexpressed in terms of electric potential. Chapter 5 calculating the vector potential recall that for the solenoid, we found that as r 0nir2 2s 1 outside the solenoid there is no magnetic eld, but the potential. Pdf thoughts on the magnetic vector potential researchgate. In this chapter we define the fourvector of the potentials of the electromagnetic field and relate it to the nonrelativistic electric scalar potential and magnetic vector potential. An educational path for the magnetic vector potential and its. Physics 3323, fall 2016 problem set 8 due oct 21, 2014 reading.
Its vector potential at point r is just as we did for v, we can expand in a power series and use the series as an approximation scheme. Magnetic vector potential an overview sciencedirect topics. Magnetic potential refers to either magnetic vector potential a or magnetic scalar potential. The vector potential, a, of the loop is given by 3. Both types of magnetic potential are alternate ways to reexpress the magnetic field b in a form that may be more convenient for calculation or analysis. In radiation problems, the goal is to determine the radiated fields electric and magnetic from an antennas, knowing what currents are. As follows from the above consideration, it is more appropriate to write the. A current flowing through a wire produce a magnetic field looping around it. Chapter 5 the vector potential in magnetostatics the magnetic eld is divergence free, and we have the vector identity r r f 0 for any vector function f, therefore if we write b r a, then we ensure that the magnetic eld is divergence free. Br ar 0 always the divergence of a curl of a vector field fr. In radiation problems, the goal is to determine the radiated elds electric and magnetic from an antennas, knowing what currents are owing on the antenna. Assume that dl is an element of length pointing in the direction of current flow of a filamentary wire that carries a current i and r is the coordinate vector. Our usual reference point for the scalar potential in electrostatics is at for the vector potential in magnetostatics we imposed the condition these reference points arise from exploitation of the builtin ambiguities in the static potentials.
We explain the distribution of the magnetic potential and how to use it when solving for the electric field. Magnetic vector potential of an infinite wire r d r v rj. Of course an isolated circuit element cannot exist by itself, so, for the magnetic vector potential from a complete circuit, the line integral of this must be calculated around. The vector potential, boundary conditions on a and b. Chapter 5 magnetostatics university of minnesota duluth.
Vector potential since the divergence of any curl is zero i. Magnetic vector potential when we derived the scalar electric. The magnetic vector potential is preferred when working with the lagrangian in classical mechanics and quantum mechanics. In this chapter we define the four vector of the potentials of the electromagnetic field and relate it to the nonrelativistic electric scalar potential and magnetic vector potential. Magnetic field, force, and inductance computations for an. The curl of the vector potential gives us the magnetic field via eq. Simple analytic expressions for the magnetic field of a. From the magnetic form of gausss law br0, it is evident that the magnetic flux density br is a solenoidal vector field. The scalar potential can thus be evaluated from the two dimensional integral 1. Consider the biotsavart law for the vector potential. Eece557 magnetic dipoles supplement to text fall 2008 2 1. Deduce the form of the vector potential a of a longin. The carried out analysis showed that the magnetic vector potential is one of the most important concepts of classical electrodynamics. Thus bvxa, which, in terms of the curvilinear coordinate system introduced in the.
The magnetic vector potential a is a vector field, defined along with the electric potential. A vector field ar is called the magnetic vector potential because of its analogous function to the electric scalar potential v r. The magnetic vector potential a can be evaluated from ar 0 4. The term magnetic potential can be used for either of two quantities in classical electromagnetism. Vector potential of a long solenoid in the poincar.
Chapter 5 calculating the vector potential recall that for the solenoid, we found that as r 0nir2 2s 1 outside the solenoid there is no magnetic eld, but the potential is non trivial. Thus, inside the solenoid the vector potential is 2 a 1 n r i. Physics of magnetic field and vector potential aascit. The curl of the magnetic vector potential is the magnetic field.
The magnetic dipole is also introduced and the biotsavart law is described. Magnetostatics and the vector potential december 8, 2015. Vector potential and magnetic induction for a circular current loop we consider the problem of a circular loop of radius a, lying in the xy plane, centered at the origin, and carrying a current i, as shown in fig. Sheet of current 2 a dla xb a xa 4 comparing these two, a reasonable candidate is a 0 2 kzx. However, the divergence of has no physical significance. It was a scalar function, related to electric eld through e rv.
With the current loop located in and symmetric with respect to the xy plane, a is given by 1. The wave equations for the electromagnetic potentials. The magnetic field is related to the current as follows. Electric and magnetic forces in lagrangian and hamiltonian. Magnetic potential simion 2019 supplemental documentation. Simple analytic expressions for the magnetic field of a circular current loop james simpson, john lane, christopher immer, and robert youngquist abstract analytic expressions for the magnetic induction and its spatial derivatives for a circular loop carrying a static current. The expression \\frac\mu i4\pi r\textbfds\, then, is the contribution \\textbfda\ to the magnetic vector potential from the circuit element \\textbfds\. Magnetic fields of a localized current distribution similar to the electrostatic multipole expansion, if we have a localized current distribution and want to know the magnetic fields far away, we can make a magnetostatic multiplole expansion and only keep the first few terms. Multipole expansion of the magnetic vector potential consider an arbitrary loop that carries a current i. That wont work for the magnetic field except where j 0, because the curl of b is not zero in general. We know from helmholtzs theorem that a vector field is fully specified by its divergence and its curl. The wires are parallel to the zaxis, and are each a distance aaway from the zaxis as shown in the gure. Vector potential, electromagnetic induction and physical meaning article pdf available in european journal of physics 314. The electromagnetic field is described by two vector fields.
I will try to explain intuition behind it with minimum mathematics. Pdf we collect together several ideas that we have found helpful in teaching the magnetic vector potential a. This makes us believe that we got a more general form of the schrodinger equation this with a. In magnetostatics where there is no timevarying charge distribution, only the first equation is needed. Another example of calculating the magnetic vector potential in a case where the current extends to in. The electric scalar and magnetic vector potentials.
The knowledge of this potential and its evolution can only permit us to calculate all the force components acting upon charges. For a stationary point charge p q6 x r, where r is. The magnetic field is merely a spatial derivative of the vector field. The quantity is known as the magnetic vector potential. Vector and scalar potentials e83 where f is an arbitrary differentiable function of x,y,z,t, then. Physics 3323, fall 2016 problem set 8 due oct 21, 2014. Scalar and vector magnetic potential steady magnetic field. If the sources charges or currents are far away, e and b solve the homogeneous maxwell equations. We have no need to fix a gauge, since for slowly varying timedependent electric and magnetic fields, the natural gauge for a is the coulomb. The vector a is called the magnetic vector potential. Pdf vector potential, electromagnetic induction and. You remember that the vector potential function has some arbitrariness. Recall that a solenoidal field is the curl of some other vector field, e.
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