Vector potential since the divergence of any curl is zero i. We explain the distribution of the magnetic potential and how to use it when solving for the electric field. 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.
Recall that a vector field is uniquely determined by specifying its divergence and curl. 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 field, force, and inductance computations for an. I will try to explain intuition behind it with minimum mathematics. In radiation problems, the goal is to determine the radiated fields electric and magnetic from an antennas, knowing what currents are. 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. 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. 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. V jr0 jr r0j d3r0 1 provided that all currents are contained within a. Magnetic potential simion 2019 supplemental documentation. Vector potential, electromagnetic induction and physical meaning article pdf available in european journal of physics 314. Simple analytic expressions for the magnetic field of a. The electromagnetic field is described by two vector fields.
The magnetic vector potential a is a vector field, defined along with the electric potential. The curl of the vector potential gives us the magnetic field via eq. Dalarsson, in tensors, relativity, and cosmology second edition, 2015. From the magnetic form of gausss law br0, it is evident that the magnetic flux density br is a solenoidal vector field. Pdf vector potential, electromagnetic induction and. Electric and magnetic forces in lagrangian and hamiltonian. The magnetic vector potential a can be evaluated from ar 0 4. Eece557 magnetic dipoles supplement to text fall 2008 2 1. 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. Vector and scalar potentials e83 where f is an arbitrary differentiable function of x,y,z,t, then. This makes us believe that we got a more general form of the schrodinger equation this with a. With the current loop located in and symmetric with respect to the xy plane, a is given by 1. This lecture introduces the concept of the magnetic vector potential, which is analogous to the electric potential. Conception of the scalarvector potential in contemporary.
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. What is the direction of the magnetic vector potential in. Now suppose the particle is bound in a strong central potential and b is relatively weak. The magnetic vector potential contributed by a length. A current flowing through a wire produce a magnetic field looping around it.
This is similar to how the electric field e can be conveniently reexpressed in terms of electric potential. For a stationary point charge p q6 x r, where r is the charge position, 4 ul for brevity lets write r x r. Vector potential of a long solenoid in the poincar. Sheet of current 2 a dla xb a xa 4 comparing these two, a reasonable candidate is a 0 2 kzx.
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. Electromagnetic fields and energy mit opencourseware. You remember that the vector potential function has some arbitrariness. The term magnetic potential can be used for either of two quantities in classical electromagnetism. In magnetostatics where there is no timevarying charge distribution, only the first equation is needed. If the sources charges or currents are far away, e and b solve the homogeneous maxwell equations. The magnetic vector potential is preferred when working with the lagrangian in classical mechanics and quantum mechanics. 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.
As follows from the above consideration, it is more appropriate to write the. It was a scalar function, related to electric eld through e rv. 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 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. The quantity is known as the magnetic vector potential. Although we cannot express the magnetic field as the gradient of a scalar potential function, we shall define a vector quantity. Physics of magnetic field and vector potential aascit. Magnetostatics and the vector potential december 8, 2015. The curl of the magnetic vector potential is the magnetic field. Recall that a solenoidal field is the curl of some other vector field, e. 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. The expression \\frac\mu i4\pi r\textbfds\, then, is the contribution \\textbfda\ to the magnetic vector potential from the circuit element \\textbfds\. 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. Magnetic vector potential of an infinite wire r d r v rj.
Br ar 0 always the divergence of a curl of a vector field fr. The same derivation1 may be followed as for the electrostatic. Chapter 5 magnetostatics university of minnesota duluth. 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. The vector potential, a, of the loop is given by 3. The scalar potential can thus be evaluated from the two dimensional integral 1. Physics 3323, fall 2016 problem set 8 due oct 21, 2014 reading.
Consider the biotsavart law for the vector potential. The magnetic dipole is also introduced and the biotsavart law is described. The magnetic field is merely a spatial derivative of the vector field. Multipole expansion of the magnetic vector potential consider an arbitrary loop that carries a current i. Classical electromagnetism university of texas at austin. The vector potential can then be written as a 1 2 b.
The electric scalar and magnetic vector potentials. 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. However, the divergence of has no physical significance. Pdf thoughts on the magnetic vector potential researchgate. 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. An educational path for the magnetic vector potential and its. 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. First visualize a simple concept from basic physics. The vector potential, boundary conditions on a and b. The wires are parallel to the zaxis, and are each a distance aaway from the zaxis as shown in the gure. The carried out analysis showed that the magnetic vector potential is one of the most important concepts of classical electrodynamics.
The knowledge of this potential and its evolution can only permit us to calculate all the force components acting upon charges. That wont work for the magnetic field except where j 0, because the curl of b is not zero in general. The magnetic field is related to the current as follows. Calculating the magnetic vector and scalar potentials gri ths. We know from helmholtzs theorem that a vector field is fully specified by its divergence and its curl. A vector field ar is called the magnetic vector potential because of its analogous function to the electric scalar potential v r. Magnetic potential refers to either magnetic vector potential a or magnetic scalar potential. Thus, inside the solenoid the vector potential is 2 a 1 n r i. The wave equations for the electromagnetic potentials.
Physics 3323, fall 2016 problem set 8 due oct 21, 2014. Thus bvxa, which, in terms of the curvilinear coordinate system introduced in the. Another example of calculating the magnetic vector potential in a case where the current extends to in. Magnetic vector potential an overview sciencedirect topics.
Scalar and vector magnetic potential steady magnetic field. This concept is useful, since it is sometimes more convenient to compute the potential and then compute the electric. Deduce the form of the vector potential a of a longin. Magnetic vector potential when we derived the scalar electric. Magnetic vector potential, a, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field. Pdf we collect together several ideas that we have found helpful in teaching the magnetic vector potential a.
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