The curl of the vector potential gives us the magnetic field via eq. Recall that a solenoidal field is the curl of some other vector field, e. Vector magnetic potential page 1 vector magnetic potential 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. A circular loop of radius carries current we place the origin at the center of the loop, with polar axis perpendicular to the plane of the loop. Schmidt department of physics and astronomy arizona state university tempe, az u. Right hand rule for force due to a magnetic field to find the direction of the force of a magnetic. In coulomb gauge, the vector potential is then given by ax 0 4. In vector calculus, a vector potential is a vector field whose curl is a given vector field. Formulation weng cho chew1, 2, invited paper abstractthe mixed vector and scalar potential formulation is valid from quantum theory to classical electromagnetics. Magnetic field of a current loop far field magnetic dipole.
The vector potential due to the green segment has a direction of the green arrow. Magnetism magnetic field of steady currents britannica. If the wire is of infinite length, the magnetic vector potential is infinite. We knowby biotsavart law that the magnetic field it produces on its axis is given. An overview of the underestimated magnetic loop hf antenna.
Faraday and joseph henry, however, found that if a current loop. Amperes law applied to the loop c and the surface leads to. Small currentloops vector potential due to a current loop the magnetic vector potential at a distant point r due to a current i flowing round a small loop c, is found by substituting jr. The magnetic field due to a current distribution satisfies amperes law.
Vector potential components in small loop antenna physics. Upon microscopic examination, it contains many tiny dipoles, with a net alignment along some direction. Applications of amp eres law i c bdl 0 z a jds 0i 1 like gauss law for electric elds amp eres law is the most e cient way of calculating. We know from helmholtzs theorem that a vector field is fully specified by its divergence and its curl. The magnetic field due to a current distribution satisfies. The process by which antennas radiate can be easily understood in terms of the.
In the presence of a magnetic field, matter becomes magnetized. Now we can calculate the vector potential inside the sphere at. Simple analytic expressions for the magnetic field of a. Theory of electromagnetic fields andrzej wolski university of liverpool, and the cockcroft institute, uk abstract we discuss the theory of electromagnetic. Direction of a contribution da is parallel to a current element idl 5. Here in this particular example, we found that the addition of an arbitrary constant vector ra axayaz r oox oy oz m to the magnetic vector potential ar was allowed, i. Vector potential and magnetic induction for a circular current loop. Vector potential electromagnetics with generalized gauge for inhomogeneous media. Fall 2002 2 multipole expansion of the magnetic vector potential consider an arbitrary loop that carries a current i.
Multipole expansion of the magnetic vector potential consider an arbitrary loop that carries a current i. Understand why vector potential is defined in a gauge. However, at the edge there is no adjacent loop to do the canceling. The net force on any close currentcarrying loop is zero. We describe the state of magnetic polarization by the vector quantity. Phy2206 electromagnetic fields small current loops 1 small currentloops vector potential due to a current loop the magnetic vector potential at a distant point r due to a current i flowing round a small loop c, is found by substituting jr. Then we can still use the concept of a vector potential a whose curl gives b.
Vectorpotentialforthemagneticfield university of texas at. Note that this is the correct form for magnetic vector potential for static elds, where. Example using spherical harmonics sp 2019 magnetic. Fitzgeralds calculation of the radiation of an oscillating. The dipole direction is pointing up from the loop for the counter.
One quick method is to realize that the integral is a vector quantity. One important objective of magnetostatics is to derive formulae for the selfinductance and mutual inductance for given current congurations. Vector potential of a current carrying circular loop physics forums. Using the vector potential is often more difficult for simple problems for the following reason. 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. Vectorpotentialforthemagneticfield university of texas. Chapter 9 sources of magnetic fields mit opencourseware. The magnetic field of a spinning, uniformly charged sphere 25 points this problem is based on problem 1 of problem set 8. This concept is useful, since it is sometimes more convenient to compute the potential and then compute the electric. A current flowing through a wire produce a magnetic field looping around it. Multipole expansion of the vector potential of a current loop the vector potential of a current ring is given by. First visualize a simple concept from basic physics. Use of the vector potential greatly facilitates calculation of the magnetic. 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.
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. As the loop slides by position 2 the flux through the loop is. 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. In gauss law we want to choose our gaussian surface so that the electric field is constant on. Here i is the current in the loop, a is the loop area, r is the radial distance from the center of the loop, and. At large distance only the first couple of terms of the multipole expansion need to be considered. Vector potential components in small loop antenna physics forums. The first term is called the monopole term and is equal to zero since the line integral of is equal to zero for any closed loop.
Physics 3323, fall 2016 problem set 8 due oct 21, 2014 reading. A few magnetic field lines produced by a current in a loop are shown in figure 1. Mar 17, 2014 consider a small, thin loop in the x,y plane centered in the origin and with radius a. Note that the retarded time is not the same for all points of the source distribution,the most distant parts of the source have earlier retarded times than the nearby ones. We are interested in the vector potential \mathbfa generated by. For example, for any closed loop l, the integral i l ad. It seems one of the best kept secrets in the amateur radio community is how well a small diminutive magnetic loop antenna can really perform in practice compared with large traditional hf antennas. The vector potential of this current loop can be written as at large distance only the first couple of terms of the multipole expansion need to be considered. The objective of this article is to disseminate some practical information about. Physics 3323, fall 2016 problem set 8 due oct 21, 2014. Vector potential electromagnetics with generalized gauge. The quantity is known as the magnetic vector potential.
Find the exact magnetic eld a distance dabove the center of a square loop of side w, carrying a current i. As the loop slides by position 1, the flux through the loop is increasing and it is pointing out of the page. Then, by symmetry, once the d 0integral is performed, the only direction. The vector potential, a, of the loop is given by 3. However, the divergence of has no physical significance. The divergence of the vector potential vanishes for static fields, since. In the present of a magnetic field, matter becomes magnetized. We consider the problem of a circular loop of radius a, lying in the xy plane, centered at the.
Thus we see that the magnetic vector potential in the vicinity of a straight wire is a vector field parallel to the wire. Magnetism magnetism magnetic field of steady currents. 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. Biot savart law and amperes law in the last lecture, we have shown that the magnetic force. The distance rqp from the current element i dz to the field. There will be additional forces on the element dlproduced by the current in other parts of its own loop. Poissons equation for a replacing b with r a in the di erential form of amp eres law. The magnetic vector potential at a distant point r due to a current i flowing round a small loop c, is found by substituting j. The vector potential of this current loop can be written as.
We define the magnetic dipole moment to be a vector pointing out of the plane of the current loop and with a magnitude equal to the product of the current and loop area. The vector potential in a certain region is given by c is a positive constant consider the imaginary loop shown. We define the magnetic dipole moment to be a vector pointing out of the plane of the. Loop free vectors, photos and psd downloads freepik. 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. Place the loop in the xy plane with its center at the origin. Applications of amp eres law university of edinburgh.
For a simple current loop of radius rin the xyplane, carrying a steady current i, the magnetic dipole momentis m i nd2x. The area vector, and thus the direction of the magnetic dipole moment, is given by a. The scalar potential for the circular current loop. When charges move in a conducting wire and produce a current i, the magnetic field at. Bx is the magnetic induction produced by this other loop. In the following examples we will calculate the magnetic fields b oh established by some. Jul 21, 2014 consider a circular loop of radius r that carrys a uniform current i.
This is analogous to a scalar potential, which is a scalar field whose gradient is a given vector field formally, given a vector field v, a vector potential is a vector field a such that. Chapter 23 magnetic flux and faradays law of induction. The present rapid development in quantum optics applications calls for. Vector potential of a current carrying circular loop. Despite ambiguity ofthe vector potential itself, some ofits properties are gauge invariant, i. Due to the cylindrical geometry, we may choose the observation point p in the xz plane without loss of generality. The vector product is required since observations show that the force acting on a moving. Vector potential of a magnetic dipole produced by a current loop.
Potential energy since the magnetic dipole wants to line up with the magnetic field, it. If, on the other hand, amperes law is applied to the loop c and the surface. Another equation one frequently sees is an expression for the in. In the last lecture we had defined the magnetic field and we had seen that a source of the. Calculating the magnetic vector and scalar potentials gri ths.
For a finite length, the potential is given exactly by equation 9. The field is equivalent to that from a tiny bar magnet a magnetic dipole. 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. Also note that the retarded potentials reduce properly to. What can you say about the direction of the magnetic field in this region. The vector potential, boundary conditions on a and b. I will try to explain intuition behind it with minimum mathematics. The vector potential and magnetic induction and of the circular current loop, of radius a in the x y plane carrying a current i, are calculated by jackson. Therefore we can represent the bfield by the vector potential. I have been starting to use vectors, and have noticed that in all of the code i see to iterate though a vector via indices, the first parameter of the for loop is always.
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