If the gauge is changed for a vector potential ax in the rotated. Given maxwells four equations, demonstrate the existence of a vector magnetic potential and a scalar electric potential. Prove that 4 vector potential is really a 4 vector. It combines both an electric scalar potential and a magnetic vector potential into a single four vector as measured in a given frame of reference, and for a given gauge, the first component of the electromagnetic four potential is conventionally taken to be the electric. The profiles of surfaces of constant a z are shown in fig. The four potential consider the vector potential a and scalar potential. Potential fourvector from eric weissteins world of physics. In the general theory of quantum electrodynamics, one takes the vector and scalar potentials as. We focus on the orthogonal case, which includes cartesian, cylindrical and spherical coordinates. The electromagnetic tensor double contravariantis given by 51, section 42 for the horizontal planeparallel capacitor the tensor is the mixed tensor is. In vector calculus, a vector potential is a vector field whose curl is a given vector field.
A reasonable guess is that momentum is a 3vector conjugate to position, so we need to find what the fourth component is to make a 4vector. Electromagnetic fields and energy mit opencourseware. Maxwells equations are reduced to a simple fourvector equation. In other words, the 4vector dot product will have the same value in every frame. That it cannot be replaced by the magnetic field in any easy way was observed by one man after the other who tried to do so.
The fact that the vector potential appears in the wave equation of quantum mechanics called the schrodinger equation was obvious from the day it was written. But avoid asking for help, clarification, or responding to other answers. On the other hand, the vector potential has only one relevant component, the azimuthal one, and the equations obtained from it are wellconditioned. A reasonable guess is that momentum is a 3 vector conjugate to position, so we need to find what the fourth component is to make a 4 vector. However, the divergence of has no physical significance.
Finding potential functions c marc conrad november 6, 2007 1 introduction given a vector. Just as e grad is the integral of the eqs equation curl e 0, so too is 1 the integral of 8. These are called vector quantities or simply vectors. Finding a from j find the vector potential for a current i along the z axis from z 1 to z 2. If vectorpotential cannot verify that v has a vector potential, it returns the vector with all three components equal to nan. Now suppose the particle is bound in a strong central potential and b is relatively weak. The magnetic vector potential university of texas at austin. The 4vector is a powerful tool because the dot product of two 4vectors is lorentz invariant. This becomes more and more apparent the more deeply we go into the quantum theory. An electromagnetic fourpotential is a relativistic vector function from which the electromagnetic field can be derived. Divergence in this section, we present the divergence operator, which provides a way to calculate the flux associated with a point in space. This is for the purpose of making all components of the four vector potential a. But life is much easier if you solve problems using 4vectors and the 4vector dot product.
Most of the textbooks i found only mention that divergence of 4 vector potential equals to zero and the dalembertian of it is a four vector current and therefore it should be a four vector. In the case of the vector potential, we can add the gradient of an arbitrary scalar function. Thus, inside the solenoid the vector potential is 2 a 1 n r i. With 67 and 68 inserted into 66, we see that maxwell has obtained the. Remember, these are also the lines of magnetic flux density, o h. So it allows you to pick and choose the frame in which the problem is easiest to solve. As measured in a given frame of reference, and for a given gauge, the first component of the electromagnetic four. This concept is useful, since it is sometimes more convenient to compute the potential and then compute the electric. Therefore, the magnetic field produced by these two straight. It is obviously important it determine how energy and momentum transform in special relativity. This article is about the general concept in the mathematical theory of vector fields. This page will introduce the four potential, and the fourcurrent notations, as well as the dalembertian, which is used when studying these topics under the theoretical framework of special relativity. 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.
With 67 and 68 inserted into 66, we see that maxwell has obtained the most general integral form of faradays law. The expression \\frac\mu i 4 \pi r\textbfds\, then, is the contribution \\textbfda\ to the magnetic vector potential from the circuit element \\textbfds\. We know from helmholtzs theorem that a vector field is fully specified by its divergence and its curl. Ab dabcos ab where ab is the angle between the vectors as in linear algebra note. The current density 4 vector the potential 4 vector there are many ways of writing the laws of electromagnetism. As mentioned, to uniquely define a vector, we must specify its divergence as well as its curl. Electric potential, lorentz invariant, magnetic vector potential. Vector potential of the coulomb gauge semantic scholar. Electrodynamicsfourvectors wikibooks, open books for. These constructs, while a little confusing for some people, are fundamental to the way in which modern physicists study electric and magnetic. Prove that 4 vector potential does really a 4 vector.
However, more recently, the physical meaning of vector potential in classical electromagnetism has been advocated in connection with a particular phenomenon of electromagnetic induction. I will try to explain intuition behind it with minimum mathematics. The combination of the scalar electric potential \phi and the magnetic vector potential a into the four vector a\mu \phi,\mathbfa \left\matrix\phi\cr a1\cr a2\cr a3\cr\right. Lecture 28 antennas and radiation and the hertzian dipole. The vector potential exists if and only if the divergence of a vector field v with respect to x equals 0. Vector analysis university of colorado colorado springs. Potential fourvector from eric weissteins world of. Vector potential article about vector potential by the. The potential 4vector there are many ways of writing the laws of electromagnetism. Another change we do is to change the normalization of the vector potential aand correspondingly b by a factor of c. Along the two straight sections of the loop, and are parallel or opposite, and thus. Gradient the gradient of a scalar field is a vector that points in the direction in which the field is most rapidly increasing, with the scalar part equal to the rate of change.
The vector potential can then be written as a 1 2 b. Recall that a solenoidal field is the curl of some other vector field, e. For this uniform electric field the potential 4vector, where and are the vector potential and the scalar potential, respectively, is given by the contravariant 4vector. From this lagrangian we can generate the 4d potential of the e ect of all. In general, we cannot guarantee the existance of such a function.
The expression \\frac\mu i4\pi r\textbfds\, then, is the contribution \\textbfda\ to the magnetic vector potential from the circuit element \\textbfds\. Gravitational four potential is a four vector function 4 vector, by which the properties of gravitational field are determined in the lorentzinvariant theory of gravitation, as well as in the covariant theory of gravitation. Scalars may or may not have units associated with them. Thanks for contributing an answer to mathematics stack exchange. The vector potential a is not unique only the curl of the vector potential is a well defined quantity i. This page will introduce the fourpotential, and the fourcurrent notations, as well as the dalembertian, which is used when studying these topics under the theoretical framework of special relativity.
Its the laplacian of ax in the x direction, laplacian of ay. Thus, if you are trying to solve for a quantity which can be expressed as a 4vector dot product, you can choose the simplest. 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. Electric and magnetic forces in lagrangian and hamiltonian. Vector calculus in curvilinear coordinates gri ths. 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. Displacement, velocity, acceleration, electric field. First visualize a simple concept from basic physics. The gravitational 4 potential includes the scalar and vector potentials of gravitational field. An electromagnetic four potential is a relativistic vector function from which the electromagnetic field can be derived. Peeter joot august 15, 2008 1 goldsteins classical mechanics, and many other texts, will introduce the four potential starting with maxwells equation in scalar, vector, bivector, trivector expanded form. For the vector potential in fluid mechanics, see stream function.
Conception of the scalarvector potential in contemporary arxiv. Vector potential article about vector potential by the free. 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. However, the most obviously lorentz invariant way is to write them in terms of the vector and scalar potentials see sect. The maxwell equations have an alternative notation, which involves two new quantities. Vector potential of vector field matlab vectorpotential. This is for the purpose of making all components of the fourvector potential a. It combines both an electric scalar potential and a magnetic vector potential into a single four vector. The combination of the scalar electric potential and the magnetic vector potential a into the fourvector. In principle, all problems can be solved without invoking the use of 4vectors. Scalars and vectors scalars and vectors a scalar is a number which expresses quantity.
That is because the 4vector dot product is an invariant it is the same in all frames. Can classi cal electrodynamics furnish correct results for the fields in a moving coordinate system, or at least offer an acceptable. However i do not see there is any tensor theorem to get this conclusion. Now that we have a test that a vector eld must pass in order to be conservative, a natural. Acos abis the component of aalong band bcos ab is the component of. Remember that we could add an arbitrary constant to without affecting e. R1, wherer1 andr2 are the position vectors of pointsp1 andp2,respectively. The curl of the vector potential gives us the magnetic field via eq. For the vector potential in electromagnetism, see magnetic vector potential. B dka delementbyelement multiply by k scalar product or dot product. The quantity is known as the magnetic vector potential. Vector potentials in gauge theories in flat spacetime. A vector field ar is called the magnetic vector potential because of its analogous function to the electric scalar potential vr. This section is reserved for advanced students, with background in electricity and magnetism, and vector differential equations problem.
6 653 1485 1285 721 1161 1108 299 589 1494 1100 42 459 1421 536 929 573 1496 1415 655 175 55 1244 1180 1215 430 159 79 90 1337 643 501 784 98 586 727 681