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Unformatted text preview: Vector Potential Mankei Tsang Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125 (Dated: May 19, 2005) The formalism of vector potential is extremely useful for solving the Maxwells equations. INTRODUCTION Lets look at the Maxwells equations again: E = (1) B = 0 (2) E = B t (3) B = J + E t (4) Imagine that you are given ( r , t ) and J ( r , t ), how do you solve for E and B using the Maxwells eqautions? It turns out that it is not easy to do so directly, but through the formalism of vector potential one can take a detour and solve for E and B indirectly. DEFINITIONS OF VECTOR AND SCALAR POTENTIALS First look at the Gausss law for magnetism, B = 0 . (5) The divergence of B is always zero, so we can always define B as the curl of another vector quantity, because the divergence of curl is always zero as well, B A (6) B = ( A ) . (7) 2 A is called the vector potential . Lets plug the definition of vector potential, Eq. (6), into Faradays law, Eq. (3), E = t ( A ) (8) parenleftBig E + A t parenrightBig = 0 (9) The curl of the quantity E + A t is zero, which means that we can define it as the gradient of a scalar function, because the curl of a gradient is always zero as well, E + A t V (10) ( V ) . (11) Hence E and B can be defined in terms of the vector potential A and the scalar potential V , E = V A t , (12) B = A . (13) MAXWELLS EQUATIONS IN TERMS OF SCALAR AND VECTOR POTENTIALS These definitions, Eqs. (12) and (13), already satisfy the Gausss law for magnetism and Faradays law. Now we can plug them into Gausss law and Amperes law and see what we get. First plug Eq. (12) into the Gausss law, Eq. (1), parenleftBig V A t parenrightBig = (14) 2 V + t ( A ) = . (15) We can also plug Eq. (12) and Eq. (13) into Amperes law, Eq. (4), ( A ) = J + t parenleftBig V A t parenrightBig (16) ( A ) 2 A = J V t 2 A t 2 (17) 2 A 2 A t 2 parenleftBig A + V t parenrightBig = J . (18) 3 With Eqs. (15) and (18), we have successfully reduced four Maxwells equations into two equations, and from six unknowns (three components for E and three components for B ) to four unknowns (one scalar V and three components for A ). I will write them again together: 2 V + t ( A ) = (19)...
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This note was uploaded on 02/04/2008 for the course EE 151 taught by Professor Psaltis during the Spring '05 term at Caltech.
 Spring '05
 Psaltis
 Electrical Engineering, Electromagnet

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