Curl of Gradient is Zero Let , , be a scalar function. Then the curl of the gradient of , , is zero, i.e. 0 Note: This is similar to the result 0 where k is a scalar. Proof ̂ , , ̂ ̂ 0 As a result, magnetic scalar potential is incompatible with Ampere’s law. For if there exists a scalar function U such that , then the curl of is 0. However Ampere’s law states that . Alternatively, we can say that magnetic scalar potential is not very useful as it can only be applied to current free region. Divergence of Curl is Zero Let , , be a vector function. Then the divergence of the curl of , , is zero, i.e. ∙ 0 Note: This is similar to the result ∙0 as is perpendicular to . Proof ̂ ∙ , , ∙ ∙ ̂ 0 The Theory of Vector Field Griffiths: Section 1.6 The Theory of Vector Fields. Theorem 1: For a Curl-less (or irrotational) vector field , the following statements are equivalent. (a) 0 everywhere (b) ∮ ∙ 0 for any closed loop. c ∙ is path independent for any given end points. d , i.e. is the gradient of some scalar function V. Proof (a) ⟺ (b). By Stokes theorem for curl, we have: ∙ ∙ ⟺0 0 ∙ ∙ where l is a closed loop (or boundary) enclosing the area A. b ⟺ (c). ∙ 0⟺ ∙via path ∙via path 0 ⟺ ∙via path ∙via path 0⟺ ∙via path ∙via path c ⇔ (d) Let 0 and . Since the integral is path independent, the result is a constant that depends only on the end point . Thus ∙ , , ⟺ ∙, , ⇔ The scalar potential is not unique and the negative sign is by convention. Any constant can be added to V with impunity, since this will not affect its gradient, i.e. where c is a arbitrary scalar constant. Theorem 2: For a Divergence-less (or solenoidal) vector field , the following statements are equivalent. (a) ∙ 0 everywhere (b) ∮ ∙ 0 for any closed surface. c ∙ is independent of surface, for any given boundary line. d , i.e. is the curl of some vector function . Proof (a) ⟺ (b). By Stokes theorem for divergence, we have: ∙∙ ∙ ⟺0 0 ∙ ∙ for any closed surface area A enclosing the volume V. b ⟺ (c). ∙ 0⟺ ∙ ∙ 0⟺ ∙ ∙ ⟺ ∙ ∙ where with both and sharing the same boundary. Also ∙ ∙ with and sharing the same boundary and . c ⇔ (d) Since the integral is area independent, the result is a constant that depends only on the boundary. Using Stokes theorem for curl, there exists a vector function such that ∙ constant ∙ ∙ where l is the boundary of A. As a result, we have The vector function is not unique. The gradient of any scalar function C can be added to without affecting the curl, since the curl of gradient is zero, i.e. Electric Scalar Potential and Magnetic Vector Potential Faraday’s law of induction states that the curl of electric field is . In electrostatic field, there is no induced magnetic field and 0 . By Theorem 1, there exists a scalar function V such that . We called V the electric scalar potential and said that electric field is conservative. Gauss’s law for magnetostatics states that the divergence of magnetic field is ∙ 0. By Theorem 2, there exists a vector function such that . We called the magnetic vector potential. Recall that by Ampere’s law, the curl of magnetic field 0 if there is current flowing in the region. As a result, magnetic field is non-conservative. The Helmholtz Theorem Griffiths: Section 1.6 The Theory of Vector Fields, Appendix B: The Helmholtz Theorem. Let be a vector function such that its divergence ∙ for some scalar function D and its curl for some vector function . Then where the scalar function V and vector function is given by 1 ′ 1 ′ ′ and ′ 4 |′| 4 |′| Solving Magnetic Field using Helmholtz Theorem Maxwell’s equations for magnetic fields states: div = ∙ 0 (Gauss’s law) curl = (Ampere’s law) Using Helmholtz Theorem, we have ⇒ where 1 0 ′ ′ C constant scalar and ′ 4 |′| 4 |′| ⇒0 Combining Ampere’s law with we have: ∙A (1) Now, we would like to choose a magnetic vector potential that is divergence-less, i.e. ∙A 0. This is possible because, just like electric scalar potential, magnetic vector potential had a built- in ambiguity also. We can add to it any function whose curl vanishes with no effect on the magnetic field. Since the curl of gradient is zero, the function that we add should be the gradient of some scalar function V, i.e. We can exploit this ambiguity freedom to make divergence-less. To do that, suppose our original potential is not divergence-less. If we add to it the gradient of some scalar function V, the new divergence is ∙ ∙ ∙ ∙ ∙ To make divergence-less, we must find the scalar function V such that ∙. But this is just Poisson equation and we know how to solve it for V. Thus, we can always choose a potential that is divergence-less and equation (1) then becomes This is again nothing but Poisson’s equation, or rather, it is three Poisson’s equation, one for each Cartesian component: Solving Electric Field using the Helmholtz Theorem Maxwell’s equations for electric fields states: div = ∙ (Gauss’s law) curl = 0 (Faraday’s law for electostatic) Using Helmholtz Theorem, we have where 1 ′ 1 0 ′ and ′ constant vector 4 |′| 4 |′| ⟹ Combining Gauss’s law ∙ with we have: ∙ ∙ ⟹ (Poisson’s Equation) (: Laplacian) .