One of Maxwell's equations in matter has the local form H = J _f + D t

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Vector calculus and field geometry

Electricity, Magnetism and Electromagnetism, Ch Displacement current, Maxwell's equations and boundary conditions

Problem

One of Maxwell's equations in matter has the local form∇×H=Jf+∂D∂t.\nabla\times\mathbf{H} = \mathbf{J}_f + \frac{\partial\mathbf{D}}{\partial t}. Let SS be an open surface with unit normal n^\hat{\mathbf{n}} and boundary ∂S\partial S traversed in the sense tied to n^\hat{\mathbf{n}} by the right-hand rule, and let If,encI_{f,\mathrm{enc}} be the free current through SS counted positive in the sense of n^\hat{\mathbf{n}}. Which one of the following is the integral form of that equation? (a) ∮∂SH⋅dl=If,enc+ddt∫SD⋅dA\displaystyle\oint_{\partial S}\mathbf{H}\cdot d\mathbf{l} = I_{f,\mathrm{enc}} + \frac{d}{dt}\int_S \mathbf{D}\cdot d\mathbf{A} (b) ∮∂SH⋅dl=If,enc+ε0ddt∫SE⋅dA\displaystyle\oint_{\partial S}\mathbf{H}\cdot d\mathbf{l} = I_{f,\mathrm{enc}} + \varepsilon_0\frac{d}{dt}\int_S \mathbf{E}\cdot d\mathbf{A} (c) ∫SH⋅dA=If,enc+ddt∫SD⋅dA\displaystyle\int_S \mathbf{H}\cdot d\mathbf{A} = I_{f,\mathrm{enc}} + \frac{d}{dt}\int_S \mathbf{D}\cdot d\mathbf{A} (d) ∮∂SB⋅dl=μ0If,enc+μ0ddt∫SD⋅dA\displaystyle\oint_{\partial S}\mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{f,\mathrm{enc}} + \mu_0\frac{d}{dt}\int_S \mathbf{D}\cdot d\mathbf{A}

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