Derivation of Oerstead's Law from Maxwell's Laws

Stoke's Theorem states  
\[\oint_C \mathbf{E} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{E}) \cdot \mathbf{n} dS\]
  for a vector field  
\[\mathbf{E}\]
  with continuous derivatives.
Let  
\[\mathbf{F}\]
  be the magnetic field strength  
\[\mathbf{H}\]
  and use Maxwell's Law  
\[\mathbf{\nabla} \times \mathbf{H} = \mathbf{j} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\]
  to get  
\[\oint_C \mathbf{H} \cdot d \mathbf{r} = \int \int_S (\mathbf{j} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}) \cdot \mathbf{n} dS\]

If the electric field is steady all time derivatives are zero, and we obtain Oersted's Law:
\[\oint_C \mathbf{H} \cdot d \mathbf{r} = \int \int_S \mathbf{j} \cdot \mathbf{n} dS\]

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