<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/e7b06146-cd0a-45a2-abcc-bd7c7f650445/Circulation.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/e7b06146-cd0a-45a2-abcc-bd7c7f650445/Circulation.png" width="40px" /> Circulation: is a line integral around a closed loop, signified by $\oint$. You need to specify the direction, clockwise or anticlockwise when viewed from above

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Stokes’ Theorem

The circulation of a vector field, $\vec B$, can be related to the surface integral of the curl of the field by Stokes’ Theorem:

$$ \oint_C \vec{B} \cdot \text{d}\vec{l} = \iint_S \left( \vec{\nabla} \times \vec{B} \right) \cdot \text{d}\vec{S} $$

• The surface $S$ is bound by the curve $C$. The theorem is only valid if the field $\vec B$ is continuous and differentiable at all points on the surface $S$
• $\text d\vec S$ is defined as:
  • $\text d\vec S=+\,\text dx \,\text dy\,\hat k$ in the anti-clockwise direction, $xy$ plane
  • $\text d\vec S=-\,\text dx \,\text dy\,\hat k$ in the clockwise direction, $xy$ plane

The theorem can also be used to define a curl

$$ \hat{n}\cdot \left( \vec{\nabla} \times \vec{B} \right) = \lim_{\delta A \to 0} \frac{1}{\delta A} \oint_C \vec{B} \cdot \text d\vec{l} $$

• where $\delta A$ is an element area, $\hat n$ is normal to the surface containing the curve $C$

Ampere’s Law

$$ \oint_C \vec B \cdot \text d \vec l = \mu_0 I $$

• This can be re-written using the current per unit area density $\vec J$

$$ \oint_C \vec B \cdot \text d \vec l = \mu_0 \iint_S \vec J \cdot \text d \vec S $$

• Then using Stokes’ Theorem

$$ \oint_C \vec B \cdot \text d \vec l = \iint_S \left ( \vec \nabla \times \vec B \right ) \cdot \text d \vec S $$

• So we find that $\vec \nabla\times\vec B=\mu_0 \vec J$

Displacement Current and Continuity Equation

The equation above does not satisfy the continuity equation, so Maxwell solved it by adding a constant $\vec J_D=\epsilon_0\frac{\text{d} \vec E}{\text dt}$ which gives

$$ \vec{\nabla} \times \vec{B} = \mu_0 (\vec{J} + \vec{J}_D) = \mu_0 \vec{J} + \mu_0\epsilon_o \frac{\text d\vec{E}}{\text dt} $$

• The continuity equation expresses the conservation of charge: