Faraday's Law

Faraday's Law

Faraday's Law of Induction and Lenz's Law

Define the flux of the magnetic field through an open surface as: \(\(\Phi_B = \iint B \bullet dA\)\)

Faraday's Law: The electromotive force (emf) induced in a circuit is \(\(\epsilon = - \frac{d\Phi_B}{dt}\)\)

The minus sign is given by Lenz's Law. The induced current will appear in such a direction that it opposes the change in flux that produced it.

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For generator,

\[ \begin{gathered} \Phi_B = BA\cos\theta = BA\cos\omega t \\\\ \Rightarrow \epsilon = BA\omega \sin\omega t \end{gathered} \]

Motional emf and Induced emf

For motional emf, we have \(\(\epsilon = \int_{-}^{+} K \bullet dl = \int_C^D (\vec{v} \times \vec{B}) \bullet dl\)\)

For induced emf, we have

\[ \epsilon = \oint \vec{E} \bullet d\vec{l} = \oint (\vec{E}_{std} + \vec{E}_{ind}) \bullet d\vec{l} \]

\[ \Rightarrow \nabla \times \vec{E} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ E_x & E_y & E_z \\ \end{vmatrix} = - \frac{\partial \vec{B}}{\partial t} \]

In spherical coordinates,

\[ \nabla \times \vec{E} = \frac{1}{r^2\sin\theta} \begin{vmatrix} \hat{r} & r\hat{\theta} & r\sin\theta\hat{\phi} \\ \frac{\partial }{\partial r} & \frac{\partial }{\partial \theta} & \frac{\partial }{\partial \phi} \\ E_r & r E_{\theta} & r\sin\theta E_{\phi} \\ \end{vmatrix} \]

In cylindrical coordinates,

\[ \nabla \times \vec{E} = \frac{1}{r} \begin{vmatrix} \hat{r} & r\hat{\theta} & \hat{z} \\ \frac{\partial }{\partial r} & \frac{\partial }{\partial \theta} & \frac{\partial }{\partial z} \\ E_r & r E_{\theta} & E_{z} \\ \end{vmatrix} \]