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What is wrong with this form of the Maxwell-Faraday equation?

$$\oint \vec{E}\ \partial \vec l= \bigcirc \hspace{-1.4em} \int \hspace{-.6em} \int \frac{\partial \vec B}{\partial t}$$

"Line integral of the electric field is equal to the double integral of partial derivative of magnetic field with respect to time".

So far as I remember the correct form used to be "Line integral of electric field is always (negative) surface integral of partially derived magnetic field..."

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2 Answers 2

The integral form of the Maxwell-Faraday law is $$ \oint\limits_{\partial S} \mathbf{E} \cdot d\boldsymbol\ell = -\frac{d}{dt} \int\limits_S \mathbf{B} \cdot \hat{\mathbf{n}}\,da.$$ If you want to apply the time derivative to the integral on the RHS, you must account for two effects that can cause a change in the magnetic flux: the time derivative of the magnetic field, and the velocity $\mathbf{v}$ of the surface $S$ through the field. This gives $$ \oint\limits_{\partial S} \mathbf{E} \cdot d\boldsymbol\ell = -\int\limits_S \frac{\partial\mathbf{B}}{\partial t} \cdot \hat{\mathbf{n}}\,da\; - \oint\limits_{\partial S} \mathbf{B} \times \mathbf{v} \cdot d\boldsymbol\ell.$$ (See, e.g., Jackson 3rd edition, eq. 5.137.) You are correct that there must be a minus sign on the RHS; this is the mathematical statement of Lenz's law.

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In your first equation are you sure that the derivative is total ? I would go for a partial derivative with respect of time. –  ChocoPouce Jul 2 '13 at 7:47

Lenz's law states that the electromotive force is equal to the negative time derivative of magnetic flux:

$$\mathcal{E} = -\frac{\partial}{\partial t} \phi_B$$.

The magnetic flux simply represents the surface integral of the magnetic field. This is important, as the area of this surface and the magnetic field at each point along this surface is now taken into account. Since the electromotive force is the line integral of the electric field along a path for electrons to flow, Lenz's law reduces to the following:

$$\oint_{\partial S} \mathbf{E} \cdot d\mathbf{\mathcal{l}} = -\frac{\partial}{\partial t} \iint_{S} \mathbf{B} \cdot d\mathbf{A}$$

The above is the true integral form of the Maxwell-Faraday equation. What you have shown both forgets about the minus sign as well as lacks the $dA$ term. The $dA$ is particularly important seeing as the time derivative of magnetic flux is what matters, and magnetic flux depends on the magnetic field as well as the area of the surface.

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