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

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