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I have been recently studying Maxwell Equations, and I wasn't able to understand properly the EMF $\zeta$. Mathematically we have $\zeta=-\dfrac{d\Phi_B}{dt}$ where $\Phi_B$ is the magnetic flux of a surface $\Sigma$.

At first, we were taught that also we have: $\zeta=\oint_{\partial\Sigma} \overrightarrow{E}\cdot \overrightarrow{dl}$ where $\overrightarrow{E}$ is the electric field and $\overrightarrow{dl}$ is an element of $\partial\Sigma$, and finally that: $\zeta=\oint_{\partial\Sigma} (\overrightarrow{v}\times\overrightarrow{B})\cdot \overrightarrow{dl}$ where $\overrightarrow{B}$ is the magnetic field and $\overrightarrow{v}$ is the velocity (is it the velocity of an element of $\partial\Sigma$?).

But after some research, I think that the first formula is used when $\Sigma$ is in rest frame (when it's not moving), and the second one is used when $\Sigma$ is moving and $\overrightarrow{B}$ is constant in respect to time.

So can you please give me a formula that generalize these two formulas? and can we derive it from$\zeta=-\dfrac{d\Phi_B}{dt}$

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The correct way to write it is to use the definition of flux

$$\oint_{\partial R} \mathbf{E}\cdot d\mathbf{r} = -\frac{d\Phi_B}{dt} $$

Using the definition of flux, we have the relation

$$ \frac{d\Phi_B}{dt} = \frac{d}{dt}\iint_{R}\mathbf{B}\cdot d\mathbf{A}. $$

We therefore have

$$ \boxed{ \oint_{\partial R} \mathbf{E}\cdot d\mathbf{r} = \frac{d}{dt}\iint_{R}\mathbf{B}\cdot d\mathbf{A}}$$


This is to be understood in a very similar way to ampere's law. You draw some curve $\partial R$ that encloses the region $R$ (usually called a "Faradian Loop"), and then the E field along $\partial R$ is given by the time derivative of the flux enclosed by $\partial R$.

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EMF for a path segment is an integral of force per unit charge that would act on it if the charge was placed sequentially on all points on that path.

For example, induced EMF is integral of induced electric field. This is a solenoidal field accompanying changing magnetic field.

Another kind of EMF is motional EMF. This is integral of force per unit charge making charges move with respect to conductor, when this conductor moves in magnetic field. This electromotive force is due to the external magnetic field, but is actually more similar to constraint force from mechanics(the charges can't move in circles in a conductor because they are limited by its walls).

Another example is chemical EMF in a Volta cell: on the macroscopic level, this is a macroscopic force due to chemical reactions; it points in a direction opposite to that of the macroscopic Coulomb electric field.

In all cases, the path is usually chosen to be some conductive path of interest, such as wire or path throught the Volta cell. But it may chosen to be any path in space. In particular, we may choose a path outside a magnet, in air or vacuum. The path may be even closed - this is often considered when induced electric field in an electric circuit is considered.

In all cases, EMF quantifies net work that would be done on a unit charge if it was pulled along the path, and all other things were frozen. So when doing the integration, the integrated quantity has to be evaluated at a single time. So EMF is a function of that time.

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