Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

5.3.1 The integral form

Integrate every term of the first Maxwell equation over an open surface $S$ bounded by the closed curve $C$ , and use Stokes's theorem (Section 2.4) on the left:

\int_{S} (\nabla \times E) \cdot d S = \oint_{C} E \cdot d l = - \int_{S} \frac{\partial B}{\partial t} \cdot d S . (5.3.1)

If the surface $S$ does not vary in time, the time derivative can be taken outside the integral,

\oint_{C} E \cdot d l = - \int_{S} \frac{\partial B}{\partial t} \cdot d S = - \frac{d}{d t} \int_{S} B \cdot d S (S fixed) . (5.3.2)

If instead the circuit $C$ , and with it the surface $S$ , moves through the field with velocity $v$ , the flux changes for a second reason, because the surface sweeps through new regions of $B$ . The rate of change of the flux is then (see the box below)

- \frac{d}{d t} \int_{S} B \cdot d S = - \int_{S} \frac{\partial B}{\partial t} \cdot d S + \oint_{C} (v \times B) \cdot d l = \oint_{C} E \cdot d l + \oint_{C} (v \times B) \cdot d l . (5.3.3)