Changing Magnetic Flux Produces an Electric Field

We have studied Faraday's law of induction in previous atoms. We learned the relationship between induced electromotive force (EMF) and magnetic flux. In a nutshell, the law states that changing magnetic field $(\frac{d \Phi_B}{dt})$ produces an electric field $(\varepsilon)$, Faraday's law of induction is expressed as $\varepsilon = -\frac{\partial \Phi_B}{\partial t}$, where $\varepsilon$ is induced EMF and $\Phi_B$ is magnetic flux. ("N" is dropped from our previous expression. The number of turns of coil is included can be incorporated in the magnetic flux, so the factor is optional. ) Faraday's law of induction is a basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF). In this Atom, we will learn about an alternative mathematical expression of the law.

Faraday's Experiment

Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current which flows through the small coil (A), creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil (B), the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer (G).

Differential form of Faraday's law

The magnetic flux is $\Phi_B = \int_S \vec B \cdot d \vec A$, where $\vec A$ is a vector area over a closed surface S. A device that can maintain a potential difference, despite the flow of current is a source of electromotive force. (EMF) The definition is mathematically $\varepsilon = \oint_C \vec E \cdot d\vec s$, where the integral is evaluated over a closed loop C.

Faraday's law now can be rewritten $\oint_C \vec E \cdot d\vec s = -\frac{\partial}{\partial t} (\int \vec B \cdot d\vec A)$. Using the Stokes' theorem in vector calculus, the left hand side is$\oint_C \vec E \cdot d\vec s = \int_S (\nabla \times \vec E) \cdot d\vec A$. Also, note that in the right hand side$\frac{\partial}{\partial t} (\int \vec B \cdot d\vec A) = \int \frac{\partial \vec B}{\partial t} \cdot d\vec A$ . Therefore, we get an alternative form of the Faraday's law of induction: $\nabla \times \vec E = - \frac{\partial \vec B}{\partial t}$.This is also called a differential form of the Faraday's law. It is one of the four equations in Maxwell's equations, governing all electromagnetic phenomena.