Time-dependent magnetic fields : Faraday's law

So far, we have considered electromagnetic phenomena when the sources are time-independent. That is, when the charges and currents are time independent and the electric and magnetic fields are time-independent. We shall now consider the situation when these quantities are time-dependent. Particularly, we shall consider the situation when the magnetic field is time dependent.

The effect of changing magnetic field is given by the Faraday's law of induction. The law states that when the magnetic flux in a closed loop changes, there is an electromotive force ( electric field ) generated in the loop. The direction of the electromotive force is such that the electric current generated by this force in the loop ( in turn ) produces magnetic induction field which opposes the change in the magnetic field ( Lenz's law ).

Figure 16: A surface enclosed by a closed curve in time dependent magnetic field.

Consider a closed loop $c$ encompassing an open surface $S$ and let $\hat n$ be the unit normal to the surface element $dS$ ( see Fig(139) ). Then the magnetic flux going through the surface is $\int_S dS \hat n \cdot \vec B(\vec r)$ and the electromotive force around the loop is $\oint_c \vec E(\vec r) \cdot d\vec l$ where $d \vec l$ is an infinitesimal vector tangential to the loop at point $\vec r$. Then Faraday's law states

$$\oint_c \vec E(\vec r) \cdot d\vec l = - k \frac{d}{dt} \int_S dS \hat n \cdot \vec B(\vec r)$$ (139)

The directions of $d \vec l$ and $\hat n$ is as shown in the figure. The negative sign on the right is to ensure that the Lenz's law is satisfied. The constant k has dimensions of inverse velocity. It turns out that it is not an arbitrary constant to be determined from experiment but translation invariance fixes its value to be $1/c$, the same constant ( inverse velocity of light ) that appears in Ampere's law.

The magnetic flux pasing through the closed loop may change because the field itself changes or because the loop is moving in space. One can think that the field is changing because of the motion of the sources. Now, we expect that the physics should not depend on whether the sources of the magnetic field are moving or whether the loop is moving. Put it in another way, the Faraday's law should remain same whether one observes from a frame in which the sourses are stationary or the frame in which the loop is stationary. Let us formulate this in mathematical terms. Consider that the loop is moving with some velocity $\vec v$¹³. The integral on the right of Eq(139) is

$$- k \left ( \int_S dS \hat n \cdot \frac{\partial \vec B(\vec r)}{\partial t} + \Delta F\right )$$

where the first term is due to the rate of change of the flux ( due to motion of the sources ) and the second is due to the motion of the loop.

Figure 17: Moving loop $c$.

We can see from the Fig(3.9) that the change of the flux in unit time due to the motion of the loop c is equal to the flux going out of the cylindrical shape. The normal to the surface element on this surface is $\vec v \times d\vec l$ and $\Delta F = \int (\vec v \times d\vec l ) \cdot \vec B(\vec r) ) = - \int d\vec l \cdot ( \vec v \times \vec B)$. Substituting this in Eq(139) we get

$$\oint_c d\vec l \cdot \Big ( \vec E(\vec r) \cdot d\vec l - k (\vec v \times \vec B(\vec r) \Big ) = - k \int_S dS \hat n \cdot \frac{\partial \vec B(\vec r) } {\partial t}$$

$$= \oint_c d \vec l \cdot \vec E'(\vec r)$$ (140)

Note that the right hand side of the equation above is just the electromotive force produced when the loop is stationary. Let us call this field as $\vec E'(\vec r)$. Consider a particle of charge $q$ stationary in the frame of the coil. The force on it is $q \vec E'(\vec r)$. When one observes this particle in the frame in which the loop is moving, the force is $q ( \vec E(\vec r) + \frac{1}{c} \vec v \times \vec B(\vec r) )$ and if we equate the two we get $k = \frac{1}{c}$.

In the argument above, we have not considered the transformation of electric and magnetic fields from one frame to another. That is, Galilean invariance has been assumed and therefore we have equated the force acting on a particle in two frames. But the point is, Galilean invariance is able to fix the constant appearing in the Faraday's law. Let us now consider that the loop is stationary and the magnetic induction is time dependent. The Faraday's law then becomes

$$\oint_c \vec E(\vec r) \cdot d\vec l = - \frac{1}{c} \frac{d}{dt} \int_S dS \hat n \cdot \frac{\partial \vec B(\vec r)}{\partial t}$$ (141)

We can convert the line integral along the loop $c$ to surface integral by means of Green's theorem:

$$\int_S dS \hat n \cdot ( \vec \nabla \times \vec E(\vec r) = \oint_c d\vec l \cdot \vec E(\vec r).$$

We then get the differential form of Faraday's law

$$\vec \nabla \times \vec E(\vec r) + \frac{1}{c} \frac{\partial \vec B(\vec r)} {\partial t}$$ (142)