Faraday's Law

The phenomenon of magnetic induction plays a crucial role in three very useful electrical devices; the electric generator (see Section 2.3.10), the electric motor (see Section 2.3.12), and the transformer (see Section 2.3.13). Without these devices, modern life would be impossible in its present form. Magnetic induction was discovered in 1830 by Michael Faraday. Joseph Henry independently made the same discovery at about the same time. Both physicists were intrigued by the fact that an electric current flowing around a circuit can generate a magnetic field. Surely, they reasoned, if an electric current can generate a magnetic field then a magnetic field must somehow be able to generate an electric current. However, it took many years of fruitless experimentation before they were able to find the essential ingredient that allows a magnetic field to generate an electric current. This ingredient is time variation.

Prior to 1830, the only known way in which to cause an electric current to flow through a conducting wire was to connect the ends of the wire to the positive and negative terminals of a battery. We measure a battery's ability to push current down a wire in terms of its voltage, by which we mean the voltage difference between its positive and negative terminals. Of course, volts are the units used to measure electric scalar potential, so when we talk about a 6 V battery, what we are really saying is that the difference in electric scalar potential between its positive and negative terminals is six volts. This insight allows us to write

$$V = \phi(\oplus)-\phi(\ominus) = -\int_{\oplus}^{\ominus} \nabla\phi\cdot d{\bf r} = \int_{\oplus}^{\ominus} {\bf E} \cdot d{\bf r},$$ (2.280)

where $V$ is the battery voltage, $\oplus$ denotes the positive terminal, $\ominus$ the negative terminal, and $d{\bf r}$ is an element of length along the wire. Of course, the previous equation is a direct consequence of ${\bf E} = -\nabla\phi$. [See Equation (2.17) and Section A.18.] Clearly, a voltage difference between two ends of a wire attached to a battery implies the presence of a longitudinal electric field that pushes electric charges along the wire. This field is directed from the positive terminal of the battery to the negative terminal, and is, therefore, such as to force electrons to flow through the wire from the negative to the positive terminal. As expected, this implies that a net positive current flows from the positive to the negative terminal. The fact that ${\bf E}$ is a conservative field (i.e., ${\bf E} = -\nabla\phi$) ensures that the voltage difference, $V$, is independent of the path of the wire between the terminals. In other words, two different wires attached to the same battery develop identical voltage differences.

Let us now consider a closed loop of wire (with no battery). The voltage around such a loop, which is sometimes called the electromotive force, or emf, is

$$V = \oint {\bf E} \cdot d{\bf r} = 0.$$ (2.281)

The fact that the right-hand side of the previous equation is zero is a direct consequence of the field equation $\nabla\times{\bf E} =-\nabla\times\nabla\phi= {\bf0}$ and the curl theorem. [See Equations (2.17) and (2.25), and Section A.22.] We conclude that, because ${\bf E}$ is a conservative field (i.e., ${\bf E} = -\nabla\phi$), the emf around a closed loop of wire is automatically zero, and so there is no current flow around such a loop.

However, in 1830, Michael Faraday discovered that a changing magnetic field can cause a current to flow around a closed loop of wire (in the absence of a battery). Of course, if current flows around the loop then there must be an emf. In other words,

$$V = \oint{\bf E} \cdot d{\bf r} \neq 0,$$ (2.282)

which immediately implies that ${\bf E}$ is not a conservative field, and that $\nabla\times{\bf E} \neq {\bf0}$. Clearly, we are going to have to modify some of our ideas regarding electric fields.

Faraday continued his experiments, and found that another way of generating an emf around a loop of wire is to keep the magnetic field constant and to move the loop. (See Section 2.3.9.) Eventually, Faraday was able to formulate a law that accounted for all of his experiments; the emf generated around a loop of wire in a magnetic field is proportional to the rate of change of the flux of the magnetic field through the loop. (See Section A.20.) Thus, if the loop is denoted $C$, and $S$ is some surface attached to the loop, then Faraday's experiments can be summed up by writing

$$V = \oint_C{\bf E} \cdot d{\bf r} = A \,\frac{\partial }{\partial t}\! \int_S {\bf B} \cdot d{\bf S},$$ (2.283)

where $A$ is a constant of proportionality. Thus, the changing flux of the magnetic field passing through the loop generates an electric field directed around the loop. This process is know as magnetic induction.

SI units have been carefully chosen so as to make $\vert A\vert = 1$ in the previous equation. So, the only question that we now have to answer is whether $A=+1$ or $A=-1$. In other words, we need to decide which way around the loop the induced emf drives the current. We possess a general principle, known as Le Chatelier's principle, that allows us to answer such questions. According to Le Chatelier's principle, every change in a physical system generates a reaction that acts to minimize the change. Essentially, this implies that the universe is stable to small perturbations. When Le Chatelier's principle is applied to the particular case of magnetic induction, it is usually called Lenz's law, after Emil Lenz who formulated it in 1834. According to Lenz's law, the current induced by an emf around a closed loop is always such that the magnetic field it produces acts to counteract the change in magnetic flux that generates the emf. From Figure 2.25, it is clear that if the magnetic field ${\bf B}$ is increasing and the current $I$ circulates clockwise (as seen from above) then the current generates a field ${\bf B}'$ that opposes the increase in the magnetic flux through the loop, in accordance with Lenz's law. The direction of the current is opposite to the sense of circulation of the current loop $C$, as determined by the right-hand rule (assuming that the flux of ${\bf B}$ through the loop is positive), so this implies that $A=-1$ in Equation (2.283). Thus, Faraday's law takes the form

$$V=\oint_C {\bf E} \cdot d{\bf r} = - \frac{\partial}{\partial t}\! \int_S {\bf B} \cdot d{\bf S}=-\frac{d{\mit\Phi}}{dt},$$ (2.284)

where ${\mit\Phi}= \int_S{\bf B}\cdot d{\bf S}$ is the magnetic flux through the loop.

Figure 2.25: Lenz's law.

Experimentally, Faraday's law is found to correctly predict the emf (i.e., $\oint {\bf E} \cdot d{\bf r}$) generated around any wire loop, irrespective of the position or shape of the loop. It is reasonable to assume that the same emf would be generated in the absence of the wire (of course, no current would flow in this case). We conclude that Equation (2.284) is valid for any closed loop $C$. Now, if Faraday's law is to make sense then it must hold for all surfaces, $S$, attached to the loop, $C$. Clearly, if the flux of the magnetic field through the loop depends on the surface upon which it is evaluated then Faraday's law is going to predict different emfs for different surfaces. Because there is no preferred surface for a general non-coplanar loop, this would not make any sense. The condition for the flux of the magnetic field, $\int_S {\bf B} \cdot d{\bf S}$, to depend only on the loop $C$ to which the surface $S$ is attached, and not on the nature of the surface itself, is

$$\oint_{S'} {\bf B} \cdot d{\bf S}' = 0,$$ (2.285)

for any closed surface $S'$. (See Section A.20.)

Faraday's law, Equation (2.284), can be converted into a field equation using the curl theorem. (See Section A.22.) We obtain

$$\nabla\times{\bf E} = - \frac{\partial {\bf B}}{\partial t}.$$ (2.286)

This field equation describes how a changing magnetic field generates an electric field. The divergence theorem (see Section A.20) applied to Equation (2.285) gives the familiar field equation

$$\nabla \cdot {\bf B} = 0.$$ (2.287)

[See Equation (2.263).] This equation ensures that the magnetic flux through a loop is a well defined quantity.

The divergence of Equation (2.286) yields

$$\frac{\partial\, (\nabla\cdot{\bf B})}{\partial t} = 0,$$ (2.288)

because $\nabla\cdot \nabla\times{\bf E} \equiv 0$. (See Section A.22.) Thus, the field equation (2.286) actually demands that the divergence of the magnetic field must be constant in time for self-consistency (this implies that the flux of the magnetic field through a loop need not be a well defined quantity, as long as its time derivative is well defined). However, a constant non-solenoidal magnetic field can only be generated by magnetic monopoles, and magnetic monopoles do not exist (as far as we are aware). (See Section 2.2.9.) Hence, $\nabla\cdot{\bf B} = 0$.

As an example of the use of Faraday's law, let us calculate the electric field generated by a decaying magnetic field of the form ${\bf B} = B_z(r,t)\,{\bf e}_z$, where

$$B_z(r,t) = \left\{ \begin{array}{lcc} B_0\,\exp(-t/\tau)&\mbox{\hspace{1cm}}&r\leq a\\ [0.5ex] 0&&r>a \end{array}\right.,$$ (2.289)

and $r$ is a cylindrical polar coordinate. (See Section A.23.) Here, $B_0$ and $\tau$ are positive constants. By symmetry, we expect an induced electric field of the form ${\bf E}(r,t)$. We also expect $\nabla\cdot {\bf E}=0$, because there are no electric charges in the problem. [See Equation (2.54).] This rules out a radial electric field. We can also rule out a $z$-directed electric field, because $\nabla\times [E_z(r)\,{\bf e}_z] = -(\partial E_z/\partial r)\,{\bf e}_\theta$, and we require $\nabla\times {\bf E} \propto {\bf B}\propto {\bf e}_z$. Hence, the induced electric field must be of the form ${\bf E}(r,t) = E_\theta(r,t)\,{\bf e}_\theta$. Now, according to Faraday's law, (2.284), the line integral of the electric field around some closed loop is equal to minus the rate of change of the magnetic flux passing through the loop. If we choose a loop that is a circle of radius $r$ in the $x$-$y$ plane then we have

$$2\pi\,r\,E_\theta(r,t) = - \frac{d{\mit\Phi}}{dt},$$ (2.290)

where ${\mit\Phi}$ is the flux of the magnetic field (in the $+z$ direction) passing through a circular loop of radius $r$. It is evident that

$${\mit\Phi}(r,t) = \left\{ \begin{array}{lcc} \pi\,r^2\,B_0\,\... ...\ [0.5ex] \pi\,a^2\,B_0\,\exp(-t/\tau)&&r>a \end{array}\right..$$ (2.291)

Hence,

$$E_\theta(r,t) = \left\{ \begin{array}{lcc} (B_0/2\,\tau)\,r\,... ... (B_0/2\,\tau)\,(a^2/r)\,\exp(-t/\tau)&&r>a \end{array}\right..$$ (2.292)