Magnetic Flux and Faraday's Law

Faraday's law forms the basis for describing how much of our modern world operates, including how electric current is generated and how modern electronic devices and components function.

Consider a crude device called a crystal radio, which essentially consists of a coil of wire and an earpiece. Why is it that the radio runs perfectly (although perhaps playing on the quiet side) without any external batteries or other power? As it turns out, the air is filled with electromagnetic waves of various spectra, including radio waves. Such waves contain oscillating magnetic (and electric) fields, and changing magnetic fields can lead to electric fields in coils of wire in the form of electric current.

As one of the fundamental relationships of electricity and magnetism, Faraday's law describe how time-varying magnetic fields produce electric fields.

Much work had been done on electrostatics by the beginning of the nineteenth century, but comparatively little had been done on magnetic fields. It wasn't around 1820, for instance, that Oested first discovered that electric current produces a magnetic field, which is arguably the beginning of magnetostatics (apart from crude studies of naturally occuring magnets in antiquity).

Thus, it is perhaps surprising that less than two decades later, Michael Faraday had already begun to describe not only changing magnetic fields (thus beginning electrodynamics) but also, as a consequence, the interaction of electric and magnetic fields. Faraday discovered that, in general, a changing magnetic field leads to an electric field, a phenomenon called electromagnetic induction (the electric field is "induced"). The quantitative description of induction is thus named Faraday's law in honor of its discover.

The simplest example of induced electric field is the current generated inside a small circular loop due to a changing magnetic field. To that extent, one must describe the magnetic field that passes through a loop. It makes sense to do so in analogy with electric flux.

As with electric flux, the orientation of the surface is generally defined such that flux (magnetic field) flowing from inside to outside counts as positive, outward flux, while flux from the outside to the inside counts as negative, inward flux. To a patch of surface, one assigns a vector \( \mathbf{a}_i \) that indicates the normal (perpendicular) to the surface. In addition, the magnitude of each \( \mathbf{a}_i \) is defined to be the area of the corresponding patch.

The magnetic flux \( \Phi_\text{patch} \) through the patch is given by the dot product, which calculates the component of \( \mathbf{B}_i \) parallel to \( \mathbf{a}_i \):

\[ \Phi_\text{patch} = \mathbf{B}_i \cdot \mathbf{a}_i \]

The total magnetic flux through a surface composed of many small patches \( \mathbf{a}_i \), meanwhile, is the sum over all patches:

\[ \Phi = \sum_i \mathbf{B}_i \cdot \mathbf{a}_i .\]

As the \( \mathbf{a}_i \) become vanishingly small, as in the case of a continuous surface, the sum is replaced with a surface integral:

\[ \Phi = \oint_S \mathbf{B} \cdot d\mathbf{a}. \]

Fortunately, the magnetic flux can often be computed without resorting to computing the integral explicitly. Generally, the calculation for magnetic flux is quite simple because one mostly considers flat loops.

A small, flat loop of wire of area \( A \) is oriented perpendicular to the uniform magnetic field inside a cylindrical solenoid with turns per unit length \( n \) and current \( I \). What is the flux through the loop?

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In this case, \( \mathbf{B} \cdot d\mathbf{a} \) is constant, so the flux is simply \( \Phi = BA = \mu_0 n I A \).

Faraday found that the induced emf \( \mathcal{E} \) (electromotive force) through a current loop was given by

\[ \mathcal{E} = - \frac{d \Phi}{d t}, \]

where \( \Phi \) is the magnetic flux through the loop (recall that emf points in the opposite direction as the voltage). In general, one determines \( \Phi \) as a function of \( t \), which allows for the calculation of \( \mathcal{E} \).

Embedded in Faraday's law is the direction of the induced current in the loop. As with the Biot-Savart law and Ampère's law, the orientation of the loop is defined such that a counterclockwise traversal of the loop (relative to \( \mathbf{a} \)) is positive. Thus, the negative sign on \( d \Phi/dt \) indicates that an increase in flux through the loop leads to a clockwise flow of current in the loop. Likewise, a decrease in flux through the loop (positive increase relative to the opposite side of the loop) leads to a counterclockwise flow of current.

It is easy to remember this sign convention using the so-called right-hand rule. If one's thumb points in the direction of magnetic flux increase, then the current flows in the opposite direction to the curl of the fingers (opposite due to the negative sign in Faraday's law).

Note that the loop itself produces a magnetic field. One can verify that the magnetic field is always produced in the direction of decreasing flux (again, due to the negative sign). In some sense, one can think of the induced magnetic field as "opposing" the change in the external magnetic field, a result often referred to as Lenz's law.

A small, flat loop of wire of area \( A \) and resistance \( R \) is oriented perpendicular to the uniform, upward-pointing magnetic field inside a cylindrical solenoid with turns per unit length \( n \) and current \( I(t) = I_0 + Bt \), where \( I_0 \) and \( B \) are constants. What is the size and direction of the current \( I_\text{loop}(t) \) induced in the loop as a function of \( t \)?

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As in the previous example, for a given \( t \), \( \mathbf{B} \cdot d\mathbf{a} \) is constant, so the flux through the loop is just

\[ \Phi(t) = BA = \mu_0 n (I_0 + Bt) A. \]

Therefore, Faraday's law yields

\[ \mathcal{E} = - \frac{d\Phi}{dt} = -\mu_0 n A B. \]

It follows that the current induced in the loop is of magnitude \( \mu_0 n A B / R \) and points clockwise (due to the negative sign), which can also be verified using the right-hand rule.

A small, flat loop of wire of area \( A \) is initially oriented parallel to a uniform magnetic field of magnitude \( B \). If the loop rotates with angular velocity \( \omega \), what is the magnitude of the induced emf \( \mathcal{E} \) in the loop?

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Although the area of the loop remains constant, the angle of the loop with respect to the field is constantly changing. The angle with respect to the field is given by \( \cos{\omega t} \), so

\[ \Phi = BA \cos{\omega t} \]

\[ \mathcal{E} = - \frac{d \Phi}{dt} = \omega A B \sin{\omega t}. \]

In effect, one can generate electricity by mechanical rotation of a wire loop in a magnetic field.

Recall that that the emf \( \mathcal{E} \) is defined in the opposite direction as the voltage, namely

\[ \mathcal{E} = \int_\text{loop} \mathbf{E} \cdot d\mathbf{s}. \]

Since

\[ \Phi = \int_S \mathbf{B} \cdot d\mathbf{a}, \]

it follows from Faraday's law that

\[ \int_\text{loop} \mathbf{E} \cdot d\mathbf{s} = - \frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{a}. \]

As was done with Ampère's law, one can invoke Stokes' theorem on the left side to equate the two integrands:

\[ \int_S \nabla \times \mathbf{E} \cdot d\mathbf{a} = - \frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{a}. \]

Again, one argue that since the relationship must hold true for any arbitrary surface \( S \), it must be the case that the two integrands are equal, and therefore

\[ \nabla \times \mathbf{E} = -\frac{d\mathbf{B}}{dt}. \]

This is the so-called differential form of Faraday's law. Whereas in electrostatics, the electric field is conservative and \( \nabla \times \mathbf{E} = 0 \), in the world of electrodynamics, the electric field is not curl-free.

[1] Griffiths, D.J. Introduction to Electrodynamics. Fourth edition. Pearson, 2014.

[2] Purcell, E.M. Electricity and Magnetism. Third edition. Cambridge University Press, 2013.