# 8.8: Faraday's Law- Electricity from Magnetism and Force

- Page ID
- 472616

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)##### Learning Objectives

- Describe methods to produce an induced voltage with a magnetic field or magnet and a loop of wire.
- Explain Faraday's Law of Induction.

The apparatus to demonstrate that magnetic fields can create currents is illustrated in Figure \(\PageIndex{1}\). When the switch is closed, a magnetic field is produced in the coil on the top part of the iron ring and transmitted to the coil on the bottom part of the ring. The galvanometer is used to detect any current induced in the coil on the bottom. It was found that each time the switch is closed, the galvanometer detects a current in one direction in the coil on the bottom. Each time the switch is opened, the galvanometer detects a current in the opposite direction. Interestingly, if the switch remains closed or open for any length of time, there is no current through the galvanometer. * Closing and opening the switch* induces the current. It is the

*in magnetic field that creates the current in the following way:*

*change**a changing magnetic field induces an electric field*, which results in the induced voltage. When this induced voltage occurs over a conducting path, as in this example, the induced voltage causes a current to flow. As a shorthand, we call the resulting current

**induced current**; the changing magnetic field does not induce the current directly but through the induced voltage and an application of Ohm's law.

An experiment easily performed and often done in physics labs is illustrated in Figure \(\PageIndex{2}\). A voltage is induced in the coil when a bar magnet is pushed in and out of it. Voltages of opposite signs are produced by motion in opposite directions, and the voltages are also reversed by reversing poles. The same results are produced if the coil is moved rather than the magnet—it is the relative motion that is important. The faster the motion, the greater the voltage, and there is no voltage when the magnet is stationary relative to the coil.

The method of inducing a voltage used in most electric generators is shown in Figure \(\PageIndex{3}\). A coil is rotated in a magnetic field, producing an alternating voltage (and current), which depends on rotation rate and other factors that will be explored in later sections. Note that the generator is remarkably similar in construction to a motor.

So we see that changing the magnitude or direction of a magnetic field produces a voltage. Experiments revealed that there is a crucial quantity called the **magnetic flux**, \(\Phi\), given by

\[\Phi=B_{\perp} A, \nonumber \]

where *\(B\)* is the magnetic field strength over an area *\(A\)*, at an angle \(\theta\) with the perpendicular to the area as shown in Figure \(\PageIndex{4}\). We need advanced mathematics to formally describe flux, but we can visualize the idea with concepts we have already introduced in this chapter. The more magnetic field lines that pass through a surface, the higher the magnetic flux. **Any change in magnetic flux \(\Phi\) induces a voltage.** This process is defined to be **electromagnetic induction**. Units of magnetic flux *\(\Phi\)* are \(\mathrm{T} \cdot \mathrm{m}^{2}\).

All induction, including the examples given so far, arises from some change in magnetic flux *\(\Phi\)*. For example, Faraday changed *\(B\)* (and, hence, *\(\Phi\)*) when opening and closing the switch in his apparatus (shown in Figure \(\PageIndex{1}\)). This is also true for the bar magnet and coil shown in Figure \(\PageIndex{2}\). When rotating the coil of a generator, the angle \(\theta\) (and, hence, *\(\Phi\)*) is changed.

Faraday’s experiments showed that the voltage induced by a change in magnetic flux depends on only a few factors. First, voltage is directly proportional to the change in flux \(\Delta \Phi\). Second, voltage is greatest when the change in time \(\Delta t\) is smallest—that is, voltage is inversely proportional to \(\Delta t\). Finally, if a coil has N turns, a voltage will be produced that is \(N\) times greater than for a single coil, so that voltage is directly proportional to \(N\). The equation for the voltage induced by a change in magnetic flux is referred to as Faraday's law of induction.

\[V=-N \frac{\Delta \Phi}{\Delta t}. \nonumber \]

The minus sign in Faraday’s law of induction is very important. The minus means that *the induced voltage creates a current I and magnetic field B that oppose the change in flux \(\Delta \Phi\)—this is known as Lenz’s law*. Faraday was aware of the direction, but Lenz stated it so clearly that he is credited for its discovery. (See Figure \(\PageIndex{1}\).)

Lenz’s law is a manifestation of the conservation of energy. The induced voltage produces a current that opposes the change in flux, because a change in flux means a change in energy. Energy can enter or leave, but not instantaneously. Lenz’s law is a consequence. As the change begins, the law says induction opposes and, thus, slows the change. In fact, if the induced voltage were in the same direction as the change in flux, there would be a positive feedback that would give us free energy from no apparent source—conservation of energy would be violated.

### Section Summary

- The crucial quantity in induction is magnetic flux \(\Phi\), defined to be \(\Phi=B_{\perp} A\), where \(B_{\perp}\) is the magnetic field strength perpendicular to the area \(A\).
- Units of magnetic flux
*\(\Phi\)*are \(\mathrm{T} \cdot \mathrm{m}^{2}\). - Any change in magnetic flux
*\(\Phi\)*induces a voltage—the process is defined to be electromagnetic induction. - Faraday’s law of induction states that the voltage induced by a change in magnetic flux is
\[V=-N \frac{\Delta \Phi}{\Delta t} \nonumber\]

when flux changes by \(\Delta \Phi\) in a time \(\Delta t\).

- If voltage is induced in a coil, N is its number of turns.
- The minus sign means that the induced voltage creates a current \(I\) and magnetic field \(B\) that
*oppose the change in flux*\(\Delta \Phi\) —this opposition is known as Lenz’s law.

### Glossary

**induced current**- the current created by a changing magnetic field through voltage induced over a conducting path

**magnetic flux**- the amount of magnetic field going through a particular area, calculated with \(\Phi=B_{\perp} A\), where \(B_{\perp}\) is the magnetic field strength perpendicular to the area \(A\)

**electromagnetic induction**- the process of inducing a voltage with a change in magnetic flux
**Faraday’s law of induction**- the means of calculating the voltage in a coil due to changing magnetic flux, given by \(V=-N \frac{\Delta \Phi}{\Delta t}\)