10: Electromagnetic Induction
- Page ID
- 5479
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 10.1: Introduction to Electromagnetic Induction
- In 1820, Oersted had shown that an electric current generates a magnetic field. But can a magnetic field generate an electric current? This was answered almost simultaneously and independently in 1831 by Joseph Henry in the United States and Michael Faraday in Great Britain.
- 10.2: Electromagnetic Induction and the Lorentz Force
- Lorentz force is underpining the observation that the movement of a meta rod through the magnetic field induces a potential difference across the ends of the rod. This is electromagnetic induction, and, seen this way, there is nothing new: electromagnetic induction is nothing more than the Lorentz force on the conduction electrons within the metal.
- 10.3: Lenz's Law
- Lenz's Law argues that when an EMF is induced in a circuit as a result of changing magnetic flux through the circuit, the direction of the induced EMF is such as to oppose the change of flux that causes it.
- 10.4: Ballistic Galvanometer and the Measurement of Magnetic Field
- A galvanometer is similar to a sensitive ammeter, differing mainly in that when no current passes through the meter, the needle is in the middle of the dial rather than at the left hand end. A galvanometer is used not so much to measure a current, but rather to detect whether or not a current is flowing, and in which direction. In the ballistic galvanometer, the motion of the needle is undamped, or as close to undamped as can easily be achieved.
- 10.5: AC Generator
- This and the following sections will be devoted to generators and motors. I shall not be concerned with – and indeed am not knowledgeable about – the engineering design or practical details of real generators or motors, but only with the scientific principles involved. The "generators" and "motors" of this chapter will be highly idealized abstract concepts bearing little obvious resemblance to the real things.
- 10.6: AC Power
- When a current I flows through a resistance R , the rate of dissipation of electrical energy as heat is IR² . If an alternating potential difference V=V₀ sin ωt is applied across a resistance, then an alternating current I=I₀ sin ωt will flow through it, and the rate at which energy is dissipated as heat will also change periodically.
- 10.7: Linear Motors and Generators
- In this section describes highly idealized and imaginary linear motors and generators, only because the geometry is simpler than for rotary motors, and it is easier to explain certain principles. Rotary motors will be discussed in the following section.
- 10.8: Rotary Motors
- Most real motors, of course, are rotary motors, though all of the principles described for our highly idealized linear motor of the Section 10.7 still apply.
- 10.9: The Transformer
- A transformer is a static electrical device that transfers electrical energy between two or more circuits through electromagnetic induction. A varying current in one coil of the transformer produces a varying magnetic field, which in turn induces a varying EMF or "voltage" in a second coil. Power can be transferred between the two coils through the magnetic field, without a metallic connection between the two circuits.
- 10.10: Mutual Inductance
- Consider two coils, not connected to one another, other than being close together in space. If the current changes in one of the coils, so will the magnetic field in the other, and consequently an EMF will be induced in the second coil. The ratio of the EMF induced in the second coil to the rate of change of current in the first is called the coefficient of mutual inductance.
- 10.11: Self Inductance
- In this section we are dealing with the self inductance of a single coil rather than the mutual inductance between two coils. If the current through a single coil changes, the magnetic field inside that coil will change; consequently a back EMF will be induced in the coil that will oppose the change in the magnetic field and indeed will oppose the change of current.
- 10.12: Growth of Current in a Circuit Containing Inductance
- This page covers the behavior of current in an inductive circuit when a switch is closed, highlighting the non-instantaneous change in current due to inductance and the role of back EMF. It presents Kirchhoff's integral equations, illustrating how current grows over time and approaches \(E/R\), characterized by the time constant \(L/R\). The page includes a problem involving multiple currents, leading to equations that describe the transient states of currents following the switch's closure.
- 10.13: Discharge of a Capacitor through an Inductance
- This page explores the dynamics of an electrical circuit with a capacitor and inductor, detailing the interplay of charge, current, and potential difference. It describes the energy oscillation between components, leading to simple harmonic motion, and notes that resistance causes dampening of these oscillations. Furthermore, it explains how oscillating fields produce electromagnetic waves, which travel at light speed and are linked to free space impedance.
- 10.14: Discharge of a Capacitor through an Inductance and a Resistance
- This page examines the behavior of a capacitor and inductor circuit during charging and discharging, focusing on how resistance influences oscillatory motion. It explains that the charge \(Q\) on the capacitor is determined by differential equations, showcasing scenarios of simple harmonic motion without resistance and damped oscillation with resistance.
- 10.15: Charging a Capacitor through and Inductance and a Resistance
- This page examines the behavior of an RLC circuit connected to a battery, detailing the time-dependent changes in current and charge influenced by resistance values. It presents differential equations relating EMF to potential differences in circuit components and discusses the resulting transient and steady-state responses. Key insights include the finite induction effects in real circuits that prevent sudden changes in current.
- 10.16: Energy Stored in an Inductance
- This page explains the energy stored in an inductor as current increases, highlighting the role of back EMF and the work required against it. It details the calculation of work over time, leading to the formula for energy storage in an inductor: \(\frac{1}{2}LI^2\). Additionally, it verifies the dimensions of this energy expression, emphasizing the connections between current, inductance, and stored energy.
- 10.17: Energy Stored in a Magnetic Field
- Energy can be stored per unit volume in a magnetic field n a vacuum.
Thumbnail: Animation showing operation of a brushed DC electric motor. (CC BY-SA 3.0; Abnormaal via Wikipedia)