14.8: Inductance (Summary)
Key Terms
| henry (H) | unit of inductance, \(\displaystyle 1H=1Ω⋅s\); it is also expressed as a volt second per ampere |
| inductance | property of a device that tells how effectively it induces an emf in another device |
| inductive time constant | denoted by \(\displaystyle τ\), the characteristic time given by quantity \(\displaystyle L/R\) of a particular series \(\displaystyle RL\) circuit |
| inductor | part of an electrical circuit to provide self-inductance, which is symbolized by a coil of wire |
| LC circuit | circuit composed of an ac source, inductor, and capacitor |
| magnetic energy density | energy stored per volume in a magnetic field |
| mutual inductance | geometric quantity that expresses how effective two devices are at inducing emfs in one another |
| RLC circuit | circuit with an ac source, resistor, inductor, and capacitor all in series. |
| self-inductance | effect of the device inducing emf in itself |
Key Equations
| Mutual inductance by flux | \(\displaystyle M=\frac{N_2Φ_2}{1_I}=\frac{N_1Φ_{12}}{I_2}\) |
| Mutual inductance in circuits | \(\displaystyle ε_1=−M\frac{dI_2}{dt}\) |
| Self-inductance in terms of magnetic flux | \(\displaystyle NΦ_m=LI\) |
| Self-inductance in terms of emf | \(\displaystyle ε=−L\frac{dI}{dt}\) |
| Self-inductance of a solenoid | \(\displaystyle L_{solenoid}=\frac{μ_0N^2A}{l}\) |
| Self-inductance of a toroid | \(\displaystyle L_{toroid}=\frac{μ_0N^2h}{2π}ln\frac{R_2}{R_1}\). |
| Energy stored in an inductor | \(\displaystyle U=\frac{1}{2}LI^2\) |
| Current as a function of time for a RL circuit | \(\displaystyle I(t)=\frac{ε}{R}(1−e^{−t/τ_L})\) |
| Time constant for a RL circuitτ | \(\displaystyle τ_L=L/R\) |
| Charge oscillation in LC circuits | \(\displaystyle q(t)=q_0cos(ωt+ϕ)\) |
| Angular frequency in LC circuits | \(\displaystyle ω=\sqrt{\frac{1}{LC}}\) |
| Current oscillations in LC circuits | \(\displaystyle i(t)=−ωq_0sin(ωt+ϕ)\) |
| Charge as a function of time in RLC circuit | \(\displaystyle q(t)=q_0e^{−Rt/2L}cos(ω't+ϕ)\) |
| Angular frequency in RLC circuit | \(\displaystyle ω'=\sqrt{\frac{1}{LC}−(\frac{R}{2L})^2}\) |
Summary
14.2 Mutual Inductance
- Inductance is the property of a device that expresses how effectively it induces an emf in another device.
- Mutual inductance is the effect of two devices inducing emfs in each other.
- A change in current \(\displaystyle dI_1/dt\) in one circuit induces an emf (\(\displaystyle ε_2\)) in the second:
\(\displaystyle ε_2=−M\frac{dI_1}{dt}\),
where M is defined to be the mutual inductance between the two circuits and the minus sign is due to Lenz’s law.
- Symmetrically, a change in current \(\displaystyle dI_2/dt\) through the second circuit induces an emf (\(\displaystyle ε_1\)) in the first:
\(\displaystyle ε_1=−M\frac{dI_2}{dt}\),
where M is the same mutual inductance as in the reverse process.
14.3 Self-Inductance and Inductors
- Current changes in a device induce an emf in the device itself, called self-inductance,
\(\displaystyle ε=−L\frac{dI}{dt}\),
where L is the self-inductance of the inductor and \(\displaystyle dI/dt\) is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law. The unit of self-inductance and inductance is the henry (H), where \(\displaystyle 1H=1Ω⋅s\).
- The self-inductance of a solenoid is
\(\displaystyle L=\frac{μ_0N^2A}{l}\),
where N is its number of turns in the solenoid, A is its cross-sectional area, l is its length, and \(\displaystyle μ_0=4π×10^{−7}T⋅m/A\) is the permeability of free space.
- The self-inductance of a toroid is
\(\displaystyle L=\frac{μ_0N^2h}{2π}ln\frac{R_2}{R_1}\),
where N is its number of turns in the toroid, \(\displaystyle R_1\) and \(\displaystyle R_2\) are the inner and outer radii of the toroid, h is the height of the toroid, and \(\displaystyle μ_0=4π×10^{−7}T⋅m/A\) is the permeability of free space.
14.4 Energy in a Magnetic Field
- The energy stored in an inductor U is
\(\displaystyle U=\frac{1}{2}LI^2\).
- The self-inductance per unit length of coaxial cable is
\(\displaystyle \frac{L}{l}=\frac{μ_0}{2π}ln\frac{R_2}{R_1}\).
14.5 RL Circuits
- When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is
\(\displaystyle I(t)=\frac{ε}{R}(1−e^{−Rt/L})=\frac{ε}{R}(1−e^{−t/τ_L})\) (turning on),
where the initial current is \(\displaystyle I_0=ε/R\)..
- The characteristic time constant \(\displaystyle τ\) is \(\displaystyle τ_L=L/R\), where L is the inductance and R is the resistance.
- In the first time constant \(\displaystyle τ\), the current rises from zero to \(\displaystyle 0.632I_0\), and to 0.632 of the remainder in every subsequent time interval \(\displaystyle τ\).
- When the inductor is shorted through a resistor, current decreases as
\(\displaystyle I(t)=\frac{ε}{R}e^{−t/τ_L}\) (turning off).
Current falls to \(\displaystyle 0.368I_0\) in the first time interval \(\displaystyle τ\), and to 0.368 of the remainder toward zero in each subsequent time \(\displaystyle τ\).
14.6 Oscillations in an LC Circuit
- The energy transferred in an oscillatory manner between the capacitor and inductor in an LC circuit occurs at an angular frequency \(\displaystyle ω=\sqrt{\frac{1}{LC}}\).
- The charge and current in the circuit are given by
\(\displaystyle q(t)=q_0cos(ωt+ϕ)\),
\(\displaystyle i(t)=−ωq_0sin(ωt+ϕ)\).
14.7 RLC Series Circuits
- The underdamped solution for the capacitor charge in an RLC circuit is
\(\displaystyle q(t)=q_0e^{−Rt/2L}cos(ω't+ϕ).\)
- The angular frequency given in the underdamped solution for the RLC circuit is
\(\displaystyle ω′=\sqrt{\frac{1}{LC}−(\frac{R}{2L})^2}\).