Inductance Physics Lesson 31 by Owen Borville 1.8.2026
Inductance is the property of a device that expresses how effectively it induces an emf (electromotive force) in another device. Mutual inductance is the effect of two devices inducing emfs in each other.
A change in current ∆I1/∆t (or dI1/dt) in one circuit induces an emf (ε2) in the second: ε2 = -M∆ I1/∆t or (-MdI1/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 ∆I2/∆t or (dI2/dt) through the second device (circuit) induces an emf (ε1) in the first: ε1 = -M∆I2/∆t or (-MdI2/dt) where M is the same mutual inductance as in the reverse process.
Current changes in a device induce an emf in the device itself, called self-inductance, which is the effect of the device inducing emf in itself. The device is called an inductor, and the emf induced in it by a change in current through it is emf = -L ∆I/∆ t (or -L dI/dt), where L is the self-inductance of the inductor and ∆ I/∆ t (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- and mutual inductance is the henry (H), where 1 H = 1 Ω * s.
The self-inductance L of an inductor is proportional to how much flux changes with current. For an N-turn inductor, L = N ∆Φ/∆ I.
The self-inductance of a solenoid is L = μ0N^2A/l where N is its number of turns in the solenoid, A is its cross-sectional area, l is its length, and μ0 = 4π x 10^-7 T*m/A is the permeability of free space.
The energy stored in an inductor Eind is Eind = 1/2LI^2.
The self inductance of a toroid is L = μ0N^2 h/2π*lnR2/R1, where N is its number of turns in the toroid, R1 and R2 are the inner and outer radii of the toroid, h is the height of the toroid, and μ0 = 4π x 10^-7 T*m/A is the permeability of free space.
RL Circuits occur when a series connection of a resistor and an inductor is connected to a voltage source, and the time variation of the current is I = I0(1-e^t/τ) (turning on) where I0 = V/R is the final current (I0 = ε/R) , therefore I(t) = ε/R(1-e^-t/τ)
The characteristic time constant τ is τ = L/R, where L is the inductance and R is the resistance. In the first time constant τ, the current rises from zero to 0.632I0, and 0.632 of the remainder in every subsequent time interval τ. When the inductor is shorted through a resistor, current decreases as I = I0e^-t/τ (turning off), or I = ε/R(e^-t/τ) Here I0 is the initial current. Current falls to 0.368Io in the first time interval τ, and 0.368 of the remainder toward zero in each subsequent time τ.
For inductors in AC circuits, when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a 90 degree phase angle. The opposition of an inductor to a change in current is expressed as a type of AC resistance.
Ohm's law for an inductor is I = V/XL, where V is the rms voltage across the inductor. XL is defined to be the inductive reactance, given by XL = 2πfL, with f the frequency of the AC voltage source in hertz.
Inductive reactance XL has units of ohms and is greatest at high frequencies. For capacitors, when a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a 90 degree phase angle.
Since a capacitor can stop current when fully charged, it limits current and offers another form of AC resistance. Ohm's law for a capacitor is I = V/XC, where V is the rms voltage across the capacitor. XC is the capacitive reactance XC = 1/2πfC. XC has units of ohms and is greatest at low frequencies.
The AC analogy to resistance is impedance Z, the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm's law: I0 = V0/Z or Irms = Vrms/Z where I0 is the peak current and V0 is the peak source voltage.
Impedance has units of ohms and is Z = √R^2+(XL-XC)^2 The resonant frequency f0 at which XL = XC is fo = 1/2π√LC
In an AC circuit, there is a phase angle 𝜙 between source voltage V and the current I, which can be found from cos 𝜙 = R/Z
𝜙 = 0 degrees for a purely resistive circuit or an RLC circuit at resonance. The average power delivered to an RLC circuit is affected by the phase angle and Pave = IrmsVrms cos 𝜙 (cos 𝜙 is called the power factor, which ranges from 0 to 1).
The energy stored in an inductor (magnetic field) U is U = 1/2LI^2 The self-inductance per unit length of coaxial cable is L/l = μ0/2π ln R2/R1
In an LC circuit, energy transferred in an oscillatory manner between the capacitor and inductor in an LC circuit occurs at an angular frequency ω = √1/LC The charge and current in the circuit are: q(t) = q0 cos(ωt + φ) and i(t) = -ωq0sin(ωt + φ)
RLC circuit: the underdamped solution for the capacitor charge in an RLC circuit = q(t) = q0e^-Rt/2L cos(ω't + φ) (charge as a function of time)
The angular frequency given in the underdamped solution for the RLC circuit is ω' = √(1/LC)-(R/2L)^2
Inductance is the property of a device that expresses how effectively it induces an emf (electromotive force) in another device. Mutual inductance is the effect of two devices inducing emfs in each other.
A change in current ∆I1/∆t (or dI1/dt) in one circuit induces an emf (ε2) in the second: ε2 = -M∆ I1/∆t or (-MdI1/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 ∆I2/∆t or (dI2/dt) through the second device (circuit) induces an emf (ε1) in the first: ε1 = -M∆I2/∆t or (-MdI2/dt) where M is the same mutual inductance as in the reverse process.
Current changes in a device induce an emf in the device itself, called self-inductance, which is the effect of the device inducing emf in itself. The device is called an inductor, and the emf induced in it by a change in current through it is emf = -L ∆I/∆ t (or -L dI/dt), where L is the self-inductance of the inductor and ∆ I/∆ t (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- and mutual inductance is the henry (H), where 1 H = 1 Ω * s.
The self-inductance L of an inductor is proportional to how much flux changes with current. For an N-turn inductor, L = N ∆Φ/∆ I.
The self-inductance of a solenoid is L = μ0N^2A/l where N is its number of turns in the solenoid, A is its cross-sectional area, l is its length, and μ0 = 4π x 10^-7 T*m/A is the permeability of free space.
The energy stored in an inductor Eind is Eind = 1/2LI^2.
The self inductance of a toroid is L = μ0N^2 h/2π*lnR2/R1, where N is its number of turns in the toroid, R1 and R2 are the inner and outer radii of the toroid, h is the height of the toroid, and μ0 = 4π x 10^-7 T*m/A is the permeability of free space.
RL Circuits occur when a series connection of a resistor and an inductor is connected to a voltage source, and the time variation of the current is I = I0(1-e^t/τ) (turning on) where I0 = V/R is the final current (I0 = ε/R) , therefore I(t) = ε/R(1-e^-t/τ)
The characteristic time constant τ is τ = L/R, where L is the inductance and R is the resistance. In the first time constant τ, the current rises from zero to 0.632I0, and 0.632 of the remainder in every subsequent time interval τ. When the inductor is shorted through a resistor, current decreases as I = I0e^-t/τ (turning off), or I = ε/R(e^-t/τ) Here I0 is the initial current. Current falls to 0.368Io in the first time interval τ, and 0.368 of the remainder toward zero in each subsequent time τ.
For inductors in AC circuits, when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a 90 degree phase angle. The opposition of an inductor to a change in current is expressed as a type of AC resistance.
Ohm's law for an inductor is I = V/XL, where V is the rms voltage across the inductor. XL is defined to be the inductive reactance, given by XL = 2πfL, with f the frequency of the AC voltage source in hertz.
Inductive reactance XL has units of ohms and is greatest at high frequencies. For capacitors, when a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a 90 degree phase angle.
Since a capacitor can stop current when fully charged, it limits current and offers another form of AC resistance. Ohm's law for a capacitor is I = V/XC, where V is the rms voltage across the capacitor. XC is the capacitive reactance XC = 1/2πfC. XC has units of ohms and is greatest at low frequencies.
The AC analogy to resistance is impedance Z, the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm's law: I0 = V0/Z or Irms = Vrms/Z where I0 is the peak current and V0 is the peak source voltage.
Impedance has units of ohms and is Z = √R^2+(XL-XC)^2 The resonant frequency f0 at which XL = XC is fo = 1/2π√LC
In an AC circuit, there is a phase angle 𝜙 between source voltage V and the current I, which can be found from cos 𝜙 = R/Z
𝜙 = 0 degrees for a purely resistive circuit or an RLC circuit at resonance. The average power delivered to an RLC circuit is affected by the phase angle and Pave = IrmsVrms cos 𝜙 (cos 𝜙 is called the power factor, which ranges from 0 to 1).
The energy stored in an inductor (magnetic field) U is U = 1/2LI^2 The self-inductance per unit length of coaxial cable is L/l = μ0/2π ln R2/R1
In an LC circuit, energy transferred in an oscillatory manner between the capacitor and inductor in an LC circuit occurs at an angular frequency ω = √1/LC The charge and current in the circuit are: q(t) = q0 cos(ωt + φ) and i(t) = -ωq0sin(ωt + φ)
RLC circuit: the underdamped solution for the capacitor charge in an RLC circuit = q(t) = q0e^-Rt/2L cos(ω't + φ) (charge as a function of time)
The angular frequency given in the underdamped solution for the RLC circuit is ω' = √(1/LC)-(R/2L)^2