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Dr. Jie Zou PHY 1361 1
Chapter 32
Inductance
Dr. Jie Zou PHY 1361 2
Outline
Self-inductance (32.1) Mutual induction (32.4) RL circuits (32.2) Energy in a magnetic field (32.3) Oscillations in an LC circuit (32.5) The RLC circuit (32.6, 33.5)
Dr. Jie Zou PHY 1361 3
Self inductance
Self induction: the changing flux through the circuit and the resultant induced emf arise from the circuit itself. The emf L set up in this case is called a self-induced emf.
L = -L(dI/dt) L = - L/(dI/dt): Inductance is a measure
of the opposition to a change in current. Inductance of an N-turn coil: L = NB/I;
SI unit: henry (H).
Due to self-induction, the current in the circuit does not jump from zero to its maximum value instantaneously when the switch is thrown closed.
Dr. Jie Zou PHY 1361 4
Mutual induction Mutual induction: Very often,
the magnetic flux through the area enclosed by a circuit varies with time because of time-varying currents in nearby circuits. This condition induces an emf through a process known as mutual induction.
An application: An electric toothbrush uses the mutual induction of solenoids as part of its battery-charging system.
Dr. Jie Zou PHY 1361 5
RL circuits An inductor: A circuit element
that has a large self-inductance is called an inductor.
An inductor in a circuit opposes changes in the current in that circuit.
A RL circuit: Kirchhoff’s rule:
Solving for I: I = (/R)(1 – e-t/) = L/R: time constant of the RL
circuit. If L 0, i.e. removing the inductance
from the circuit, I reaches maximum value (final equilibrium value) /R instantaneously.
0dt
dILIR
Dr. Jie Zou PHY 1361 6
Energy in a magnetic field Energy stored in an
inductor: U = (1/2)LI2. This expression represents
the energy stored in the magnetic field of the inductor when the current is I.
Magnetic energy density: uB = B2/20 The energy density is
proportional to the square of the field magnitude.
I ILIIdILLIdIdUU
dt
dILI
dt
dUdt
dILIRII
dt
dILIR
0 0
2
2
2
1
0
Dr. Jie Zou PHY 1361 7
Oscillations in an LC circuit Total energy of the circuit: U = UC + UL =
Q2/2C + (1/2)LI2. If the LC circuit is resistanceless and non-
radiating, the total energy of the circuit must remain constant in time: dU/dt = 0.
We obtain
Solving for Q: Q = Qmaxcos(t + ) Solving for I: I = dQ/dt = - Qmaxsin(t +
)
Natural frequency of oscillation of the LC circuit:
QLCdt
Qd 12
2
LC
1
Dr. Jie Zou PHY 1361 8
Oscillations in an LC circuit-from an energy point of view
Dr. Jie Zou PHY 1361 9
The RLC circuit The rate of energy
transformation to internal energy within a resistor: dU/dt = - I2R
Equation for Q:
Compare this with the equation of motion for a damped block-spring system:
Solving for Q: Q = Qmaxe-Rt/2Lcos(dt)
02
2
C
Q
dt
dQR
dt
QdL
02
2
kxdt
dxb
dt
xdm
