CHPTR 30 Notes: Inductance

Inductance

  • Inductance is a key concept in electromagnetic theory, utilizing coils of wire to explore fundamental interactions between magnetic fields and electric currents.

Basics of Inductance

  • A changing current in a coil of wire induces an electromotive force (emf) in an adjacent coil due to changing magnetic flux.
  • A changing current in a coil of wire will also induce an emf in that same coil, which is defined as an INDUCTOR.
  • The interaction between the coils results in a phenomenon known as MUTUAL INDUCTANCE.

Mutual Inductance

  • Mutual inductance can have significant effects in electric circuits:
      - Can act as a nuisance, where variations in current in one circuit can induce unwanted emfs in nearby circuits.
      - Can be beneficial, as it is the principle behind transformers, which rely on inductance to transfer energy between circuits.

Mathematical Representation of Mutual Inductance

  • Recall Faraday’s Law regarding induced emf:
      - The induced emf ( 1) is given by:
    extemf=−NdextΦBdtext{emf} = -N \frac{d ext{Φ}_B}{dt}
      - For mutually induced emfs, the relations are:
        - extemf2=−Mdi1dtext{emf}_2 = -M \frac{di_1}{dt}
        - extemf1=−Mdi2dtext{emf}_1 = -M \frac{di_2}{dt}
      - Where:
        - M=N2extΦBM = N_2 ext{Φ}_B
        - i1=N1extΦBi_1 = N_1 ext{Φ}_B
        - i2i_2
  • Mutual inductance (M) is measured in Henries (H), defined as:
      - 1H=1WbA=1VsA=1extΩsA1H = 1 \frac{Wb}{A} = 1 \frac{Vs}{A} = 1 \frac{ ext{Ωs}}{A}

Self-Inductance

  • Self-inductance occurs when a coil generates an opposing self-induced emf as the current within it increases or decreases.
  • The self-induced emf is proportional to the rate of change of current in the inductor.

Characteristics of Self-Inductance

  • In a single isolated circuit, the current flowing sets up a magnetic field, leading to a change in magnetic flux if current changes.
  • The self-induced emf is mathematically expressed as:
      - extemf=−Ldidtext{emf} = -L \frac{di}{dt}
      - Where the self-inductance (L), also in Henries (H), is defined as:
        - L=NextΦBiL = \frac{N ext{Φ}_B}{i}

Energy in Inductors

  • The potential energy (U) stored in an inductor of inductance (L) is expressed as:
      - U=12LI2U = \frac{1}{2} LI^2
  • In the context of an ideal toroidal solenoid, where the magnetic field is entirely confined within its core:
      - The magnetic energy density (u) is given by:
        - u=Uextvolume=B22extµ0u = \frac{U}{ ext{volume}} = \frac{B^2}{2 ext{µ}_0}
      - When the coils have a material of permeability (Km):
        - u=B22extµu = \frac{B^2}{2 ext{µ}}
        - Where Km=extµextµ0Km = \frac{ ext{µ}}{ ext{µ}_0}

R-L (Resistor-Inductor) Series Circuits

  • The primary function of an inductor within a circuit is to oppose variations in current through that circuit.
  • When a switch in the circuit is closed, current does not immediately rise to a steady value; it increases exponentially.

Behavior When Switch is Closed

  • According to Kirchhoff’s Loop Rule, the relationship can be expressed as:
      - −extemf−iR−Ldidt=0- ext{emf} - iR - L \frac{di}{dt} = 0

Time Constants in R-L Circuits

  • In R-L circuits, the current establishment is controlled by growth and decay:
      - Time constant describes how quickly current builds:
        - Current growth when the switch is closed:
    i(t)=I0(1−e−RLt)i(t) = I_0 (1 - e^{-\frac{R}{L}t})
        - Current decay (assuming only L and R in the circuit):
    i(t)=I0e−RLti(t) = I_0 e^{-\frac{R}{L}t}
  • RC and RL circuits emphasize exponential growth and decay characteristics.

L-C (Inductor-Capacitor) Series Circuits

  • L-C series circuits are crucial components in a myriad of electronic devices:
      - Applications include radio equipment, graphics tablets, oscillators, filters, tuners, amplifiers, and frequency mixers.

Conservation of Energy in L-C Circuits

  • The conservation of energy principle in an oscillating LC circuit relates charges and currents as follows:
      - 12CQ2+12Li2=extconstant\frac{1}{2} C Q^2 + \frac{1}{2} L i^2 = ext{constant}
      - (Refer to Table 30.1 for comparison to mass-spring oscillations from PHY2048)

R-L-C (Resistor-Inductor-Capacitor) Circuits

  • An RLC circuit sees a resistor dissipating energy, leading to damped harmonic oscillations.

  • The behavior can be represented mathematically as:
       - 12Ld2idt2+Rdidt+12Ci=0\frac{1}{2} L \frac{d^2 i}{dt^2} + R \frac{di}{dt} + \frac{1}{2} C i = 0
      - Where the damped oscillation frequency is:
    extω=1extLC−R24L2ext{ω} = \frac{1}{ ext{LC}} - \frac{R^2}{4L^2}

  • This concludes the essential notes on inductance, mutual inductance, self-inductance, energy storage in inductors, and dynamics of R-L, L-C, and R-L-C circuits.