 # Lectures (Ch. 30) Inductance and Self-inductunce

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Lectures 18-19 (Ch. 30) Inductance and Self-inductunce
Mutual inductunce Tesla coil Inductors and self-inductance Toroid and long solenoid Inductors in series and parallel Energy stored in the inductor, energy density 7. LR circuit 8. LC circuit 9. LCR circuit

Mutual inductance Virce verse: if current in coil 2 is changing, the changing flux through coil 1 induces emf in coil 1.

Units of M Joseph Henry ( ) Typical magnitudes: 1μH-1mH

Examples where mutual inductance is useful

Tesla coil Estimate. Nikola Tesla (1856 –1943) [B]=1T to his honor

Example: M=? Mutual inductance may induce unwanted emf in nearby circuits. Coaxial cables are used to avoid it.

Self-inductance

Thin Toroid Long solenoid
Thin solenoid with approximately equal inner and outer radius. Long solenoid

Example. Toroidal solenoid with a rectangular area.

Inductors in circuits

Energy stored in inductor
Compare to

Magnetic energy density
Let’s consider a thin toroidal solenoid, but the result turns out to be correct for a general case Compare to: Energy is stored in E inside the capacitor Energy is stored in B inside the inductor

Example. Find U of a toroidal solenoid with rectangular area

LR circuit, storing energy in the inductor

Energy conservation law
Power output of the battery =power dissipated in the resistor + the rate at which the energy is stored in inductor General solution Initial conditions (t=0) Steady state (t→∞)

LR circuit, delivering energy from inductor
ε t The rate of energy decrease in inductor is equal to the power input to the resistor.

Oscillations in LC circuit

Oscillations in LC circuit

Compare to mechanical oscillator
F x

General solution q t i t

i t q t

Energy conservation law
UC+UL=const UC UL UL UC t Q -Q q T/2

Example. In LC circuit C=0.4 mF, L=0.09H.
The initial charge on the capacitor is 0.005mC and the initial current is zero. Find: (a) Maximum charge in the capacitor (b) Maximum energy stored in the inductor; (c) the charge at the moment t=T/4, where T is a period of oscillations.

Example. In LC circuit C=250 ϻF, L=60mH.
The initial current is 1.55 mA and the initial charge is zero. 1) Find the maximum voltage across the capacitor . At which moment of time (closest to an initial moment) it is reached? 2) What is a voltage across an inductor when a charge on the capacitor is 1 ϻ C? q

Example. In LC circuit C=18 ϻF, two inductors are placed in parallel: L1=L2=1.5H and mutual inductance is negligible. The initial charge on the capacitor is 0.4mC and the initial current through the capacitor is 0.2A. Find: (a) the current in each inductor at the instant t=3π/ω, where ω is an eigen frequency of oscillations; (b) what is the charge at the same instant? (c) the maximum energy stored in the capacitor;(d) the charge on the capacitor when the current in each inductor is changing at a rate of 3.4 A/s.

LCR circuit Characteristic equation Critical damping

a) Underdamped oscillations:
b) Critically damped oscillations: c) Overdamped oscillations:

Example. The capacitor is initially uncharged
Example. The capacitor is initially uncharged. The switch starts in the open position and is then flipped to position 1 for 0.5s. It is then flipped to position 2 and left there. 1) What is a current through the coil at the moment t=0.5s (i.e. just before the switch was flipped to position 2)? 2) If the resistance is very small, how much electrical energy will be dissipated in it? 3) Sketch a graph showing the reading of the ammeter as a function of time after the switch is in position 2, assuming that r is small. 10µF 25Ω 1 2 50V r 10mH A 3)

Induced oscillations in LRC circuit, resonance
~ Q At the resonance condition: an amplitude greatly insreases