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Chapter 30. Mutual Inductance. Consider a changing current in coil 1 We know that B 1 = m 0 i 1 N 1 And if i 1 is changing with time, dB 1 /dt= m 0 N 1 d(i 1 )/dt But a changing B-field across coil 2 will initiate an EMF 2 such that EMF 2 =-N 2 A 2 dB1/dt

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mutual inductance
Mutual Inductance

Consider a changing current in coil 1

We know that

B1=m0i1N1

And if i1 is changing with time,

dB1/dt=m0N1 d(i1)/dt

But a changing B-field across coil 2 will initiate an EMF2 such that

EMF2=-N2A2 dB1/dt

Since dB1/dt is proportional to di1/dt then the

Where M is the mutual inductance which is based on the sizes of the coils, and the number of turns

mutual mutual inductance
Mutual-mutual Inductance

But it could be that the changes are happening in coil 2. Then

It turns out that this value of M is identical to the previously discussed M so

my favorite unit the henry
My favorite unit—the henry
  • The Henry (H) is the unit of inductance
  • Equivalent to:
    • 1H=1 Wb/A = 1 V*s/A = 1 W*s = 1 J/A2
  • H is large unit; typically we use small units such as mH and mH.
self inductance
Self Inductance
  • But a coil of wire with a changing current can produce an EMF within itself.
  • This EMF will oppose whatever is causing the changing current
  • So a coil of wire takes on a special name called the inductor
inductor
Inductor

Electrical symbol

  • Definition of inductance is the magnetic flux per current (L)
  • For an N-turn solenoid, L is
    • L= NF/I
    • N turns= (n turns/length)*(l length)
    • The near center solution of inductance depends only on geometry
inductor7

i (increasing)

i (decreasing)

High potential

Low potential

VL

acts like a

VL

acts like a

Low potential

High potential

Inductor

Electrical symbol

Again, the EMF acts to oppose the change in current

rl circuits

R

A

S

B

V

L

RL Circuits
  • Initially, S is open so at t=0, i=0 in the resistor, and the current through the inductor is 0.
  • Recall that i=dq/dt
switch to a
Switch to A

Initially, the inductor acts against the changing current but after a long time, it behaves like a wire

R

A

S

H

i

B

V

L

L

voltage across the resistor and inductor

R

A

S

B

V

L

Voltage across the resistor and inductor

Potential across resistor, VR

Potential across capacitor, VC

At t=0, VL=V and VR=0

At t=∞, VL=0 and VR=V

l r another time constant
L/R—Another time constant
  • L/R is called the “time constant” of the circuit
  • L/R has units of time (seconds) and represents the time it takes for the current in the circuit to reach 63% of its maximum value
  • When L/R=t, then the exponent is -1 or e-1
  • tL=L/R
switch to b
Switch to B
  • The current is at a steady-state value of i0 at t=0

R

A

S

B

V

L

energy considerations
Energy Considerations

Rate at which energy is supplied from battery

Rate at which energy is stored in the magnetic field of the inductor

Energy of the magnetic field, UB

energy density u
Energy Density, u
  • Consider a solenoid of area A and length, l

Energy stored at any point in a magnetic field

Energy stored at any point in a magnetic field

l c oscillator the heart of everything
L-C Oscillator – The Heart of Everything

C

L

If the capacitor has a total charge, Q

starting points
Starting Points

Charge q

Current i

t

The phase angle, f, will determine when the maximum occurs w.r.t t=0

The curves above show what happens if the current is 0 at t=0

energy considerations18
Energy considerations
  • A quick and dirty way to solve for i at any time t in terms of Q & q
  • At t=0, the total energy in the circuit is the energy stored in the capacitor, Q2/2C
  • At time t, the energy is shared between the capacitor and inductor
    • (q2/2C)+(1/2 Li2)
  • Q2/2C= (q2/2C)+(1/2 Li2)
give me an r
Give me an “R”!

Consider adding a resistor, R to the circuit

The resistor dissipates the energy. For example, consider a child on a swing. His/her father pushes the child and gets the child swinging. In a perfect system, the child will continue swinging forever.

The resistor provides the same action as if the child let their feet drag on the ground. The amplitude of the child’s swing becomes smaller and smaller until the child stops.

The current in the LRC circuit oscillates with smaller and smaller amplitudes until there is no more current

mathematically
Mathematically

When oscillation stops due to R, critically damped

If R is small,

underdamped

Very large values of R, overdamped

why didn t i use a voltage source
Why didn’t I use a voltage source?
  • The practical applications of the LC, LR, and LRC circuits depend on using a sinusoidally varying voltage source:
    • An AC voltage source