-Self Inductance-Inductance of a Solenoid
-RL Circuit-Energy Stored in an Inductor
AP Physics C Mrs. Coyle
• Induced emf and induced current are caused by changing magnetic fields.
Self-InductanceSwitch Open: No current and no magnetic field
Switch Closed: current increases, creates an increasing magnetic field which in turn induces an induced emf
• The direction of the induced emf is opposite the direction of the emf of the battery
• Gradually the net current increases to an equilibrium value
• This effect is called self-inductance– Because the changing flux through the circuit and
the resultant induced emf arise from the circuit itself
• The emf εL is called a self-induced emf
Inductors• In general wires between resistors cause a small
self inductance which is ignored.
• However, some circuit elements such as solenoids can cause a significant inductance, L. These objects are called inductors.
• Symbol for an inductor:
Inductance• Induced emf:
• L is a proportionality constant called the inductance of the coil and it depends on the geometry of the coil and other physical characteristics
• The inductance is a measure of the opposition to a change in current
IL
dε Ldt
I LεL
d dt
Inductance Units
• The SI unit of inductance is the henry (H)
• Named for Joseph Henry
AsV1H1
Inductance in a Solenoid (Coil)
The polarity of the induced emf, from Lenz’s Law, opposes the change in magnetic flux.
Inductance of a Solenoid• Assume a uniformly wound solenoid having N
turns and length ℓ– Assume ℓ is much greater than the radius of the
solenoid• The interior magnetic field is uniform:
I Io oNB μ n μ
Inductance of a Solenoid
• Faraday’s Law
• Equate eq. 1 and 2 solve for L for a solenoid:
• Remember also :
I
BNL
IL
dε Ldt
Bdεdt
I LεL
d dt
(eq.1)
(eq.2)
Inductance of a Solenoid
• The magnetic flux through each turn is
• Note that L depends on the geometry of the object
IB oNABA μ
I
BNL
2oμ N ALI Io o
NB μ n μ
RL Circuit (Resistor and Inductor)• Kirchhoff’s loop rule for:
0II dε R Ldt
1I Rt Lε eR
LR
1I t τε eR
Time Constant,
RL Circuit Current
• The inductor affects the current exponentially• The current does not instantly increase to its
final equilibrium value• If there is no inductor, the exponential term
goes to zero and the current would instantaneously reach its maximum value as expected
1I Rt Lε eR
RL Circuit, Time Constant
= L / R
When T=
I= 0.632 Ieq,
then, the current has reached 63.2% of its equilibrium value.
1I t τε eR
I 11ε eR
LR Circuit, Charging
dIRI L 0dt
t /I 1 eR
LR
Prove:
LR Circuits, discharging
dIRI L 0dt
t /oI I e
LR
Prove:
Energy Stored in an Inductor, U• The rate at which energy is being supplied
by the battery (Power=IV)2 II I I dε R L
dt
21U LI2
0
II IU L d
IIdU dLdt dt
Remember for a capacitor:21U CV
2
Ex: RL Circuit
a) What happens JUST AFTER the switch is closed?
b) What happens LONG AFTER switch has been closed?
c) What happens in between?
Note: At t=0, a capacitor acts like a regular wire; an inductor acts like an open wire.
After a long time, a capacitor acts like an open wire, and an inductor acts like a regular wire.
Ex: RL Circuits
Immediately after the switch is closed, what is the potential difference across the inductor?(a) 0 V(b) 9 V(c) 0.9 V
• Immediately after the switch, current in circuit = 0.• So, potential difference across the resistor = 0• So, the potential difference across the inductor = E9 V
10
10 H9 V
Ex: RL Circuits• Immediately after the
switch is closed, what is the current i through the 10 resistor?(a) 0.375 A(b) 0.3 A(c) 0
• Long after the switch has been closed, what is the current in the 40 resistor?(a) 0.375 A(b) 0.3 A(c) 0.075 A
40
10 H
3 V
10
• Immediately after switch is closed, current through inductor = 0.• Hence, current trhough battery and through 10 resistor is i = (3 V)/(10) = 0.3 A
• Long after switch is closed, potential across inductor = 0.
• Hence, current through 40 resistor = (3 V)/(40) = 0.075 A
Ex: RL Circuits
• How does the current in the circuit change with time?
0dtdiLiRE
tLR
eREi 1
“Time constant” of RL circuit = L/R
i
i(t) Small L/R
Large L/R
t
• The switch has been in position “a” for a long time.
• It is now moved to position “b” without breaking the circuit.
• What is the total energy dissipated by the resistor until the circuit reaches equilibrium?
• When switch has been in position “a” for long time, current through inductor = (9V)/(10) = 0.9A.
• Energy stored in inductor = (0.5)(10H)(0.9A)2 = 4.05 J• When inductor “discharges” through the resistor, all this stored
energy is dissipated as heat = 4.05 J.
9 V
10
10 H
Ex: Energy Stored in a B-Field
Energy Density of a Magnetic Field• U = ½ L I 2
• Aℓ is the volume of the solenoid
• Magnetic energy density, uB :
• This applies to any region in which a magnetic field exists (not just the solenoid)
2 221
2 2oo o
B BU μ n A Aμ n μ
2
2Bo
U BuA μ
2oμ N AL
o
BIN
Example 32.5: The Coaxial Cable
• Calculate L for the cable• The total flux is
• Therefore, L is
• The total energy is
ln2 2I Ib
o oB a
μ μ bB dA drπr π a
ln2I
B oμ bLπ a
221 ln
2 4II oμ bU Lπ a
Prob.#3
A 2.00-H inductor carries a steady current of 0.500 A. When the switch in the circuit is opened, the current is effectively zero after 10.0 ms. What is the average induced emf in the inductor during this time?
Ans: 100V
Prob#5 A 10.0-mH inductor carries a current I = Imax
sin ωt, with Imax = 5.00 A and ω/2π = 60.0 Hz. What is the back emf as a function of time?
Ans: (18.8V)cos (377t)
Prob.# 7
An inductor in the form of a solenoid contains 420 turns, is 16.0 cm in length, and has a cross-sectional area of 3.00 cm2. What uniform rate of decrease of current through the inductor induces an emf of 175 μV?
Ans: -0.421A/s
Prob.#14
Calculate the resistance in an RL circuit in which L = 2.50 H and the current increases to 90.0% of its final value in 3.00 s.
Prob.#16
dIRI L 0dt
t /oI I e Show that is a solution
of the differential equation
where and Io is the current
at t=0.
LR
Prob. #20
A 12.0-V battery is connected in series with a resistor and an inductor. The circuit has a time constant of 500 μs, and the maximum current is 200 mA. What is the value of the inductance?
Prob.# 24
A series RL circuit with L = 3.00 H and a series RC circuit with C = 3.00 μF have equal time constants. If the two circuits contain the same resistance R, (a) what is the value of R and (b) what is the time constant?
Prob.#26
• A) What is the current in the circuit a long time after the switch has been in position a?
• B) Now the switch is thrown from a to b. Compare the initial voltage across each resistor and across the inductor.
• C) How much time elapses before the voltage across the inductor drops to 12.0V?
12.0
2.00 H
12 V
1200
Prob.#31
An air-core solenoid with 68 turns is 8.00 cm long and has a diameter of 1.20 cm. How much energy is stored in its magnetic field when it carries a current of 0.770 A?
Prob.#33
On a clear day at a certain location, a 100-V/m vertical electric field exists near the Earth’s surface. At the same place, the Earth’s magnetic field has a magnitude of 0.500 × 10–4 T. Compute the energy densities of the two fields.
Prob.# 36
A 10.0-V battery, a 5.00-Ω resistor, and a 10.0-H inductor are connected in series. After the current in the circuit has reached its maximum value, calculate (a) the power being supplied by the battery, (b) the power being delivered to the resistor, (c) the power being delivered to the inductor, and (d) the energy stored in the magnetic field of the inductor.