Post on 19-Dec-2015
Physics 1402: Lecture 21Today’s Agenda
• Announcements:– Induction, RL circuits
• Homework 06: due next MondayHomework 06: due next Monday
• Induction / AC current
Recap from the last Chapter:
• Time dependent flux is generated by change in magnetic field strength due motion of the magnet
• Note: changing magnetic field can also be produced by time varying current in a nearby loop
Faraday's Law of Induction
v
BN S
v
BS N
BdI/dt
Can time varying current ina conductor induce EMF inin that same conductor ?
Self-Inductance• The inductance of an inductor ( a set of coils in some
geometry ..eg solenoid, toroid) then, like a capacitor, can be calculated from its geometry alone if the device is constructed from conductors and air.
• If extra material (eg iron core) is added, then we need to add some knowledge of materials as we did for capacitors (dielectrics) and resistors (resistivity)
• Archetypal inductor is a long solenoid, just as a pair of parallel plates is the archetypal capacitor.
r <<
l
l
r
N turns
SI UNITS for L : Henry
d
A
- - - - -
+ + + +
Calculationl
r
N turns
• Long Solenoid:
N turns total, radius r, Length l
For a single turn,
The total flux through solenoid is given by:
Inductance of solenoid can then be calculated as:
This (as for R and C) depends only on geometry (material)
RL Circuits
• At t=0, the switch is closed and the current I starts to flow.
• Loop rule:
Note that this eqn is identical in form to that for the RC circuit with the following substitutions:
RI
a
b
L
I
RC: RCRL:
Lecture 21, ACT 1• At t=0 the switch is thrown from position b to
position a in the circuit shown:
– What is the value of the current I a long time after the switch is thrown?
(a) I = 0 (b) I = / 2R (c) I = 2 / R
a
b
R
L
II
R
(a) I = 0 (b) I = / 2R (c) I = 2 / R
1A
• What is the value of the current I immediately after the switch is thrown?
1B
RL Circuits
• To find the current I as a fct of time t, we need to choose an exponential solution which satisfies the boundary condition:
• We therefore write:
• The voltage drop across the inductor is given by:
τRL = LR
R
a
b
L
I I
RL Circuit ( on)
t
I
0
RL/R 2L/R
VL
0t
Current
Max = R
63% Max at t=L/R
Voltage on L
Max = /R
37% Max at t=L/R
RL Circuits• After the switch has been in
position a for a long time, redefined to be t=0, it is moved to position b.
• Loop rule:
• The appropriate initial condition is:
• The solution then must have the form:
R
a
b
L
I I
RL Circuit ( off)
0
-
VL
t
L/R 2L/R
t
I
0
R Current
Max = R
37% Max at t=L/R
Voltage on L
Max = -
37% Max at t=L/R
Review: RC Circuits
(Time-varying currents)• Discharge capacitor:
C initially charged with Q=C
Connect switch to b at t=0.
Calculate current and charge as function of time.
• Convert to differential equation for q:
C
a
b+ +
- -
R
I I
• Loop theorem
Review: RC Circuits
(Time-varying currents)
• Trial solution:
q = Ce- /t RC
• Check that it is a solution:
!
Note that this “guess” incorporates the
boundary conditions:
C
a
b+ +
- -
R
I I• Discharge capacitor:
Review: RC Circuits
(Time-varying currents)
• Current is found from differentiation:
Conclusion:
• Capacitor discharges exponentially with time constant τ= RC
• Current decays from initial max value (= -/R) with same time constant
• Discharge capacitor:
C
a
b+
- -
R+
I I
q = Ce- /t RC
0
-R
I
t
RC 2RC
Discharging Capacitor
t
q
0
C
Current
Max = -/R
37% Max at t=RC
Charge on C
Max = C
37% Max at t=RC
Energy of an Inductor• How much energy is stored in
an inductor when a current is flowing through it?
• Start with loop rule:
• From this equation, we can identify PL, the rate at which energy is being stored in the inductor:
• We can integrate this equation to find an expression for U, the energy stored in the inductor when the current = I:
R
a
b
L
I I
• Multiply this equation by I:
Where is the Energy Stored?• Claim: (without proof) energy is stored in the Magnetic field
itself (just as in the Capacitor / Electric field case).
• To calculate this energy density, consider the uniform field generated by a long solenoid:
l
r
N turns
• The inductance L is:
• We can turn this into an energy density by dividing by the volume containing the field:
• Energy U:
Mutual Inductance• Suppose you have two coils
with multiple turns close to each other, as shown in this cross-section
• We can define mutual inductance M12 of coil 2 with respect to coil 1 as:
Coil 1 Coil 2
B
N1 N2
It can be shown that :
Inductors in Series• What is the combined (equivalent)
inductance of two inductors in series, as shown ?
a
b
L2
L1
a
b
LeqNote: the induced EMF of two inductors now adds:
Since:
And:
Inductors in parallel• What is the combined (equivalent)
inductance of two inductors in parallel, as shown ?
a
b
L2L1
a
b
LeqNote: the induced EMF between points a and be is the same !
Also, it must be:
We can define:
And finally:
LC Circuits
• Consider the LC and RC series circuits shown:
LCC R
• Suppose that the circuits are formed at t=0 with the
capacitor C charged to a value Q. Claim is that there is a qualitative difference in the time development of the currents produced in these two cases. Why??
• Consider from point of view of energy!
• In the RC circuit, any current developed will cause energy to be dissipated in the resistor.
• In the LC circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor!
RC/LC Circuits
RC:
current decays exponentially
C R
i
Q
-it
0
0 1
+++
- - -
LC
LC:
current oscillates
i
0 t
i
Q+++
- - -
LC Oscillations(quantitative)
• What do we need to do to turn our qualitative knowledge into quantitative knowledge?
• What is the frequency of the oscillations?
LC+ +
- -
LC Oscillations(quantitative)
• Begin with the loop rule:
• Guess solution: (just harmonic oscillator!)
where: • determined from equation
• , Q0 determined from initial conditions • Procedure: differentiate above form for Q and substitute into
loop equation to find .
LC+ +
- -
i
Q
remember:
Review: LC Oscillations
• Guess solution: (just harmonic oscillator!)
where: • determined from equation
• , Q0 determined from initial conditions
LC+ +
- -
i
Q
1
which we could have determinedfrom the mass on a spring result:
Lecture 21, ACT 2• At t=0 the capacitor has charge Q0; the resulting
oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I . – What is the relation between 0 and 2 , the
frequency of oscillations when the initial charge = 2Q0 ?
(a) 2 = 1/2 0 (b) 2 = 0 (c) 2 = 2 0
1A
LC
+ +
- -Q Q=
t=0
Lecture 21, ACT 2• At t=0 the capacitor has charge Q0; the
resulting oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I .
(a) I = I (b) I = 2 I (c) I = 4 I
• What is the relation between I and I , the maximum current in the circuit when the initial charge = 2Q0 ?
1B
LC
+ +
- -Q Q=
t=0