2415_web_Lec_23_AC_Circuits_II

magnetic field is created.
• The inductor’s energy is stored in the field
with
energy density
1 2
B2
/ 0.
LR Circuits
• Suppose we have a
•.
R
L
steady current flowing
from the battery
I
through the LR circuit
shown.
Switch
• Then at t = 0 we flip the
switch…
V0
• This just takes the battery out of the circuit.
LR Circuits
• The decaying current • .
raise the current from zero to I takes total
work
I
U
LIdI
1 2
LI 2
0
• This energy is stored in the inductor exactly
as
U
1 2
CV
2
is
stored
in
a
capacitor.
Energy is Stored in Fields
• Suppose we have two coils close to each other. A changing current in coil 1 gives an emf in coil 2:
E2 M21dI1 / dt
• Evidently we will also find:
E1 M12dI2 / dt
inductance: when the current
I1 in coil 1 changes, it gives rise to an emf E 2 in coil 2.
• The mutual inductance M21 is defined by: M21 N221 / I1
where 21 is the magnetic
• The self-inductance L is defined by:
E L dI dt
• and symbolized by:
• Unit: for E in volts, I in amps L is in henrys (H).
Mutual Inductance
• We’ve already met mutual
• If an increasing current I is flowing through
an inductance L, the emf LdI/dt is opposing
the current, so the source supplying the
current is doing work at a rate ILdI/dt, so to
AC Circuits II
Physics 2415 Lecture 23 Michael Fowler, UVa
Today’s Topics
• Review self and mutual induction • LR Circuits • LC Circuits
Definition of Self-Inductance
generates an emf
R
L
E LdI / dt
and this drives the current through the resistance:
I Switch
LdI / dt IR
V0
• This is our old friend dx / dt ax
which has solution x x0eat .
flux through a single loop of
coil 2 from current I1 in coil 1.
E2
N2
d 21 dt
M 21
dI1 dt
• .Coil 1:
N1 loops
Coil 2: N2 loops
Coil 2 Coil 1
Mutual Inductance Symmetry
• Remember that for such a system, emf in one
coil will be generated by changing currents in
both coils:
E1
Байду номын сангаас
L1
dI1 dt
M
dI2 dt
E2
M
dI1 dt
L2
dI2 dt
Energy Stored in an Inductance
LR Circuits
• The equation
LdI / dt IR
has solution
I
I e(R/L)t 0
I0et /
so the decay time:
L/R
A
•.
I
I0 I(t) 0.37I0
0
• Remarkably, it turns out that M12 = M21
• This is by no means obvious, and in fact quite difficult to prove.
Mutual Inductance and Self Inductance
• For a system of two coils, such as a transformer, the mutual inductance is written as M.
• When a capacitor is charged, an electric field is created.
• The capacitor’s energy is stored in the field
with
energy
density
1 2
0
E
2.
• When a current flows through an inductor, a
• For any shape conductor, when the current changes there is an induced emf E opposing the change, and E is proportional to the rate of change of current.