Inductance- II
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Transcript Inductance- II
Inductance- II
Plan:
Energy Storage in a Magnetic
field
Energy density and the
magnetic field
Recap
di
L
dt
di
L L
dt
• Units : volt-second/ampere
(Henry)
N B
L
i
NB 0 N h b
L
ln
i
2
a
2
Toroid
• Solenoid
L n2 A
0
l
Filling the magnetic materials
can increase the inductance
L m L0
Inductors in series
Leq L1 L2
• Inductors in parallel
1
1 1
Leq L1 L2
Energy storage in magnetic
field
• Energy is stored in the electric
field due to the charges.
• Similar manner, there is energy
stored around a current carrying
wire, where magnetic field
exists.
Example
• Work done in separating the
current carrying wires is stored
in the energy of the magnetic
field of the wire.
• This energy can be recovered
by allowing the wires to move.
Energy in Magnetic fields
• When the current is turned on, it has to
work against the back emf.
• This energy is recovered if the current
is turned off.
• This energy can be regarded as the
energy stored in the magnetic field.
• Work done on a charge dq against
the back emf in one trip around
the circuit
W dq
•Work done per unit time
dW
I
dt
dW
dI
LI
dt
dt
The energy stored in the
magnetic field.
• Work done, from zero to build a
current I
W LI dI
1 2
W LI
2
Energy density and magnetic
field
UB
uB
volume
UB
uB
AL
1 2
LI
2
uB
AL
1 2
LI
2
uB
AL
• For a solenoid
L l 0 n A
2
1 2
2
I l 0 An
uB 2
AL
uB
1
2 0
B
2
• A solenoid plays a role for magnetic field
similar to that of the parallel plate capacitor
for the electric fields.
uB
1
2 0
Solenoid
B
2
1
2
uB 0 E
2
Capacitor
A length of cupper wire carries a current of 10
A, uniformly distributed. Calculate (a)
magnetic energy density just outside the
surface of the wire. The wire diameter is 2.5
mm.
0 I
B
2r
1 0 I
uB
2 0 2r
2
uB =
3
1J/m
Find the magnetic energy density of a
circulating
electron
in
the
hydrogen
atom.Electron circulates about the nucleus in a
circular path of radius 5.29 x 10-11m at a
frequency f of 6.60 x 1015 Hz (rev/s).
uB
B
1
2 0
B
2
q
I qf
T
0 I
B= 12.6 T
2R
uB = 6.32 x 107 J/m3
A long coaxial cable carries a current I
(the current flows down the surface of
the inner cylinder, radius a, and back
along the outer cylinder, radius b. Find
the magnetic energy stored in a section
of length l.
I
a
b
I
0 I
ˆ
B
2r
uB
1
2 0
1 0 I
uB
2 0 2r
0 I
uB 2 2
8 r
2
B
2
2
• Energy in a cylindrical shell of
length l, radius r and thickness
ds is
2
0 I
dU 2 2 dr rd dz
8 r
0 I
U 2 2 2lr dzdr
8
r
0 a
l b
2
0 I
U 2 2 2lr dzdr
8 r
0 a
l b
2
0 I l b
U
ln
4
a
2
1 2
U LI
2
0l b
L
ln
2 a
0 I
dzd r
2r
0 a
l b
b
Area element
is dr dz
r
a
Calculate the energy stored in a
section of length l of the
solenoid.
1 2
W LI
1
2
2 2
2
W 0 n R lI
L 0n R l
2
2
2
Calculate the energy stored in the
toroidal coil.
1 2
uB
B
2 0
0 NI
B
2r
UB
1
2 0
B
d
2
UB
1
2 0
2 b
B d
2
2
0 NI
UB
dr rdh
2 0 0 a 2r
1
1
b
2 2
UB
0 n I h ln
4
a
A long wire carries a current I
uniformly distributed over a cross
section of the wire.
(a) Find the magnetic energy of a
length l stored within the wire.
0 I r
B
2
2 R
1 0 Ir
uB
2
2 0 2R
2
0 I
2
uB 2 4 r
8 R
2
R l 2
0 I 2
U B 2 4 r dr rd dz
8 R
0 0 0
2
0 I l
UB
16
2
Find the inductance of the length
l of the wire associated with the
flux inside the wire
1 2
U B LI
2
0 I l
UB
16
2
0l
L
8
Two long parallel wires, each radius
a, whose centers are at a distance d
apart carry equal current in opposite
directions. Neglecting the flux within
the wire themselves, find the
inductance of a length l of such a
pair of wires.
I
a
d-y
d
y
I
a
d a
a
0 I 0 I
ldy
2y 2 d y
0 Il
d a
d a
ln y a lnd y a
2
0l d a
L
ln
a
A uniform magnetic field B fills a
cylindrical volume of radius R. A metal
rod of length L is placed as shown. If B
is changing at the rate dB/dt, find the
emf that is produced by the changing
magnetic field and that acts between the
ends of the rod.
Facts
• For electrostatics E dl 0
Time varying
magnetic fields
E
d
l
A
B
E
d
l
0
Path
dependent
Electric field at a distance r from
center
d
E 2r BA
dt
r dB
E
2 dt
Along the triangle via path
AOBA
O B A
E dl E dl E dl E dl
A
O
B
A
E dl 0 0 E dl
A
B
A
d
B A 0 0 E dl
dt
B
O
B
A
d
B A 0 0 E dl
dt
B
dB
A
E dl
dt B
A
• A is the area of the triangle AOB
If you choose other path
Via
Path
ACBA
B
A
C
E
d
l
B
AC
A
E dl E dl
B
B
A
d
B da E dl E dl
dt
AC
B
A
d
B da RE E dl
dt
B
Important laws
divE
0
divB 0
dB
Curl E
dt
CurlB 0 J