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CPHY 122
Class Notes 15
Instructor: H. L. Neal
1 Energy of Electric and Magnetic Fields
In this section we calculate the energy stored by a capacitor and an inductor. It is most proÖtable to think of the energy in these cases as being stored in the electric and magnetic Öelds produced respectively in the capacitor and the inductor. From these calculations we compute the energy per unit volume in electric and magnetic Öelds. These results turn out to be valid for any electric and magnetic Öelds ñnot just those inside parallel plate capacitors and inductors! Let us Örst consider a capacitor. Recall that the energy stored is
UE =
q^2 2 C
Assuming that we have a parallel plate capacitor, letís insert the formula for the capacitance of such a device C =
0 A
d
Let us further recall that the electric Öeld in a parallel plate capacitor is
E = = 0 = q=( 0 A);
so that q = 0 EA
and
UE =
(E 0 A)^2
2( 0 A=d)
0 E^2 Ad 2
The combination Ad is just the volume between the capacitor plates. The energy density in the capacitor is therefore
uE =
UE
Ad
0 E^2
This formula for the energy density in the electric Öeld is speciÖc to a parallel plate capacitor. However, it turns out to be valid for any electric Öeld.
A similar analysis of a current increasing from zero in an inductor yields the energy density in a magnetic Öeld. The work done by the generator in time dt is
dW = Edq = EIdt
so that the power is
P = dW dt
= EI = LI
dI dt
d dt
LI^2
This implies that
W =
LI^2 :
But
U = W
LI^2 :
Or U =
LI^2 + constant.
The constant term is usually ignored. Now recall that for a solenoid
B = 0
N
`
I
L = 0
N 2
`
A
Putting this int0 to the equation for U gives
UB =
N 2
`
A
`
0 N
B
The solution of this equation is of the form
I (t) = Imax cos (!t + ) ;
where ! =
p LC
I(t)
ω t
I max
φ
Created using UNREGISTERED Top Draw 3.10 Nov 26,'106 12:55:56 PM
3 RLC Circuits
C (^) L
switch
Q (^) max
I
R
Created using UNREGISTERED Top Draw 3.10 Nov 26,'106 10:36:00 AM Once the switch is closed we must have
EL = q C
+ IR
or d^2 I dt^2
R
L
dI dt
LC
I = 0:
This equation has three types of solutions: (1) underdamped, (2) overdamped, and (3) critically damped.
For underdamping
I (t) = Imax exp
R
2 L
t
cos (!dt + )
!d =
s 1 LC
R
2 L
For overdamping
I = A 1 e ^1 t^ + A 2 e ^2 t
1 ; 1 =
R
2 L
s R 2 L
LC
