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Energy Storage in Electric and Magnetic Fields: Capacitors and Inductors, Study notes of Physics

Class notes on the calculation of energy stored in electric and magnetic fields, focusing on capacitors and inductors. It includes formulas for energy density in electric and magnetic fields, and discusses kirchhoff's rules and the solution of rlc circuits.

Typology: Study notes

Pre 2010

Uploaded on 08/04/2009

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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
pro…table to think of the energy in these cases as being stored in the electric and magnetic
elds produced respectively in the capacitor and the inductor. From these calculations we
compute the energy per unit volume in electric and magnetic elds. These results turn out
to be valid for any electric and magnetic elds –not just those inside parallel plate capacitors
and inductors!
Let us rst consider a capacitor. Recall that the energy stored is
UE=q2
2C:
Assuming that we have a parallel plate capacitor, let’s insert the formula for the capacitance
of such a device
C=0A
d:
Let us further recall that the electric eld in a parallel plate capacitor is
E==0=q=(0A);
so that
q=0EA
and
UE=(E0A)2
2(0A=d)=0E2Ad
2:
The combination Ad is just the volume between the capacitor plates. The energy density in
the capacitor is therefore
uE=UE
Ad =0E2
2:
This formula for the energy density in the electric eld is speci…c to a parallel plate capacitor.
However, it turns out to be valid for any electric eld.
1
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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