Environmental Engineering Reference

In-Depth Information

kg/m
3
,

=

2E

(
−

11

)

farad/meter, a dielectric strength of E

=

6

.

5E

(

9

)

V/m, and density

ρ
=

1E

(

3

)

3E(8) J/m
3
and the free energy per unit mass is

so that the free energy per unit volume is
F

/V
=

4

.

J/kg.

High-energy density capacitors are being developed for potential electric vehicle use, with

energy densities of about 10 Wh/kg

f

=

4

.

3
E

(

5

)

0.036 MJ/kg. The electric power input and output requires

high and variable electric potentials requiring power conditioning equipment to deliver the lower

voltage and higher current needed for traction motors. Electric failure of the capacitor dielectric

can present safety problems.

=

4.4.2

Magnetic Energy Storage

It is possible to store energy in the magnetic field produced by a current flowing in a conducting

wire. In the sketch of Figure 4.7(b), a magnetic inductor consists of a long cylinder of material, of

cross-sectional area
A
and length
L
, around which a coil of electric wire having
N
turns carries a

current
I
. Ampere's law relates the
magnetic induction B
11

in the material to the current flowing

in the wire,

=
µ

NI

L

B

(4.11)

where

is the magnetic permeability of the material.
12
To determine the energy stored in the

inductor, we first note that an increment of magnetic induction
dB
, caused by an increment of

current
dI
in a time interval
dt
, is related to the potential difference

µ

φ

between the ends of the

coil by Faraday's law of magnetic induction,

A
dB

dt
=
N

(4.12)

During this time interval, electric power of amount

φ

I
is expended in increasing the magnetic

induction, so that the free energy increase is

d
NI

L

2

AL

2

=
µ

AL

2

dB
2

=
φ

=

=

dF

Idt

NAI dB

µ

(4.13)

B
2

2

2

F

V
=

µ
=
µ(

NI

)

2
L
2

This relation may be expressed alternatively in terms of the
inductance

L

of the coil,
13

which is

N
2
A

the ratio

φ/(

dI

/

dt

)
=
µ

/

L
by equation (4.13), so that

I
2

2

=
L

F

(4.14)

11
The units of magnetic induction are weber/meter
2

=
volt second/meter
3
. (See Table A.1.)

12
The units of magnetic permeability are henry/meter
=
volt second/ampere meter. (See Table A.1.)

13
The unit of inductance is the henry
=
volt second/ampere. (See Table A.1.)