Energy density in electrostatic field

1. Subject Code :151002
Name Of Subject :Engineering Electromagnetics
Name of Unit :Energy and potential
Topic :Energy density in electrostatic field
Name of Faculty : Miss. Tanvi Shah
Mr. Niraj Tevar
Name of Students : (I) Savalia Avani(100870111020)
(ii) Patel Jay (100870111021)

2. Energy Density
Definition:
Energy density is the amount of energy stored in a 
given system or region of space per unit mass.
Often  only  the useful or  extractable  energy  is 
quantified,  which  is  to  say  that  chemically 
inaccessible  energy  such  as rest mass energy  is 
ignored.
Sub: EM Topic: Energy Density in Electrostatic Field

3. Consider  a point charge
Q
1
 transferred 
from infinity to position r
1
i
n the system.  It  takes no
work to  bring  the  first 
charge  from infinity since 
there is no electric field to 
fight  against  (as 
the system is  empty  i.e. 
charge free).
 Hence, W
1
= 0 J  
Sub: EM Topic: Energy Density in Electrostatic Field

4. Now bring in another point
charge
Q
2
 from infinity to position r2 
in the system. In this case we 
have to do work against the
electric field generated by the 
first charge Q
1
.
Hence, W
2
= Q
2
V
21
where V
21
 is the electrostatic
potential at point r
2
due to Q
1
.
- Work done W
2
 is also given 
as:
Sub: EM Topic: Energy Density in Electrostatic Field

5. And then the work required to bring Q3 to a distance
R13 from Q1 and distance R23 from Q2 is
W
3
 = Q
3
 V
31
 + Q
3
 V
32
 = Q
3
 ( V
31
 + V
32 
)
where V
31
and V
32
are electrostatic potential at point 
r
3
 due to Q
1
 and Q
2
respectively.
The work done is simply the sum of the 
work done against the electric field generated
by point charge Q
1
and Q
2
 taken in isolation: 

Sub: EM Topic: Energy Density in Electrostatic Field

6. Thus the total work done in assembling the 
three charges is given as:
   W
E
 = W
1
 + W
2
 + W
3
          = 0 + Q
2
 V
21
 + Q
3 
( V
31
 + V
32
 )
• Also total work done ( W
E
 ) is given as:
Sub: EM Topic: Energy Density in Electrostatic Field

7. If the charges were positioned in reverse order,
then the total work done in assembling them is given
as:
  W
E
 = W
3
 + W
2
+ W
1
         = 0 + Q
2
V
23
 + Q
3
( V
12
+ V
13
)
where V
23
 is the electrostatic potential at point r
2
 due 
to Q
3
and V
12
 and V
13
 are electrostatic potential at 
point r
1
 due to Q
2
 and Q
3
 respectively.
Sub: EM Topic: Energy Density in Electrostatic Field

8. Adding the above two equations we have,
 2W
E
 = Q
1
 ( V
12
 + V
13
) + Q
2
 ( V
21
 + V
23
) + Q
3
 ( V
31
 + V
32
)= 
Q
1
 V
1
 + Q
2
 V
2
 + Q
3
 V
3
Hence, W
E
 =1 / 2 [Q
1
V
1
 + Q
2
V
2
 + Q
3
V
3
]
where V
1
, V
2
 and V
3
are total potentials at position r
1
, 
r
2
 and r
3
respectively.
The result can be generalized for N point charges as:
Sub: EM Topic: Energy Density in Electrostatic Field

9. The above equation has three interpretation:

a) This equation represents the potential energy of 
the system.

b) This is the work done in bringing the static 
charges from infinity and assembling them in the 
required system.

c) This is the kinetic energy which would be released 
if the system gets dissolved i.e. the charges returns
back to infinity.
Sub: EM Topic: Energy Density in Electrostatic Field

10. In place of point charge, if
the system has continuous charge distribution ( line,
surface or volume charge), then the total work done 
in assembling them is given as:
Sub: EM Topic: Energy Density in Electrostatic Field

11. Since ρ
v
 =   . D and E = -   V,
∇ ∇
Substituting the values in the above equation, work 
done in assembling a volume charge distribution in
terms of electric field and flux density is given as:

The above equation tells us that the potential 
energy of a continuous charge distribution is stored 
in an electric field. 
Sub: EM Topic: Energy Density in Electrostatic Field

12. The electrostatic energy density w
E
is defined as:
Sub: EM Topic: Energy Density in Electrostatic Field