ABSTRACT
W CV QC= =12 122 2 (4.3) This relation can be proved in several ways-by calculating (i) Joule’s heat generated by this charge, (ii) the energy of a charged capacitor by gradually increasing the distance between the two electrodes, and (iii) the electromagnetic field energy stored under the space between the two electrodes. 4.3.1 Calculating Joule’s Heat Generated by ChargesJoule’s heat generated by the charges can be calculated with the help of Fig. 4.2.Starting from I V
R = 1 2
(4.4)
t QI QV R
= = 2 (4.5) W = Pt = I2 Rt (4.6) I
Figure 4.2 RC circuit. Substituting Eqs. 4.4 and 4.5 in Eq. 4.6, we get W R V R
= ◊Ê ËÁ
ˆ ¯˜
Ê ËÁ
ˆ ¯˜2 2 2 (4.7) W QV= 1
2 (which is Eq. 4.2)
4.3.2 Calculating Energy of Charged CapacitorThe energy of a charged capacitor can be calculated by gradually increasing the distance between the two electrodes (if W ≠ 0), as exhibited in Fig. 4.3. +Q –Q
d Figure 4.3 A conductor system. We know that F QE= 1
2 (4.8) W F Q E= =d d1
2 ( ) (4.9)
Thus, a charged capacitor naturally has electromagnetic energy as W QV= 1 2
(which is Eq. 4.2) 4.3.3 Calculating Electromagnetic Field Energy Stored
We calculate the electromagnetic field energy density W stored under the space between two electrodes as W E H dV E V V
= +Ê ËÁ
ˆ ¯˜
@Ú 12 1
2 2 2e m e (4.10)
W E S ES E= =1 2
2e d e d( ) (4.11)From Eq. 3.14, we have E n dS Q S Ú ◊ = 1e total or E S Q Q E S= fi =
e e (4.12)
Substituting Eq. 4.12 in Eq. 4.11, we finally obtain W QV= 1 2
(which is Eq. 4.2) We may be annoyed by the factor 0.5 in Eq. 4.2 when we learn the potential energy of a point charge in an electric field: W = qV (which is Eq. 3.6) This is neither a mystery nor a contradiction. The electric field and the electric potential in a capacitor are produced by the charge itself and are zero when there is no stored charge. In the case of a point charge under an electric field, the electric field exists no matter whether the charge in question exists or not. This difference of the energy between a charge in free space and this charge under two electrodes is reflected as the difference of factor 0.5.
