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1.6.2 
Power and Energy in two-terminal Elements
Consider a general two-terminal element shown in Fig. 1.6-4. Assume that at the instant t both v(t) 
and i(t) are positive. 
This implies that positive charges are flowing into the element at that instant. Positive charges 
moving through a voltage drop will lose their potential energy to the element. If the element is 
a resistance, then, the energy lost by positive charges moving from higher potential end to lower 
potential end will appear as heat in the resistive element. If the element is a capacitor, then, the energy 
Fig. 1.6-4 

Passivesign
convention
+
–
v
(
t
)
i
(
t
)
Fig. 1.6-3 

Fourdifferentchoicesforreferencedirectionsandcorrespondingstatements
ofOhm’slaw
v
(
t
)
a
(i)
b
+
–
v
(
t
) = 
Ri
(
t
)
i
(
t
)
–
v
(
t
)
a
(ii)
b
+
v
(
t
) = –
Ri
(
t
)
i
(
t
)
v
(
t
)
a
(iii)
b
+
–
v
(
t
) = –
Ri
(
t
)
i
(
t
)
+
–
v
(
t
)
a
(iv)
b
v
(
t
) = 
Ri
(
t
)
i
(
t
)
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1.32


CircuitVariablesandCircuitElements
lost by these charges will get stored in the capacitor as electrostatic energy storage. If the element is 
an inductance, the energy lost by charges will appear as increase in energy stored in the magnetic field 
in the element. If the element is a source, the energy lost by the charges will be absorbed by the source 
element and stored inside in some other form of energy.
Let us assume that 
D
q coulombs of positive charge crossed the left terminal into the element in 
a time interval 
D
t centered around the time instant t. Then, the energy lost by these charges will be 
D
E 
=
v(t) 
×
D
q Joules since 1 C passing through a voltage drop of v(t) V will lose v(t) Joules of energy. 
Energy lost by the charge is equal to energy delivered to the element. Therefore, the energy delivered 
to the element over the time interval 
D
t is v(t) 
×
D
q Joules. The average rate at which this energy is 
delivered to the element is given by 
v t
q
t
( )
∆
∆
Joules/second. The instantaneous rate at which energy is 
delivered to an element is defined as the instantaneous power delivered to the element and is denoted 
by p(t). Therefore,
p t
E
t
v t
q
t
v t
i t
t
t
( )
lim
( ) lim
( )
( )
=
=
×
=
×
→
→
∆
∆
∆
∆
∆
∆
0
0
Joules/sec.
The unit Joules/sec is given the name ‘Watts’ and denoted by ‘W’.
Thus, the instantaneous power delivered to a two-terminal element is given by

p(t)=v (t)i (t)Wattswherev (t)andi (t)arethevoltageacrosstheelementandcurrent
throughtheelementasperpassivesignconvention.
Since instantaneous power p(t) is the instantaneous rate at which energy is delivered to the element, 
total energy E(t) that was delivered to the element from t 
=
-∞
to the current instant t is given by 
E t
p t t
t
( )
( ) .
=
−∞
∫
d
[Note: We are using the symbol E(t) to denote the time-varying electromotive force 
of a voltage source as well as the total energy delivered to a two-terminal element. The symbol will 
have to be interpreted contextually.] The relation between energy delivered to a two-terminal element 
and power delivered to it is summarized below.
p t
dE t
dt
v t i t
E t
p t dt
v t i t dt
t
t
( )
( )
( ) ( )
( )
( )
( ) ( )
=
=
=
=
−∞
−∞
∫
∫
W
J
(1.6-1)
Power delivered by a two-terminal element is obviously the negative of power delivered to it. 
Therefore,
Instantaneouspowerdeliveredbyatwo-terminalelement=
-
v (t)i (t)W,wherev (t)and
i (t)areinstantaneousvoltageandinstantaneouscurrentoftheelementasperpassive
signconvention.
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Power and Energy Relations for Two-Terminal Elements