potential of point b and we have
V a ϩ Ᏹ Ϫ ir ϭ V b ,
or
V b Ϫ V a ϭ Ᏹ Ϫ ir.
( 2 7 - 8 )
To evaluate this expression, we need the current i. Note that the circuit is the
same as in Fig. 27-4a, for which Eq. 27-4 gives the current as
(27-9)
Substituting this equation into Eq. 27-8 gives us
(27-10)
Now substituting the data given in Fig. 27-6, we have
(27-11)
Suppose, instead, we move from a to b counterclockwise, passing through
resistor R rather than through the battery. Because we move opposite the
current, the potential increases by iR.Thus,
V a ϩ iR ϭ V b
or
V b Ϫ V a ϭ iR.
( 2 7 - 1 2 )
Substituting for i from Eq. 27-9, we again find Eq. 27-10. Hence, substitution of
the data in Fig. 27-6 yields the same result, V b Ϫ V a ϭ 8.0 V. In general,
V b Ϫ V a ϭ
12 V
4.0 ⍀ ϩ 2.0 ⍀
4.0 ⍀ ϭ 8.0 V.
ϭ
Ᏹ
R ϩ r
R.
V b Ϫ V a ϭ Ᏹ Ϫ
Ᏹ
R ϩ r
r
i ϭ
Ᏹ
R ϩ r
.
778
CHAPTE R 27 CI RCU ITS
To find the potential between any two points in a circuit, start at one point and
traverse the circuit to the other point, following any path, and add algebraically
the changes in potential you encounter.
Potential Difference Across a Real Battery
In Fig. 27-6, points a and b are located at the terminals of the battery. Thus, the
potential difference V b Ϫ V a is the terminal-to-terminal potential difference V
across the battery. From Eq. 27-8, we see that
V ϭ Ᏹ Ϫ ir.
( 2 7 - 1 3 )
If the internal resistance r of the battery in Fig. 27-6 were zero, Eq. 27-13 tells
us that V would be equal to the emf Ᏹ of the battery — namely, 12 V. However,
because r ϭ 2.0 ⍀, Eq. 27-13 tells us that V is less than Ᏹ. From Eq. 27-11, we
know that V is only 8.0 V. Note that the result depends on the value of the current
through the battery. If the same battery were in a different circuit and had a
different current through it, V would have some other value.
Grounding a Circuit
Figure 27-7a shows the same circuit as Fig. 27-6 except that here point a is directly
connected to ground, as indicated by the common symbol
. Grounding a circuit usually means connecting the circuit to a conducting path to Earth’s surface
(actually to the electrically conducting moist dirt and rock below ground). Here,
such a connection means only that the potential is defined to be zero at the
grounding point in the circuit. Thus in Fig. 27-7a, the potential at a is defined to
be V a ϭ 0. Equation 27-11 then tells us that the potential at b is V b ϭ 8.0 V.
V a ϩ Ᏹ Ϫ ir ϭ V b ,
or
V b Ϫ V a ϭ Ᏹ Ϫ ir.
( 2 7 - 8 )
To evaluate this expression, we need the current i. Note that the circuit is the
same as in Fig. 27-4a, for which Eq. 27-4 gives the current as
(27-9)
Substituting this equation into Eq. 27-8 gives us
(27-10)
Now substituting the data given in Fig. 27-6, we have
(27-11)
Suppose, instead, we move from a to b counterclockwise, passing through
resistor R rather than through the battery. Because we move opposite the
current, the potential increases by iR.Thus,
V a ϩ iR ϭ V b
or
V b Ϫ V a ϭ iR.
( 2 7 - 1 2 )
Substituting for i from Eq. 27-9, we again find Eq. 27-10. Hence, substitution of
the data in Fig. 27-6 yields the same result, V b Ϫ V a ϭ 8.0 V. In general,
V b Ϫ V a ϭ
12 V
4.0 ⍀ ϩ 2.0 ⍀
4.0 ⍀ ϭ 8.0 V.
ϭ
Ᏹ
R ϩ r
R.
V b Ϫ V a ϭ Ᏹ Ϫ
Ᏹ
R ϩ r
r
i ϭ
Ᏹ
R ϩ r
.
778
CHAPTE R 27 CI RCU ITS
To find the potential between any two points in a circuit, start at one point and
traverse the circuit to the other point, following any path, and add algebraically
the changes in potential you encounter.
Potential Difference Across a Real Battery
In Fig. 27-6, points a and b are located at the terminals of the battery. Thus, the
potential difference V b Ϫ V a is the terminal-to-terminal potential difference V
across the battery. From Eq. 27-8, we see that
V ϭ Ᏹ Ϫ ir.
( 2 7 - 1 3 )
If the internal resistance r of the battery in Fig. 27-6 were zero, Eq. 27-13 tells
us that V would be equal to the emf Ᏹ of the battery — namely, 12 V. However,
because r ϭ 2.0 ⍀, Eq. 27-13 tells us that V is less than Ᏹ. From Eq. 27-11, we
know that V is only 8.0 V. Note that the result depends on the value of the current
through the battery. If the same battery were in a different circuit and had a
different current through it, V would have some other value.
Grounding a Circuit
Figure 27-7a shows the same circuit as Fig. 27-6 except that here point a is directly
connected to ground, as indicated by the common symbol
. Grounding a circuit usually means connecting the circuit to a conducting path to Earth’s surface
(actually to the electrically conducting moist dirt and rock below ground). Here,
such a connection means only that the potential is defined to be zero at the
grounding point in the circuit. Thus in Fig. 27-7a, the potential at a is defined to
be V a ϭ 0. Equation 27-11 then tells us that the potential at b is V b ϭ 8.0 V.