42. The voltage across the 30kΩresistor in Fig. 2860 is measured with (a) a 50kΩvoltmeter, (b) a 250kΩvoltmeter, and (c) a digital meter with 10MΩresistance. To two significant figures, what does each read?
43. In Fig. 2861 what are the meter readings when (a) an ideal voltmeter or (b) an ideal ammeter is connected between
666
CHAPTER 28
points
A
and
B
?
+
–
30 V
10 k
Ω
20 k
Ω
A
B
FIGURE
2861 Problem 43.
Solution
(a) An ideal voltmeter has infinite resistance, so
AB
is still an open circuit (as shown on Fig. 2861) when such a voltmeter is
connected. The meter reads the voltage across the
20 k
Ω
resistor (part of a voltage divider), or
(
)
(
)
30
20
20
10
V
=
=
20 V
(see Equation 282a or b). (b) An ideal ammeter has zero resistance, and thus measures the current through the
points
A
and
B
when shortcircuited (i.e., no current flows through the 20 k
Ω
resistor). In Fig. 2861, this would be
I
AB
=
30
10
3
V
mA.
=
Ω =
(Such a connection does not measure the current in the original circuit, since an ammeter should be
connected in series with the current to be measured.)
Problem
45. Show that the quantity
RC
has the units of time (seconds).
Solution
The SI units for the time constant,
RC
, are
(
)( )
(
)(
)
(
)(
)
,
Ω
F
V A
C V
s C
C
s
=
=
=
=
=
=
as stated.
Problem
47. Show that a capacitor is charged to approximately 99% of the applied voltage in five time constants.
Solution
After five time constants, Equation 286 gives a voltage of
V
e
C
=E
'
=

=

×


1
1
6 74
10
99 3%
5
3
.
.
of the applied
voltage.
Problem
49. Figure 2862 shows the voltage across a capacitor that is charging through a
4700
Ω
resistor in the circuit of
Fig. 2829. Use the graph to determine (a) the battery voltage, (b) the time constant, and (c) the capacitance.
CHAPTER 28
667
FIGURE
2862 Problem 49 Solution.
Solution
(a) For the circuit considered, the voltage across the capacitor asymptotically approaches the battery voltage after a long time
(compared to the time constant). In Fig. 2862, this is about 9 V. (b) The time constant is the time it takes the capacitor
voltage to reach
1
63 2%
1

=

e
.
of its asymptotic value, or 5.69 V in this case. From the graph,
τ
'
15
.
.
ms
(c) The time
constant is
RC
, so
C
=
=
15
4700
0 319
.
.
.
ms
F
=
Ω
μ
Problem
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 Winter '14
 Volt, Resistor, Electrical resistance, Series and parallel circuits, resistor R1