SUMMARY
Current
S
(Section 19.1) Current
is the amount of charge flowing through a
conductor per unit time. The SI unit of current is the ampere, equal
to 1 coulomb per second 1 1 A 5 1 C s 2 . If a net charge DQ flows
through a wire in time Dt, the current through the wire is
I 5 DQ Dt (Equation 19.1).
E
/
/
Resistance and Ohm’s Law
(Section 19.2) In a conductor, the resistance R is the ratio of voltage
to current: R 5 V I (Equation 19.2). The SI unit of resistance is
the ohm 1 V 2 , equal to 1 volt per ampere. In materials that obey
Ohm’s law, the potential difference V between the ends of a conductor is proportional to the current I through the conductor; the
proportionality factor is the resistance R.
For a given conducting material, resistance R is proportional to
length and inversely proportional to cross-sectional area. For a specific material, this relationship can be expressed as R 5 r 1 L A 2
(Equation 19.3), where r is the resistivity of that material.
Resistance and resistivity vary with temperature; for metals,
they usually increase with increasing temperature.
S
E
DQ
Current I 5
Dt
I
I
L
/
Resistance: R 5
S
E
a
A
L
Also, R 5 r ,
A
b
I
Vab
Lower
potential
Higher
potential
Vab
I
where r 5 resistivity of
material.
/
Electromotive Force and Circuits
(Section 19.3) A complete circuit is a conductor in the form of a
loop providing a continuous current-carrying path. A complete
circuit carrying a steady current must contain a source of electromotive force (emf), symbolized by E. An ideal source of emf maintains a constant potential difference Vab 5 E (Equation 19.5), but
every real source of emf has some internal resistance r. The terminal potential difference Vab then depends on current: Vab 5 E 2 Ir
(Equation 19.7).
r
a
E 5 emf; r 5 internal resistance
E
b
+
I
Ideal emf source: Vab 5 E; r 5 0
Real emf source: Vab 5 E 2 Ir
emf source
R
Energy and Power in Electric Circuits
(Section 19.4) A circuit element with a potential difference V and a
current I puts energy into a circuit if the current direction is from
lower to higher potential in the device and takes energy out of the
circuit if the current is opposite. The power P (rate of energy transfer) is P 5 VI (Equation 19.9). A resistor R always takes energy
out of a circuit, converting it to thermal energy at a rate given by
P 5 Vab I 5 I 2R 5 Vab2 R (Equation 19.10).
/
Resistors in Series and in Parallel
(Section 19.5) When several resistors R1 , R2 , R3 , care connected in
series, the equivalent resistance Req is the sum of the individual
resistances: Req 5 R1 1 R2 1 R3 1 c. (Equation 19.12). When
several resistors are connected in parallel, the equivalent resistance
Req is given by
1
1
1
1
5
1
1
1 c.
Req
R1
R2
R3
(19.13)
R1
a
I
R1
x
R2
y
R3
b
I
a
I
R2
R3
b
I
Loop 1
Junction
+
I2
I1
+
Kirchhoff’s Rules
(Section 19.6) Kirchhoff’s junction rule is based on conservation of
charge. It states that the algebraic sum of the currents into any
junction must be zero: gI 5 0 (Equation 19.14). Kirchhoff’s loop
rule is based on conservation of energy and the conservative
nature of electrostatic fields. It states that the algebraic sum of the
potential differences around any loop must be zero: gV 5 0
(Equation 19.15). Be especially careful with signs when using
Kirchhoff’s rules.
Loop 2
E
I1 1 I2
At any junction: SI 5 0
Loop 3
R
E
Around any loop: SV 5 0
Electrical Measuring Instruments
(Section 19.7) The ideal behavior for a meter is for it to measure the
circuit quantities of interest without changing or disturbing them.
An ammeter always measures the current passing through it. An
ideal ammeter would have zero resistance, so that including it in a
branch of a circuit would not affect the current in that branch. A
voltmeter always measures the potential difference between two
points. An ideal voltmeter would have infinite resistance, so that no
current would flow through it.
Resistance–Capacitance Circuits
(Section 19.8) When a capacitor is charged by a battery in series
with a resistor, the current and capacitor charge are not constant.
The charge varies with time as q 5 Qfinal 1 1 2 e 2t/RC 2 (Equation 19.17). In a time t 5 RC, there is a significant change in the
charge on the capacitor. This time is called the time constant, or
relaxation time, and is the same for charging or discharging.
*Applications of Currents
(Sections 19.9 and 19.10) The conduction of nerve impulses is basically an electrical process. Currents through the body as small as
0.1 A can be fatal because they interfere with this process (which
also occurs in heart and other muscle cells).
In house wiring, one line entering the house is always neutral,
or at the same voltage as the ground (to which it is connected). The
other one or two wires are hot. The maximum current available
from an individual circuit is limited by the resistance of the wires;
if they carry too much current, I 2R power loss causes them to overheat. Protection against overloading of circuits is provided by fuses
or circuit breakers.
+
E
i
i, q
1q
R
q versus t
i
2q
C
O
i versus t
t