Dr. Alain Brizard
College Physics II (PY 211)
Electric Current and Resistance
Textbook Reference: Chapter 25 – sections 1-8.
• Electric Battery
An ideal electric battery is a “black box” with two electrodes (known as terminals labeled
+ and −) kept at a fixed potential difference V ; by convention, the negative terminal has
potential zero and the positive terminal has potential V . The Figure below shows the
symbol for an ideal battery and, by convention, the battery causes electric charges to
flow from the positive terminal to the negative terminal (of course, charges flow from the
negative terminal to the positive terminal inside the battery).
A simple electric circuit is composed of an electric battery connected to various electric
devices; the circuit is said to be closed when electric charges are allowed to flow continuously
from the positive terminal to the negative terminal (when charges cannot flow, the circuit
is said to be open). When an ideal battery is connected to several electric devices (see
Figure below),
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the potential difference across each device drops as we follow the path from the positive
terminal to the negative terminal. Hence, in the Figure above, the potentials are Va = V ,
Vb = Vc < Va , and Vd = 0, with
V = (Va − Vb ) + (Vc − Vd ) = V1 + V2 ,
where V1 and V2 are the voltage drops across devices 1 and 2, respectively.
• Electric Current
The flow of electric charge in any path of the electric circuit is called electric current
(symbol I) defined as the rate with which charges cross any point of the path [SI unit:
Ampere (A = C/s)].
• Ohm’s Law & Resistance
The electric current I flowing through a closed circuit including a battery with variable
potential V will generally increase with an increase in potential. The Figures below show
current-voltage (I-V) curves of various electric devices.
Some electric devices (known as Ohmic devices) exhibit linear I-V curves (see left Figure
above). The constant slope of the current-versus-voltage curve for an Ohmic device is known
as its conductance; alternately, the constant slope of the voltage-versus-current curve for
an Ohmic device is known as its resistance (labeled R). For an Ohmic device, the voltageversus-current curve is known as Ohm’s Law:
V = R I,
where the SI unit of resistance is the ohm (symbol: Ω) defined as Ω = V/C. Non-ohmic
devices are described by I-V curves with non-constant resistance (or conductance); see
middle and right Figures above).
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Ohmic devices are known as resistors. The Figure below shows the symbol for a resistor
and emphasizes the fact that the voltage drops as we follow the current I from left to right
from voltage Va to voltage Va − RI in accordance to Ohm’s Law.
Resistors have typical values of electric resistance from 1 ohm to several millions of
ohms (MΩ).
• Resistivity
The resistance of a typical resistor is proportional to the longitudinal length ` of the
resistor along which the current flows and is inversely proportional to the cross-sectional
area A of the resistor. The remaining constant of proportionality, known as resistivity ρ,
depends on the material used for the resistor, i.e., R = ρ `/A, with the SI unit of resistivity
as Ω · m. Conductors have low resistivities [(1 − 100) × 10−8 Ω · m] while insulators have
very high resistivities (> 109 Ω · m).
Lastly, the conductivity σ = 1/ρ is defined as the inverse of resistivity, with SI unit
(Ω·m)−1. The temperature dependence of resistivity is such that ρ(T ) = ρ0 [1+α (T −T0)],
where ρ0 is the resistivity at temperature T0 and the constant α is independent of T if ∆T
is not too large. For example, copper has a resisitivity of 1.68 × 10−8 Ω · m at 20 o C; a
1-meter copper wire with a 1-mm diameter (area A = π d2 /4 ' 0.8 × 10−6 m2) has an
electric resistance of R = ρ`/A ' 2 × 10−2 Ω.
• Electric Power
The electric power transformed by an electric device is
dU
dQ
= V
= V I.
dt
dt
For an Ohmic device with resistance R, the rate of energy dissipation is
P =
P = R I2 =
3
V2
.
R
For example, the current I flowing through a 60-watt lightbulb connected to a 120-volt
power supply is I = P/V = 0.5 A.
• Alternating Current
When the power supply V (t) is a periodic function of time, the current flowing through
a resistor is also a function of time I(t) = V (t)/R. While a battery provides directcurrent (DC or dc) electric power, an alternating-current (AC or ac) power supply is often
represented by a sinusoidal function of time: V (t) = V0 sin ωt, where V0 denotes the peak
voltage and f = ω/(2π) denotes the frequency (e.g., 60 Hz). The peak current I0 = V0 /R
denotes the peak current flowing through the circuit.
The time-average dissipated power is
P = R I02 (sin ωt2) =
1
V2
2
R I02 = R Irms
= rms ,
2
R
√
where
√ Irms = (1/ 2) I0 denotes the root-mean-square (rms) value of the current and Vrms =
(1/ 2) V0 denotes the root-mean-square (rms) value of the voltage (e.g., 110 V).
Reading Assigment: Sec. 25-8 Microscopic View of Electric Current
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