ELECTRIC CIRCUITS
Electromotive force(emf) of a battery is the maximum voltage
that can be produced by the battery. It is measured with no
current flowing in the circuit.
A car battery has an emf of 12.0 v and a flashlight battery has
an emf of 1.5 v. Flashlight battery is really a misnomer since it
consists internally of only one pair of electrodes. The car
battery has 6 sets of electrodes(plates) arranged in series.
Electric current is defined as the rate at which charge flows
past a point in a conductor. The equation for average or
constant current is:
I = Δq/Δt
where I is current, Δq is the amount of charge, and Δt is the
time for that charge to pass by. The unit of electric current is
the ampere(A) which is one coulomb per second.
A defibrillator is used during a heart attack and passes 18.0
amps of current through a patient in 2.0 ms. How much charge
passes through the patient?
If the defibrillator voltage is 2500v, how much energy is
delivered to the patient?
Two types of electric current are direct current(dc) and
alternating current(ac). Dc is produced by batteries and the
current always flows in the same direction. In the case of ac,
the current changes direction(+ to -) periodically. In the U.S.
we use 60 Hz ac.
Today we know that electric current is carried by the electrons
that move in conductors. Early pioneers in the study of
electricity, however, thought that electric current was carried
by positive charge. Conventional current describes the flow of
electricity from the positive terminal to the negative terminal
of a source of current.
Using this idea does not affect the amount or rate of energy
transfer. We still use the same equations. Note in this text, I
stands for conventional current.
Ohm's Law
Ohm's Law relates current, voltage and resistance of a circuit.
It states that the current in a circuit is directly proportional to
the voltage across the circuit and inversely proportional to the
resistance in the circuit. The equation is:
I = V/R
V= IR
The equation may be seen in either form. Resistance is defined
as the tendency of a circuit component to oppose the flow of
electric current. The SI unit of resistance is the Ohm(Ω) which
is one volt per ampere.
A device that provides resistance in an electric circuit is called
a resistor.
Above, we see a drawing of a flashlight and its bulb along with
the circuit diagram.
Example
The filament of a light bulb has a resistance of 580 ohms. A
voltage of 120 v is connected across the filament. How much
current flows in the filament?
The resistance of a circuit component depends on several
factors. For most materials the resistance is directly
proportional to the length, inversely proportional to the crosssectional area, and directly proportional to the resistivity.
Resistivity is a proportionality constant that is a characteristic
of any conductor. It does vary with temperature.
The equation relating these variables is:
R = ρL/A
where ρ is resistivity. The metric unit for resistivity is the ohmmeter.
Example
A cylindrical copper cable carries a current of 1200 A. There is
a potential difference of .016 v between two points on the
conductor that are 0.24 m apart. Find the radius of the cable.
(ρ = 1.72 x 10-8 ohm m)
Superconductors are materials whose resistivity goes to zero at
very low temperatures called the critical temperature. For
most common metals, the critical temperature is less than 10
Kelvins. Some copper oxide complexes have a critical
temperature as high as 175 K.
Electric Power
Electric power, just like mechanical power, is work per unit of
time. Since the energy transferred depends on the voltage and
the amount of charge, the equation can be written as:
P = W/t = qV/t
P = VI
where V is voltage and I is current.
Since according to Ohm's Law, V = IR,
P = I2R
P = V2/R
Any of these equations can be used to calculate the power
expended in a circuit component.
Example
An electric heater is used to heat small amounts of water and
consists of a 15 ohm coil immersed directly in the water to be
heated. it operates at 120 V. How long does it take to heat 0.50
kg of water from 13° C to 100° C? c = 4186 j/kgC°
Alternating Current
The second type of electric current is alternating current in
which the direction of the current changes periodically. In the
U.S., 60 complete cycles of change occur every second. This is
the reason it is called 60 cycle or 60 Hz AC.
In the diagram above, we can see how the voltage across the
circuit varies from +V0 to -V0 in a sinusoidal fashion. In fact,
we can write an equation relating the voltage and time using
the sine function.
V = V0sin2πft
where V is the voltage as a function of time, V0 is the maximum
magnitude of the voltage, f is the frequency in Hz and t is the
time in seconds.
Example
For a frequency of 60 Hz, what is the smallest value of time
when the voltage = one half of the peak value?
In an AC circuit that has only resistance components, the
current oscillates in phase with the voltage. The equation for
current as a function of time in such a circuit is:
I = I0sin2πft
since I0 = V0/R.
The power in an AC circuit is calculated as P = VI and varies
with time as both current and voltage do. The equation is:
P = I0V0sin22πft
You can see that the power always has a positive value even
though the voltage and current can have negative values. This
happens because sin22πft(a squared term) is always positive.
The diagram above shows that the average power is ½ of the
maximum or peak power. A type of average voltage and
average current can be derived from the equation for average
power.
Pave = (½)I0V0
Pave = I0 · V0 = Irms · Vrms
√2 √2
where rms stands for root mean square. Irms and Vrms can be
calculated from the maximum values for current and voltage.
In the US, maximum AC voltage from a wall outlet is 170 volts
which makes the rms voltage 120 volts.
Example
A light bulb is connected to a 120 V outlet. The equation for
the current is I = (0.707 A)sin[(314 Hz)t]. (a)Find the frequency
of the alternating current. (b)Determine the resistance of the
bulb's filament. (c) Find the average power consumed by the
light bulb.
Series Wiring
A series circuit is wired so that there is only one path for the
electric current to follow. Each device in the circuit has the
same current through it.
In the circuit shown above, we can see how the current must
flow through each of the light bulbs. Since charge cannot
accumulate at any one of the light bulbs, it must continue to
flow at the same rate through all three.
The equivalent resistance for a series circuit is simply the sum
of the individual component resistances. The equation is:
Req = R1 + R2 + R3
This means that we could replace the three resistors in series
with Req and the current in the circuit would not be changed.
Example
Three resistors, 25, 45, and 75 ohms are connected in series
and a .51 amp current passes through them. Find the
equivalent resistance and the potential difference across the
three resistors.
Parallel Wiring
When devices are wired so that the same voltage is applied
across each device, they are said to be in parallel. One
disadvantage to series wiring is that all components in the
circuit must work for current to flow in the circuit. With
parallel wiring each branch forms a complete circuit
independently of the others.
In the diagram above, you can see that each light bulb forms a
complete circuit whether the others are included or not.
As in a series circuit, the resistors in a parallel circuit can be
replaced with an equivalent resistance without changing the
total voltage or current in the circuit.
In a parallel circuit the total current equals the sum of the
currents in each branch.
It = I1 + I2 + I3
We can replace each current symbol with its corresponding
V/R.
Vt/Req = V1/R1 + V2/R2 + V3/R3
Since the total voltage and the voltages across each resistor are
all the same, we can divide out all the voltages leaving:
1/Req = 1/R1 + 1/R2 + 1/R3
This equation can be used to find the equivalent resistance in a
parallel circuit.
Example
A resistor(resistance = R) is connected first in parallel and then
in series with a 2.00 ohm resistor. A battery delivers 5 times as
much current to the parallel circuit as it does to the series
circuit. Find the 2 possible values for R.
Circuits Wired Partly in Series and Partly in Parallel
When analyzing a circuit with both parallel and series
components, determine the equivalent resistance for one part
of the circuit at a time until you find the overall equivalent
resistance.
Example
Find the equivalent resistance between points A and B in the
diagram.
Internal Resistance
Any device in a circuit will have some resistance to the flow of
electric current. Even devices that supply voltage to the circuit
have resistance. In this case, it is called internal resistance.
In many cases this resistance is too small to affect the output
voltage of the device. However, the larger the current, the
larger the voltage drop caused by the internal resistance.
Terminal voltage is the voltage measured between two
terminals of a battery.
When no current is flowing, the terminal voltage is equal to the
emf(maximum voltage) of the battery. The terminal voltage
with current flowing is equal to the emf minus the product of
the current and the internal resistance.
Example
A battery has an emf of 12.0 v and an internal resistance of
0.15 ohms. Find the terminal voltage when the battery is
connected to a 1.50 ohm resistor.
Kirchoff's Rules
1. Junction Rule - the total current entering a circuit junction
must equal the total current leaving a circuit junction.
In the example, 7 A flows into the junction so a total of 7 A
must flow out.
2. Loop Rule - the sum of all the potential drops and rises
around a closed loop in a circuit must add up to zero. This
results from the Law of Conservation of Energy.
Each potential drop can be calculated by multiplying the
magnitude of the resistance by the magnitude of the current
through it.
To solve a problem using Kirchoff's rules follow these steps:
1. Draw the circuit diagram showing the current in each
branch of the circuit. If you get the current direction wrong in
any particular branch, your answer for that branch will be
negative. Simply reverse the current direction.
2. Use conventional current which always flows from positive
to negative. Mark both ends of each resistor + or - to help you
see the current direction.
3. Apply the junction rule and the loop rule to the circuit to
obtain as many independent equations as you have variables.
4. Solve the system of equations you generated in part 3.
Example
Determine the voltage across the 5.0 ohm resistor in the
diagram. Which end of the resistor is at the higher potential?
A galvanometer is a device that can be used to detect a voltage
or an electric current. The device is wired with a large series
resistance to be used as a voltmeter to measure voltage. It is
wired with a very small parallel resistance(shunt) to be used as
an ammeter to measure current.
A voltmeter is always wired parallel to the device over which a
voltage is to be measured.
An ammeter is always wired in series with the part of the
circuit in which current is to be measured.
Capacitors in Series and Parallel
Capacitors wired in parallel act like on larger capacitor. If
they are wired in parallel the area of the plates is effectively
increased therefore causing the combination to have a larger
total capacitance than any individual capacitor in the circuit.
The equation is:
CT = C1 + C2 + C3
The total charge on these capacitors will also be equal to the
sum of the individual charges while the voltage across each
capacitor will be the same.
In the case of capacitors arranged in series in a circuit, the
charge on each capacitor is the same while the voltage drops
are different and depend on the values of the individual
capacitances.
The equation for the equivalent capacitance when they are in
series is:
1/CT = 1/C1 + 1/C2 + 1/C3
Example
Determine the equivalent capacitance between A and B for the
capacitors in the diagram.