Download Notes Ch 17 – Current and Resistance

Ch. 20 – Electric Circuits
Electromotive force and current
 In a closed circuit, current is driven by a “source of emf” which is either a battery or a generator. Batteries convert
chemical potential energy into electrical energy and generators convert mechanical energy into electrical energy. Emf
(symbolized with  or V; units are volts) stands for electromotive force although the term force is used incorrectly
since emf is the source that maintains the potential difference (voltage) in the circuit. The maximum potential
difference of a battery is the emf of the battery. In a closed circuit, the battery creates an electric field within and
parallel to the wire, directed from the positive toward the negative terminal. The electric field exerts a force on the
free electrons causing them to move producing electric current. The diagram below shows the physical and schematic
diagram for one of the simplest circuits.
Current (symbolized with I) is the rate at which charge flows.
I=q/t units are C/s=A (ampere)
By convention, the direction of the flow of current is the same as the direction of the flow of positive charge which, in
metals, is opposite the direction of the flow of electrons. We use this convention since it is consistent with our earlier
use of a positive test charge for defining electric fields and potentials. Current is classified as either dc or ac based on
the motion of the current. In a direct current (dc) circuit, current moves around the circuit in the same direction at all
times. Batteries and dc generators produce direct currents. In an alternating current (ac) circuit, current moves first
one direction and then the opposite, changing direction from moment to moment. Ac generators (power companies)
produce alternating currents.
Ohm’s law
 When a potential difference (voltage) is applied across the ends of many metallic conductors, the current is directly
proportional to the applied voltage. This statement is known as Ohm’s law although it is not a fundamental law of
nature since it is only valid for certain materials and within a certain range.
VI
R is the proportionality constant for the above relationship which is the resistance of the conductor.
V=IR
Rearranging the above equation for R shows that the units for resistance are V/A =  (ohm). Experiments show that
most metals have a constant resistance over a wide range of applied voltages (graph (a) below). Such materials are
said to be ohmic since they obey Ohm’s law. Nonohmic materials do not obey Ohm’s law (graph (b) below).
Resistance and resistivity
 As charge moves through a circuit, it encounters resistance, or opposition to the flow of the current. Resistance is the
electrical equivalent of friction. In our circuit above, the wires and the light bulb would be considered resistances,
although usually the resistance of the wires is neglected. The resistance of a material is proportional to the length and
inversely related to the cross-sectional area.
L
RL/A
R
A
 is the proportionality constant known as the resistivity of the material and has units of ohm-meters ( m). Every
material has a characteristic resistivity that depends on its electronic structure and temperature (Table 20.1 in your
book). In metals, the resistivity increases with increasing temperature, whereas in semiconductors the reverse is true.
Superconductors are a class of metals and compounds whose resistance goes to virtually zero below certain
temperatures (critical temperature).

Example 1: The five resistors shown below have the lengths and cross-sectional areas indicated and are made of
material with the same resistivity. Which has the greatest resistance?

A resistor is a simple circuit element that provides a specified resistance. Resistors are represented by a zigzag line in
circuit diagrams (a straight line represents an ideal conducting wire, or one with negligible resistance). Resistors can
be used in circuits to control the amount of current in a conductor.
Electrical energy and power
 As current moves through a circuit, electrical energy is transformed into thermal energy due to collisions with atoms in
the resistor. The amount of heat produced in joules per second is equal to the power in the resistor. You should
remember from previous chapters the power is the rate at which work is done, or the rate at which energy is
transferred.
P
Work
time
Since W=qV and I=q/t and V=IR, by substitution it can be shown that
P  IV  I 2 R 
V2
R
Of course the unit for power is the joule/second, or watt.
Schematic diagrams and circuits
 A schematic diagram is a diagram that depicts the construction of an electrical apparatus or circuit using symbols to
represent the different circuit elements (emf, resistors, capacitors, wires, switches, ammeters, voltmeters, etc.). The
diagram at the beginning of the notes illustrates a simple circuit that causes a light bulb to shine. You may wonder
what would happen if you connect another bulb to the battery. It actually depends upon how you connect the bulb,
whether you have one path (series connection) or create another path (parallel connection). We will analyze each of
these connections separately to see their effect on the circuit. First, a few definitions.
 An ammeter is an instrument used to measure current. As shown below left, ammeters must be inserted into a
circuit so that the current passes directly through it (in series). A good ammeter is designed with a sufficiently
small resistance, so the reduction in current is negligible whenever the ammeter is inserted.
 A voltmeter is an instrument used to measure the potential difference (voltage) between two points. As shown
below right, voltmeters must be inserted in parallel in the circuit. A good voltmeter is designed with a large
resistance so the effects on voltage are negligible when it is inserted into the circuit.
V
A
+

-
Internal resistance is resistance in a battery or generator. In a battery, this is due to the chemicals and in a
generator it is due to the resistance in the wires and other components. As shown below, internal resistance is
treated as resistor connected in series with the external circuit.
Resistors in series
 Two or more resistors of any value placed in a circuit in such a way that the same current passes through each of them
is called a series connection. A series connection will have a single path between two points. A break in the circuit of
a series connection will disconnect all elements. Christmas lights often use series wiring which is why if one bulb goes
out the entire strand goes out. The diagram below shows a series connection with two resistors, R 1 and R2. When
resistors are connected in series, the total resistance of the circuit increases and the current decreases.

Rules for series connections
1. Current through all resistors in series is the same, because any charge that flows through the first resistor must
also flow through the second.
Itotal = I1 = I2 = I3.
The total current in the circuit is
I total 
2.
3.
Vtotal
Rtotal
The potential difference across the entire connection (total voltage) is equal to the sum of potential drops
(voltages) across each resistor.
Vtotal=V1+V2
And the voltage divides proportionally among the resistances according to Ohm’s law.
V2  I 2 R2
V1  I1 R1
The equivalent resistance is the sum of the individual resistances.
Req=R1+R2
Resistors in parallel
 Two or more resistors of any value placed in such a way that each resistor has the same potential difference is called a
parallel connection. A parallel connection will have a junction creating separate paths between two points. A break in
one of the paths does not affect the other path. Household circuits are generally connected so appliances, light bulbs,
etc. are connected in parallel so each gets the same voltage and each can be operated independently. The diagram
below shows a parallel connection with two resistors, R 1 and R2. When resistors are connected in parallel, the total
resistance decreases and the current coming into the junction increases, although the current through each path remains
the same.

Rules for parallel connections
1. The potential difference (voltage) across each resistor is the same.
Vtotal=V1=V2
2. Current coming into a junction is the sum of the currents in each path.
Itotal = I1 + I2
The total current in the circuit is
I total 
Vtotal
Rtotal
The current in each path is
I1 
3.
V1
R1
I2 
V2
R2
The equivalent resistance is the reciprocal of the sum of the individual resistances reciprocal.
1
1
1
 
Req R1 R2
The equivalent resistance is always less than smallest resistance in group.
Combined Series-Parallel Circuits
 Most circuits today use both series and parallel wiring to utilize the advantages of each type. Circuits containing
combinations of series and parallel circuits can be understood by analyzing them in steps. When determining the
equivalent resistance for a complex circuit, you must simplify the circuit into groups of series and parallel resistors and
then find the equivalent resistance for each group until the circuit is reduced to a single resistance. Work your way
backwards finding all potential drops and currents across the individual circuits as shown in the diagram below.

Example 2: Light bulbs of fixed resistance 3.0  and 6.0 , a 9.0 V battery, and a switch S are connected as shown in
the schematic diagram below. The switch S is initially closed.
a. Calculate the current in bulb A.
b. Which light bulb is brightest? Justify your answer.
c. Switch S is now opened. By checking the appropriate spaces below, indicate whether the brightness of
each light bulb increases, decreases, or remains the same. Explain your reasoning for each light bulb.
i.
Bulb A: The brightness
Explanation:
increases
decreases
remains the same
ii.
Bulb B: The brightness
Explanation:
increases
decreases
remains the same
iii. Bulb C: The brightness
Explanation:
increases
decreases
remains the same
Example 3: Three resistors are arranged in a circuit as shown above. The battery has an unknown but constant emf and
negligible internal resistance.
a. Determine the equivalent resistance of the three resistors. The current I in resistor R3 is 0.40 ampere.
b.
Determine the emf (Voltage) of the battery.
c.
Determine the potential difference across resistor R1.
d.
Determine the power dissipated in resistor R1.
e.
Determine the amount of charge that passes through resistor R2 in one minute.
Kirchhoff’s rules and complex DC circuits
 Electrical circuits that contain many paths or multiple sources of emf are analyzed using Kirchhoff’s rules (laws).
Kirchhoff’s rules are an application of two fundamental laws: the junction rule is based on the law of conservation of
charge and the loop rule is based on the law of conservation of energy. Note that Kirchhoff’s rules are not needed to
solve most problems on the physics B test.
1. The junction rule states that the sum of the currents entering any junction must equal the sum of the currents
leaving that junction.
 To use the junction rule, assign symbols and directions to all currents entering and leaving a junction and set
them equate them. Don’t worry about direction because if you happen to guess the wrong direction, the end
result will be negative, but the magnitude will be correct.
2. The loop rule states that the sum of the potential drops (voltages) across all elements around any closed circuit
loop must be zero. This is because any charge that moves around a closed loop in a circuit must gain as much
energy as it loses (change in potential energy is zero).
 To use the loop rule
1. Choose a direction (clockwise or counterclockwise) to traverse a loop.
2. If a resistor is traversed in the direction of the current, then V=-IR; if opposite to the current, then V=IR.
3. If emf is traversed in direction of emf (from – to +), then V=+; if opposite, then V=-.
12
Example 5: The circuit shown above is constructed with two batteries and three resistors. The connecting wires may be
considered to have negligible resistance. The current I is 2 amperes.
a. Calculate the resistance R.
b.
Calculate the current in the
i. 6-ohm resistor
i. 12-ohm resistor
c.
The potential at point X is 0 volts. Calculate the electric potential at points B. C, and D in the circuit.
d.
Calculate the power supplied by the 20-volt battery.
Capacitors in parallel
 When capacitors are in parallel, the potential drop across each capacitor is the same as the battery and the battery
moves a charge that is the sum of the charges moved for each of the capacitors.
qbattery=q1+q2
Vbattery=V1=V2
Ceq=C1+C2
Capacitors in series
 When capacitors are in series, charge on each capacitor in is the same and the battery only moves an amount of
charge equal to the charge on one of the capacitors because the charge passes by induction from one capacitor to the
next one. Also, the terminal voltage of the battery is equal to the sum of the individual potential differences across
each capacitor.
qbattery=q1=q2
Vbattery=V1+V2
1
1
1


Ceq C1 C2

Remember from Chapter 19 that to calculate energy stored by a capacitor use
Uc= ½CV2 = ½QV
and to find total energy stored by a combination of capacitors add the individual values for each capacitor, or simply
do one calculation using the equivalent capacitance.
RC circuits (circuits that contain a resistor(s) and a capacitor(s))
 Current in the circuit and charge stored by the capacitor vary exponentially with time. After the switch is closed,
graph below left, the battery begins to charge the plates of the capacitor and continues to charge the capacitor to the
maximum. Current is initially at a maximum but decreases as the charge on the plates increases. Once charged, the
current is zero. The reverse occurs when the capacitor is discharged, graph below right.
Example 6: A circuit contains two resistors (10  and 20 ) and two capacitors (12 F and 6 F) connected to a 6 V
battery, as shown in the diagram above. The circuit has been connected for a long time.
a. Calculate the total capacitance of the circuit.
b.
Calculate the current in the 10  resistor.
c.
Calculate the potential difference between points A and B.
d.
Calculate the charge stored on one plate of the 6 F capacitor.
e. The wire is cut at point P . Will the potential difference between points A and B increase, decrease, or
remain the same?
_____increase _____decrease
_____remain the same
Justify your answer.
AC circuits
 To analyze AC circuits, you use the same basic techniques that you used to analyze DC circuits, but there is an
important distinction to note. The direction of current across a resistor has no effect on the behavior of it (electric
energy will be converted to heat energy either way current flows), but the heating effect produced by an alternating
current with a maximum value of Io is not the same as that produced by a direct current. This is because the magnitude
of the alternating current is continually changing, as shown in the graphs below. To find the average power dissipated
you must use the rms (root mean square) value for the current and voltage.
Safety devices
 Fuses and circuit breakers are safety devices that open a circuit to prevent circuit overloads that can occur when too
many appliances are turned on at the same time or a short circuit occurs. A short circuit occurs when a path is formed
that has a very low resistance; large current results which may damage appliances or cause a fire. A fuse is a short
piece of metal that melts from the heating effect of the current if the current is too large. A circuit breaker is an
automatic switch that uses a bimetallic strip to open the switch if the current exceeds some set value. A ground-fault
interrupter (often found in electrical outlets in bathrooms or near sinks) detects small differences in current and opens
the circuit preventing shock if tripped.