Summary reading for the Electric Circuits Model

Electric Circuit Model
Summary Reading
Circuit representations:
Definition of circuit
A circuit is a continuous conducting loop including something to push charge through the
circuit (like a battery or a generator). A closed circuit is one in which charges are flowing. An
open circuit is one that has an intentional break in the loop, usually with a switch. A short
circuit has a low resistance path that may bypass elements in a circuit and allow huge amounts
of charge to flow in the circuit (dangerous).
The circuit on the left is a closed circuit; charge flows around the complete loop and the bulb
lights. The middle circuit is an open circuit since the switch has broken the path so charge cannot
flow and light the bulb. The circuit on the right is a short circuit. Charge will not flow through
the bulb since the direct conducting path between the battery terminals has much less resistance.
Since the resistance is so low, huge amounts of charge will flow, making the wires hot and
quickly dissipating the energy stored in the battery.
Physical layout vs. schematic diagram
A physical layout diagram shows the actual arrangement of elements in a circuit as if a
picture had been taken. A schematic diagram uses symbols for circuit elements and the lines
connecting elements show direct electrical connections. These may or may not be wires. For
example, in the left schematic diagram above, we don’t know if there is a wire attaching the bulb
to the positive terminal of the battery or if the bulb is actually touching the battery.
Charge flow model: (Current)
Charges move through conductors due to electric attraction and repulsion.
A battery is a device in which a chemical reaction (oxidation-reduction) separates charge.
When the battery is connected to a circuit, the electric field in the wires connecting the positive
and negative terminals of the battery causes charges to flow in the wires. The conventional
definition for the direction of charge flow is positive charge flowing away from the positive
terminal toward the negative terminal. (This definition was proposed before the electron was
discovered, and since the description still works, it is still used today.)

Charges do not get “used up” in a circuit
Charges are not created or destroyed (conservation of charge) so it is important to picture them
as little particle-like objects that flow in and out of circuit devices. The charges themselves do
not make bulbs light. We will see later that it is the energy that each charge carries that lights the
bulb.
Adding obstacles inhibits the flow of charge (series circuits)
A series circuit consists of a single conducting loop. In a series circuit the charge flows
equally through each circuit element, and equally through each wire.
In a series circuit, when more obstacles are added it is harder for charges to flow, and as a
result the flow rate decreases throughout the circuit. Each obstacle (i.e. a light bulb) resists the
flow of charge and has a measurable property called resistance. Charge flow is inversely
proportional to the resistance of the circuit.
In the three series circuits above, we can see the inverse relation between charge flow and
resistance. The left circuit is our reference circuit. The middle circuit has twice the resistance of
the left circuit, (assuming the bulbs have identical resistance) therefore the middle circuit has
only half the charge flow of the left circuit. The circuit on the right has three times the resistance
of the left circuit and has one-third the charge flow from the battery and throughout the circuit.
Adding pathways allows more charge to flow (parallel circuits)
A parallel circuit consists of more than one conducting loop. This means that the flow of
charge is not equal everywhere in the circuit, since there will be junctions where the charge flow
splits to take different paths around the circuit.
Note that the parallel circuits above are identical since they both show the same electrical
connections. Both show that the charge flowing from the battery splits, some travels through
each bulb, the charges rejoin and flow back into the opposite terminal of the battery.
Adding additional pathways (branches or loops) to the circuit makes it easier for charge to
flow since there are more ways for charge to flow around the circuit. Adding branches increases
the charge flow from the battery and decreases the overall resistance of the circuit. In parallel
circuits, each branch is also independent of one another. Any branch can be turned on or off
without affecting the operation of the other branches. (This is why your house is wired in
parallel.)

In all three parallel circuits above, the flow of charge through each branch is identical. The
flow of charge from the battery is not identical. If the flow in the left circuit is X, then the flow
from (and into) the battery in the middle circuit is 2X. In the right circuit the flow would be 3X,
assuming that the bulbs have identical resistance.
Quantitative combinations of resistors
The squiggly line in the schematics below represents a generic resistor (like a light bulb or
heater). The unit for electrical resistance is the Ohm, represented by the Greek letter omega, Ω .
In series circuits, the total resistance is additive: R total = R 1 + R 2 + R 3 + . . .
Each additional resistor in the series makes charge flow more difficult. In the series circuit
shown above, the total resistance equals the sum of the two resistances, or 10 Ω.
In parallel circuits, the total conductivity is additive: (Conductivity is the reciprocal of
resistivity.) 1/R total = 1/R 1 + 1/R 2 + 1/R 3 . . .
Each additional pathway decreases the resistance and makes charge flow easier. In the parallel
circuit shown above, the total resistance equals the reciprocal of the sum of the reciprocals of the
branch resistances, or 2.5 Ω.
Current (in Amperes) is defined as the number of charges (in Coulombs) flowing past a
point in the circuit each second
Electrical current is the fancy name for "rate of flow of charge". One Coulomb of charges
passing a point in a circuit each second is one Ampere of current (one Coulomb is 6.25 x 10 18
charges). The common abbreviation for current in equations is the capital letter "I" to minimize
confusion with Coulomb, charge, and capacitance. The unit for current, Ampere, is abbreviated
"A".
Junction rule: (conservation of charge)
In parallel circuits there are junctions where the charge flow splits or rejoins. The number of
charges flowing into a junction must equal the number flowing out of the junction. In other
words, the sum of the currents entering the junction must equal the sum of the currents leaving
the junction.
The resistance of each branch determines how the current will split when it gets to a junction.
More current will flow along the branch with less resistance.

In the circuit above, the charges will split unevenly at the junction. Twice as many charges
flow through the single bulb branch as flow though the two-bulb branch. For example, 3
Amperes of current might flow out of the battery. The current would split at the junction so that
2 A travels through the single bulb and 1 A would flow through the two-bulb branch. The current
would rejoin at the other junction and 3 A would flow back into the battery.
Electric Field Energy model: (Potential)
Energy stored in the electric field is transferred to circuit resistors as charges move
through the field.
In order for charges to move through a circuit, energy is required. The energy ultimately
comes from the chemical reactions in the battery that separate charges. The separated charges
store energy in the electric field in the circuit. As charges move through the field, the field loses
energy, which is continually replaced by the battery until the chemical reactions in the battery
reach completion. Energy that the field loses is transferred to electrical devices in the circuit
(bulbs, resistors, motors, etc . . .) The field loses very little energy for each charge that passes
through connecting wires because the resistance in connecting wires is so small. The field loses
much more energy for each charge that passes through resistive circuit elements.
Electric Potential tells the “energy density” of the electric field in Joules per Coulomb at a
point in the circuit.
For example, the negative terminal of a battery has an electric potential of zero: positive
charges at the negative terminal of the battery will not enter the circuit. Positive charges at the
positive terminal of the battery are at a high potential, that is, the field can lose lots of energy as
the positive charges leave the positive battery terminal and pass through the circuit. The larger
the potential difference between the terminals of the battery, the greater the amount of energy the
field will lose for each Coulomb of charge that passes through the circuit.
Potential difference, or Voltage, is the number of Joules of electric field energy required to
move one Coulomb of charge between two points in a circuit
Measuring a potential difference between two points in a circuit implies the existence of an
electric field. As the field moves charges, the field loses energy. For example, a 1.5 Volt battery
supplies 1.5 Joules of energy to the electric field for each Coulomb of charge that travels through
the circuit. In a series circuit with two identical bulbs connected to a 1.5 volt battery, the
potential difference across each lightbulb is 0.75 V. In other words, as charges pass through the
fields in the bulbs, 0.75 Joules of energy will be transferred to each lightbulb from the field for
every Coulomb of charge that passes through them.

Increasing the potential difference increases the flow of charge: (current is proportional to
voltage)
When batteries are connected in series, there is a larger potential difference than when a single
battery is used. The resulting electric field through the circuit is stronger, causing more charges
to move.
The circuit on the left will light the bulb. In the middle circuit, there are two batteries. The
voltage is doubled and the current will double. The circuit on the left has triple the voltage of the
circuit on the left, and three times as many charges will flow through the bulb in the right circuit
each second. The current is directly proportional to the voltage.
Loop rule: the battery voltage minus the voltage drops around a circuit loop equals zero
(conservation of energy)
The series circuit on the left has a 12 Volt battery. The field will lose a total of 12 Joules of
energy for each Coulomb of charge that passes through the circuit, transferring 6 Joules of
energy to the first resistor and 6 Joules of energy to the second resistor as one Coulomb of charge
passes through them. So the voltage drop across each resistor is 6 Volts. The loop rule is satisfied
by 12 V = 6 V + 6 V.
In the parallel circuit on the right, both resistors have direct electrical connections to the
battery, and both resistors are in their own loop. The voltage of the battery has to be equal to the
voltage drop across each resistor, regardless of its resistance, since the field will lose 12 Joules of
energy for every Coulomb of charge passing through a branch. Here we use the loop rule twice:
12V across the battery = 12 V across the left resistor and 12V across the battery = 12 V across
the right resistor.
Now we will look at the loop rule when the resistors do not have equal values. In the series
circuit, the field will lose twice as much energy as charges pass through the 10 Ohm resistor as is
lost as charges pass through the 5 Ohm resistor. Consequently, for every charge that passes
through the circuit, one third of the field’s energy will be transferred to the 5 Ohm resistor and
two-thirds to the 10 Ohm resistor. The loop rule is satisfied by 12 V across the battery = 4 V
drop across the 5 Ohm resistor and an 8 Ohm drop across the 10 Ohm resistor.
In the parallel circuit, the voltage drop across each resistor is still equal to the battery voltage.
In this case, the current will not be the same in each branch, but the amount of energy the field
will lose for every Coulomb of charge passing through the branches is the same.

Ohm's law: Voltage = Current x Resistance
In the previous sections we have established that current is inversely proportional to resistance
and current is directly proportional to voltage. Combining these relations we get current =
voltage/ resistance. Solving for resistance: resistance = voltage/current. Solving for voltage,
voltage = current x resistance, or V = IR.
Power is the amount of energy transferred each second: Power (in watts) = Voltage x
Current
The actual brightness of a light bulb depends on the number of charges flowing through the
bulb each second (current), as well as the energy lost by the field per charge (voltage). The
product of current and voltage is a rate of energy transfer, power.
Example of energy transfer (power) in series and parallel circuits:
In the series circuit, the current is the same through both resistors, however the voltage drop is
not. Using Ohm’s law, we can determine that the current in the circuit is I = V/R = 12 V/15 Ω =
0.8 A. So the rate of electric field energy transferred to the 5 Ohm resistor is P = IV = 0.8 A x 4
V = 3.2 Watts. The energy transfer rate to the 10 Ohm resistor is P = IV = 0.8 A x 8 V = 6.4
Watts. Note that if the resistors were light bulbs, the 10 Ohm resistor would correspond to the
brighter bulb.
In the parallel circuit, the voltage is the same across each branch. The current flowing through
the 5 Ohm branch is V/R = 2.4 A, more than the current flowing in the 10 Ohm branch, V/R =
1.2A. So the energy transferred to the 5 Ohm resistor is P = IV = 2.4 A x 12 V = 28.8 Watts and
the energy transferred by the 10 Ohm resistor is P = IV = 1.2 A x 12 V = 14.4 Watts. Note that in
this case, the 5 Ohm resistor would correspond to the brighter bulb.