DC Circuits

Electric Circuits

Electric circuits transfer energy. Electrical energy is converted into light, heat, sound, mechanical work, etc. The byproduct of any circuit is always heat.

Electricity: The Mouse/Cheese Analogy You must be a member of this site to access it.

DC current (direct current)

a steady flow of current in one direction

AC current (alternating current)

direction of current flow changes many times a second. In the US, the frequency of change is 60 Hz. Therefore, the current changes direction 120 times per second.

emf (electromotive force)

source of energy moving electrons through the circuit. For a battery, the maximum potential difference that exists between the terminals is referred to as the emf (e) of the battery.

electric current (symbol is I; SI unit is the ampere, or A)

flow of charge, or current = charge/sec

1 A = 1 C/ sec

Consider a circuit composed of wire connecting the terminals of a battery. Within the battery, a chemical reaction occurs that transfers electrons from one terminal to another. Because of the positive and negative charges existing on the battery terminals, a potential difference (voltage) exists between them. The battery creates an electric field within and parallel to the wire, directed from the positive toward the negative terminal. This field exerts a force on the free electrons, causing them to move. This movement of charge is known as an electric current. The current is how much charge flows in a unit of time,

I = Dq /Dt

conventional current

flow of positive charge

electric power (symbol is P; SI unit is watt)

the rate of doing electrical work. The amount of energy per unit of time (power) is the amount of charge, Dq, that comes from the battery in a time Dt across a potential difference V. Or, P = (Dq / Dt) V. Since Dq / Dt is the current I, power is the product of current and voltage.

P = VI = (IR)I

P = I²R = V²/R

Watt

SI unit of power; 1 Watt = 1 Joule/sec = 1 VoltAmp

electrical energy (symbol is EE; SI unit is Joule)

EE = P t

EE = V I t
Total power The total power in a series combination of light bulbs and in a parallel combination of light bulbs is simply the sum of the individual wattages. For example, two 60 W light bulbs have a dissipate 120 W in a series combination as well as in a parallel combination.

Electric companies sell you electrical energy. Your energy consumption is computed by expressing power in kilowatts and time in hours. Energy is sold to you in units of kW-hr.

AP Multiple Choice & Free Response Questions on Electrical Energy:

They like to ask questions about heaters! They will tell you the wattage of a heater. Remember this is the unit of power (P=Vi). They will ask how much energy the heater will use in a certain amount of time (Remember - electrical energy=Vit).
As part of a free response question, they will give you the wattage of a heater and ask you how much thermal energy it dissipates in a certain amount of time.
They will ask you to calculate how much the electrical energy costs to operate a device of known current and voltage for a known amount of time.

resistance (symbol is R,; SI unit is ohm or W)

opposition to current flow

1 W = 1 V/ 1 A

Animation representing a simple circuit You must be a member of this site to access it.

Ohm’s law: for a given resistance, the potential difference is proportional to the current flow; V = I R

Ohm's Law Virtual Lab

The current that a battery pushes through a wire can be compared to the flow of water that a pump pushes through a pipe. Greater pump pressures cause greater water flow rates; greater voltages cause greater currents. In a pipe, other things determine the water flow rate besides just the pump pressure; in a circuit, other things determine the current magnitude besides just the voltage. Longer and narrower pipes cause greater resistance to the water flow; longer and smaller diameter wires cause greater resistance to the current. Resistance of a solid conductor depends upon:

nature of the material
length of the conductor
cross-sectional area of the conductor
temperature

Factors That Affect Resistance7nbsp; You must be a member of this site to access it.

Resistivity Resistivity is an inherent property of a material. Its unit is W meter. Its symbol is r. Insulators have large resistivities; conductors have small resistivities. Semiconductors, such as germanium and silicon, have intermediate resistivities. The resistivity also depends upon temperature. In metals, the resistivity increases with increasing temperature; in semiconductors, the reverse is true.

r = r_o[1 + a(T - T_o)]
where a is the temperature coefficient of resistivity, T and T_o are the temperature and initial temperature, respectively.

Resistance and resistivity The resistance, R, of a material of length, L, and cross-sectional area, A, is given by

R = r L / A

AP Multiple Choice Questions on Resistivity:

They will give a picture of two wires with different cross-sectional areas and ask you by how much one wire's cross-sectional area differs from anothers (another example of how they love ratios!).

Critical Temperature. The temperature below which the resistivity of a class of materials goes to zero. Below this temperature, such materials are called superconductors. Since these materials have zero resisitivity below the critical temperature, they offer no resistance to electric current. Once a current is established in a superconducting material below its critical temperature, it continues indefinitely with no need of an outside voltage source. Metals are superconducting at temperatures slightly above absolut zero. Some ceramics have been developed with critical temperatures as high as 133.5 K.

Terminal voltage (V_ab) When using a battery, what is really measured is the voltage delivered to the circuit, or the voltage between the two terminals. When no current is drawn from the battery, the terminal voltage is equal to the emf e. When a current is drawn, the voltage between the terminals is called the terminal voltage, the actual voltage delivered to the circuit.

Internal resistance (r)When a current is drawn from a battery, over time, the voltage delivered to the circuit by the battery drops below its listed emf. A battery itself has some internal resistance due to the chemical reactions moving charges from one terminal to another.

V_ab = e - Ir

AP Free Response Questions about Internal Resistance:

You may be given a battery in a circuit that contains an internal resistance. They may ask you to calculate the voltage delivered to the circuit.
They will duplicate an experimental situation for a circuit consisting of a variable resistor and a battery with internal resistance. You are given data terminal voltage vs current data as the resistance is varied. From this information, you will be expected to calculate the emf and the internal resistance of the battery.

Experimentally determine the relationship between current, voltage, and resistance

galvanometer: measures small currents
ammeter: measures current

ammeters and galvanometers are connected in a circuit, positive to positive and negative to negative; they have very low resistance so that the current flow through them is a maximum; they are connected in series

voltmeter: measures potential difference between two points in a circuit

voltmeters are connected in a circuit, positive to negative and negative to positive. They have very high resistance so that the current flow through them is a minimum; they are connected in parallel

Circuit symbols

circuit symbols

AP Free Response Questions on Meters:

You will be asked to draw the correct placement for an ammeter and a voltmeter in a circuit.
You will be given meters and other equipment. You must design an experiment that will take the appropriate current and voltage readings to calculate resistance. Usually, these are related to ultimately determining a temperature.

Series Circuits

Series circuit: resistors are connected so that there is only one path for the current to flow through the resistors; the current is the same at all points - current is constant in series
effective resistance (equivalent resistance): the resistance of a single resistor that could replace all the resistors in a series circuit
Kirchoff’s second law: in any closed circuit loop, the potential energy drops of the individual electrical devices equals the total energy of the circuit; this is a statement of the law of conservation of energy

As resistors are added in series, total resistance increases and total current decreases.

Steps in simplifying series circuits:

find the effective resistance of the circuit
find the total current using Ohm’s law
apply Ohm’s law to each individual resistor to determine the individual resistor’s voltage drop

Series Circuit applet

Series Circuit Animation You must be a member of this site to access it.

Parallel Circuits

Parallel circuits: resistors in parallel have the same voltage drop across them. The sum of the currents in each parallel branch equals the total current entering the parallel branch of resistors. Voltage is constant in parallel.
Kirchoff’s first law: the sum of the currents entering a point is equal to that of the currents leaving the point.

In a parallel circuit, all the resistors in parallel can be replaced with one equivalent resistance that carries the same current and has the same voltage drop across it.

As resistors are added in parallel, total resistance decreases and total current increases.

Power companies maintain a house voltage of 120 V. House appliances are connected in parallel. The more appliances on a circuit, the lower the total resistance, the greater the current. Fuses protect against circuit overloading.

Steps in simplifying parallel circuits:

find the equivalent resistance
use Ohm’s law to find total current
apply Ohm’s law to each resistor to find the current in that branch

A href="http://physics.uwstout.edu/physapplets/javapm/java/resist4/index.html">Parallel Circuits Virtual Lab

Parallel Circuit Animation You must be a member of this site to access it.

Complex Circuits

Complex circuits are a combination of resistors in parallel and in series.

steps in simplifying complex circuits:

determine the equivalent resistance of each set of resistors in parallel
determine the total resistance of the circuit
determine total current
calculate voltage drops across all series resistors
calculate currents in each parallel branch

AP Multiple Choice Questions

This is a big percentage of the AP test!
They will give you diagrams of two resistors arranged in series or in parallel. Then they will give you the same diagrams, but the resistors will be double what they were. They will ask you which combination of resistors yields the largest or the smallest equivalent resistance.
They will give you a diagram showing a complex circuit consisting of a resistor in series with two or more resistors in parallel and ask you the potential difference across one of the parallel resistors.
They will give you a diagram of two resistors in parallel and ask you what happens to the current (or certain ammeter reading) or voltage (or certain voltmeter reading) when another equivalent resistor is added in parallel.
They will give you diagrams of two or more resistors &/or a capacitor arranged in series, in parallel, or in a complex networ and ask you which circuit will store more energy or dissipate the most power (Remember, since P=Vi the circuit with the greates possible combination of voltage and current will dissipate the most power. Remember, since stored energy=1/2 qV, the circuit which stores the most energy has the greatest voltage with the greatest charge stored on its plates.).
They may ask you to calculate the equivalent resistance between two points in a circuit.
In a circuit consisting of at least one unknown resistor, you may be asked to calculate the value of the resitance that would yield a specific current at a known voltage.
You need to know what units are equivalent to. In other words, what is a Watt equivalent to? What is a Joule equivalent to?

AP Free Response Questions

They like to give you various objects to put into circuit. You will be required to draw the circuit diagram. You might be required to construct a circuit using all the objects in which the current is as large or as small as possible. Or, one in which the circuit will have a prescribed function.
One of your objects could be a variable resistor. You will be asked to determine its resistance in the circuit as you have drawn it.
Predict how the brightness of a circuit containing a light bulb will change if a variable resistor's resistance is increased or decreased. Predict how the power dissipated in another object would change if the variable resistor's resistance is increased or decreased.
You may be given a circuit diagram of a complex circuit and asked to calcuate a specific current or voltage. You may be asked to calculate equivalent resistance, circuit current, or the power dissipated in the circuit.
You may be given a circuit with serveral light bulbs and asked how the brightenss of the bulbs would changed by screwing in or unscrewing specific light bulbs. Remember, treat the light bulbs as if they were a resistor (if nothing else, assign them a resistance of say 5 ohms each!). Calculate the current through each part of the circuit for each situation described. The brightness of the bulb is directly related to the current.

Kirchoff's Laws Circuits

Some circuits cannot be analyzed using the above methods. They might contain multiple voltage sources or have arrangements of resistors too complex for analysis using the above methods. They are best using Kirchoff's point rule and loop rule.

Point (or junction) ruleThe total current directed into a point (or junction) must equal the total current directed out of the point( or junction).

Loop ruleFor a closed loop, the total of all the potential rises is the same as the total of all the potential drops.

Steps used to analyze a circuit using Kirchoff's laws:

Draw the direction of the currents in each branch. The direction is arbitrary. If you have chosen the wrong direction, your current will turn out negative in your solution. (Remember, currents must enter and leave the point (or junction). They can't all enter or all leave.)
Choose an arbitrary direction in which to traverse the loops of your circuit. (They must either be traversed all in a clockwise or all in a counterclockwise direction.)
If you traverse a resistor in the direction of the designated current flow, it is a voltage drop. The amount of the voltage drop is given by -iR.
If you traverse a resistor opposite the direction of the designated current flow, it is a voltage rise. The amount of the voltage rise is given by +iR.
If you traverse an emf in the direction of the emf (conventional current is + to -), it is a voltage rise, or +e
If you traverse an emf opposite the direction of the emf (conventional current is + to -), it is a voltage drop, or -e
To analyze you circuit, apply the point (or junction) rule. You will write an equation showing the currents that enter and leave a point. (For example, if there are three currents that share a common point, you can specify that two enter and one leaves. Or, i₁ + i₂ = i₃)
To analyze your circuit, apply the loop rule. Starting at a point, traverse each closed loop in the designated direction, adding the voltage rises and drops. This sum of rises and drops equals zero. You need to write one less loop equation than you have unknown currents. In other words, if you have three unknown currents, you need to write two loop equations.
Algebraically solve your equations for the unknown currents. (Or, you may use a graphing calculator to do so. On the TI-86 or TI-85, you may use the simultaneous equation feature. On the TI-83, you may use the matrix feature.)

AP Free Response Questions on Kirchoff Law Circuits:

You may be given a circuit diagram containing more than one voltage source. The calculations that they ask require your being able to calculate the currents through the branches of the circuit.

Electric Circuits Sample Problems

Electric Circuits Homework

AP Objectives - DC Circuits