Resistance and DC circuits Lab manual
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Resistance and DC circuits
Lab manual
Introduction
In this online lab experiment, you will learn about the resistance of a conductor and how resistors are used in direct current (DC) circuits in conjunction with other circuit elements to regulate electrical properties such as the current, voltage, and power.
Parts 1 and 2 look at how the resistance of a conductor will vary with both temperature and size/shape. Part 3 involves the analysis of simple circuit networks made up of resistors and batteries to understand how resistors can be used to limit current, divide voltage, or dissipate energy in an electrical circuit.
Theory
Electric current
A closed loop containing a conducting path for electric charges to flow is called an electric circuit. The electric current represents the movement of charges though a given cross-sectional area as a function oftime and is measured in amperes (A), where 1 A is equivalent to moving one coulomb (C) of charge past a region per second. In electric circuits, the charge carriers that move are electrons and they flow through the circuit due to a potential difference, or voltage, that is set up by a source. The voltage appears as a uniform electric field in the wire which applies an electric force on the electrons to push them along the circuit’s path.
Resistivity and resistance
In a conducting material, the current density represents the amount of current per unit area and if this value is proportional to the electric field that generates the current, that material is considered ohmic. The proportionality constant that relates the electric field to the current density is known as the resistivity, , and is a property that quantifies how strongly a material can resist electric current flow. Most metal and metal alloys have very low resistivity values and are able to conduct electricity.
Measuring the current density and electric field in a conductor is complicated so instead we measure related properties for a conductor such as the current (related to current density) and the voltage across its length (related to the electric field). For an ohmic conductor, the ratio of the voltage to the current within it is known as its resistance, , which is measured in Ohms (Ω):
=
This can be re-arranged to the form = and is often called Ohm’s law for electric circuits.
Resistance of a wire as afunction of size and temperature
For a cylindrical conductor, such as a length of wire, its resistance is a function of both its length, , and its cross-sectional area, , and is given by the expression:
=
where is the resistivity of the conducting material measured in units of Ω ∙ m.
The resistivity of a conducting material depends on its temperature which, for a solid, is a measure of its thermal energy. When a conductor has high thermal energy, ions in the material will vibrate with greater amplitude which increases the likelihood of them colliding with drifting electrons thereby reducing the flow of current. Over a moderate temperature range (up to about 100 °C), the resistivity of a conductor is proportional to the change in temperature and therefore so is the resistance of a conductor. We can model this relationship using the equation:
() = 0 [1 + ( − 0)]
where 0 is the resistance of the conductor at temperature 0 = 20° and is a constant factor known as the temperature coefficient of resistivity measured in units of (°C)−1 .
DC circuits
In direct-current (DC) circuits, the direction ofthe current does not change with time. Some common examples are flashlights or the wiring in a car. Any combination ofvoltage sources and resistors will make up a DC circuit and the electrical properties for circuit elements such as current, voltage, and power are independent oftime.
Combinations of resistors
Fig. 1 - Three resistors connected in series
When resistors in a circuit are connected in series, as shown in Figure 1, there is only one current path therefore the current, , is the same in all of them. However, the voltage across each resistor may not be the same but the voltage across the entire combination, , will be the sum of the three individual voltages. By applying Ohm’s law, we can write each voltage in terms ofthe current and resistance:
= + + = (1 + 2 + 3)
The ratio / is equal to an equivalent resistance of a single resistor which could replace the combination of 1, 2, and 3 in series. Therefore, our rule for adding resistances for multiple resistors series is the sum of all resistances:
, = 1 + 2 + 3 + ⋯ |
Fig. 2 - Three resistors connected in parallel
Ifwe have resistors connected in parallel, as shown in Figure 2, the current through each one may not be the same. However, the voltage between the terminals of each resistor must be the same and is equal to . In general, the current is different through each resistor therefore the total current must be equal to the sum ofthe three currents in the resistors:
= 1 + 2 + 3 = ( + + )
We find that the ratio / is the reciprocal of the equivalent resistance for a single resistor that can replace the three resistors in parallel and our rule for adding resistors in parallel is thus:
1 1 1 1
|
For many simple circuits, it is possible to simplify the network of resistors that may be in different combinations of series or parallel connections into a single equivalent resistor and it is always advised to do so.
Kirchhoff’s rulesfor circuit analysis
For any practical resistor network that may not be reduced, we use Kirchhoff’s loop and junction rules to analyse our circuit. Consider the circuit shown in Figure 3 that contains two power supplies with voltages 1 and 2 and resistor . Points and are called junctions where three (or more) conducting wires meet.
Fig. 3 - Electrical circuit that cannot be reduced
The junction rule states that the sum of all currents into a junction is equal to zero: ∑ = 0. This rule is based on conservation of electrical charge which means that charge cannot accumulate at a junction so the total charge entering a junction per unit time must equal the charge leaving per unit time. The sign convention used for the junction rule is that current entering the junction is considered positive and current leaving it is negative. Assigning a direction to the current in a wire is arbitrary and may lead to a solution with a negative value and this is discussed further below.
If we consider the currents 1, 2, and 3 with directions shown in Figure 3, we can write:
1 + 2 − 3 = 0 → 1 + 2 = 3
A loop in a circuit is any closed conducting path. The loop rule states that the sum ofthe voltages around any loop is equal to zero: ∑ = 0. As we travel around a loop while measuring voltages across successive circuit elements, the algebraic sum ofthe voltages must be zero or else we would not be able to define a value for the potential at any point.
There is a specific sign convention for using the loop rule which must be followed explicitly. Figure 4 shows the three possible loops in our previous circuit. It is important to define the direction of travel for our loops.
Fig. 4 - Three loops in a circuit
Loop 1 is the exterior loop that goes through the supply 1 and resistor and travels in a counterclockwise (CCW) direction. Loop 2 is the smaller loop on the left side and goes through both supplies and traveling CCW while loop 3 is the smaller loop on the right that goes through supply 2 and the resistor, also traveling CCW.
1) When traveling through a power supply from negative to positive, take a positive voltage +
2) When traveling from positive to negative, take a negative voltage − .
3) When traveling through a resistor against the direction of current, take a positive + = + .
4) When traveling through a resistor with the direction of current, take a negative − = −
We can write now write the loop rule equations for our three loops using the correct signs:
Loop 1: Loop 2: Loop 3:
−1 + 3 = 0
− 1 + 2 = 0
− 2 + 3 = 0
When using Kirchhoff’s rules to analyse circuits, it is often the case where the direction of current is not given and it is up to you to assign a direction. It is possible that when solving a problem, you calculate a negative value for a current which means that the actual direction of that current is opposite the one you assigned (or was suggested). By simultaneously solving the equations from the junction and loop rules, you should be able to find all required unknown values in your circuit analysis problems.
Energy and power
Electric circuits are a means for transferring energy from one place to another. As electrons move in a circuit, electric potential energy is transferred from a voltage source to a device in which that energy is stored or converted to another form. For example, a resistor transforms electric energy into heat.
The rate at which energy (measured in J) is transferred to or from a circuit element is known as the power and is measured in watts (where 1 W is equal to 1 J/s). A voltage source delivers energy to the circuit at a rate given by the following equation:
=
where is the current leaving the source. Energy can also be delivered to a voltage source (using the same equation) from a larger source such as when a car’s alternator charges its battery.
For a resistor, the power delivered can be calculated using several equations depending on what quantities are known:
= = 2 =
The energy is said to be dissipated in the resistor and leads to an increase in temperature for which there are many applications such as resistive heating devices like a toaster.
Suggested reading
Suggested reading |
|
Serway, R. A., Jewett, J. W., Physics for Scientists and Engineers with Modern Physics, 9th edition. Brooks/Cole (2013). |
Chapters 27 and 28 |
Young, H. D., Freedman, R. A., University Physics with Modern Physics, 14th edition. Addison-Wesley (2014). |
Chapters 25 and 26 |
2022-02-26