Physics 1102 Laboratory Manual

From

PHYSICS 1102 LABORATORY MANUAL

and

PHYSICS 2126 LABORATORY MANUAL

Olan E. Kruse

Revised, 2006, by Paul H. Cox

Version 2.1

Copyright © 2006, Olan E. Kruse and Paul H. Cox

Circuit Problem-Solving: 1. Definitions

by Paul H. Cox

A large part of this laboratory course is the study of electrical circuits. We will deal only with DC (Direct Current) circuits, where the direction, at least, of the electrical flows does not change unless the circuit is changed. In this study we will observe currents and potentials in circuits; we begin by discussing what these are.

A. Circuits.

Electric charges exert forces on each other; other forces can also act on charged objects. Hence charges will generally be in motion. This motion may be haphazard, or it may be along a prepared path which allows the charges to return to their starting point, following a complete, or “closed,” circuit. If the path is broken, as by opening a switch, it becomes an “open circuit”; if an alternate, easier, path becomes available, that is a “short circuit” (even if it is geometrically longer). Short circuits are dangerous, first, because they are normally unintended, and either the current through the short, or the absence of the intended current, may have side effects; and second, because, by being an electrically easier path, they lead to overloads. (A short-circuit path through you can be especially dangerous.)

Circuits are usually discussed using circuit diagrams, in which standard symbols are used for various circuit elements. Connecting wires are shown as simple lines, normally drawn only parallel to the page edges. The actual length of a connecting wire has no necessary relation to the length in the diagram; only relevant is that the wires connect the appropriate points. If two wires meet and are joined, a small dot (symbolizing a blob of solder) may be used to emphasize that; if they are not joined but must be shown crossing, then a small arch is used in one of the wires.

Batteries, or similar sources of the energy input to a circuit, are indicated by two unequal short parallel lines; the longer of the lines indicates the positive or higher-potential terminal.

Resistors are indicated by zigzag lines; capacitors by two equal short parallel lines; inductors by a looping line (projected view of a helix). A variable component may be indicated by an arrow crossing the symbol at an angle.

Meters are often indicated by a circle enclosing a letter: I for a current meter or ammeter, V for a potential meter or voltmeter. G indicates a galvanometer, a current meter used for small currents and (unlike typical ammeters) able to display negative as well as positive currents.

B. Current.

Electric current is the time rate of flow of electric charge. This can be due to motion of either positive or negative charge; however, if positive- and negative-charge objects have the same velocity, the currents will be opposite.

In most everyday situations, the moving charges are electrons. At a microscopic level, one would therefore start with electron current, describing the actual motion of objects; however, historically, the positive and negative labels were assigned long before a microscopic view was possible. Electric current is normally discussed as if only positive charge moved; thus, the electric current is opposite the electron current. (I have been told that some presentations use “electric current” to mean electron current; formulas from such sources will have extra minus signs when current is involved.)

Operational note: A current meter normally measures the current through the meter, so an ammeter must be connected so that the relevant current flows through it; that is, in series with the component or group of components.

C. Potential.

Under most circumstances, there are frictional effects in electrical circuits. Hence maintaining an electric current will require an input of energy to balance the frictional losses. Since electrostatic forces are conservative, this input will generally not be electrostatic; it may involve chemical, magnetic, or mechanical forces. However, most such sources of energy share the characteristic that they transfer a fixed amount of energy per unit of charge that passes through them.

Electric potential, which is sometimes defined as electric potential energy per unit charge, is thus the second quantity which is important to circuit problems. However, it must be observed that potential energies are defined only in relation to an amount of work done, and work done gives only change in energy, not total energy. Hence potential (colloquially called voltage, from its unit, 1 volt = 1 joule/coulomb) must always be expressed for one point with respect to another point. Also, the voltage associated with a particular circuit element is the “voltage across” the element; that is, the potential at one terminal minus the potential at the other. Referring to "voltage of" indicates confused thinking. Only when a general reference has been designated is it legitimate to give a value for potential at a point.

In many real-world problems, a reference level for potential is provided by the Earth, which is effectively a conductor so large that any single circuit can make only negligible changes in its average energy level. A connection, either direct or indirect, to the Earth (as a large conductor) is referred to as a “ground.” When such a connection is present, it is a reference point for voltage, with V = 0.

Operational note: Since only potential difference is defined, a voltmeter must be connected to two points; that is, across a component (in parallel with it) or group of components. The potential difference between the two points will be what is measured.

D. Resistance.

In most materials when electric currents flow, only a small fraction of the electric charges in the material are actually moving. In ordinary conductors, which are metals, typically only one or two electrons per atom are free enough to move about in the material; the rest of the electrons are bound to their nuclei, which in turn are held in place by the inter-atomic forces which give the metal its structure.

When an electric field is present in the material, so that there is a potential difference across it, this field gives an electric force on all the charges, but not all the charges move because this is not the only force. Even for those charges which do move, the motion is not motion with sustained constant acceleration, because the charges interact with each other much like particles in a gas. The acceleration remains constant only during the times between collisions, while the frequent collisions randomly redistribute velocities. The result is usually that a small drift velocity is added to the large random velocities which undisturbed charges exhibit; this drift velocity is proportional to the electric field, and is given roughly by the electric field’s acceleration multiplied by the time between collisions.

From the drift velocity, v = kF = k(-e)E, where k is a proportionality constant which is a property of the material, we can determine the current, using the number density, n, of the charges (number of electrons (-e) per volume) and the volume AΔℓ occupied by the charges which will cross a given area in time Δt:

We may note at this point that I/A defines current density J, and this equation therefore asserts that current density is proportional to electric field. The proportionality constant, here expressed in terms of elementary charge e and two material-dependent properties, is called conductivity, σ; we have J = σE, or I = σEA.

Now the electric field is related to the potential difference V across the object by E = V/ℓ (assuming uniform field), so or

ρ = 1/σ is called resistivity, and depends on the material; the combination ρℓ/A = ℓ/(σA) is the resistance of the object, R, so this equation, known as Ohm’s Law, is V = IR. This relates the voltage across an element whose resistance is R, to the current through that element. Note that a larger value of R corresponds to requiring a larger voltage to obtain a given current, or obtaining a smaller current for a given voltage, hence the name ‘resistance’: this value describes how the element resists charge flows.

Circuit Problem-Solving: 2. Combinations

by Paul H. Cox

The simplest combinations of circuit elements are those which are known as series combinations and parallel combinations.

Two or more elements are in series with each other, if one terminal of each is connected only to one terminal of the next, so that any circuit path must follow through all of them if it includes any of them. This leads directly to the relationships that hold for all series combinations; the charge or current through each component is the same, while the potential difference across the combination is the (algebraic) sum of the potential differences across the elements. Thus, in a flashlight using two standard 1.5-V batteries, these are connected in series, so that they are equivalent to a single 3-V battery, if they are properly placed, in what can be called series-aiding alignment. If one is reversed, they are in series-opposing alignment, and the equivalent potential difference available is

(1.5 V) + (-1.5 V) = 0, which is not going to be useful.

Two or more elements are in parallel with each other, if both terminals of each are connected directly to corresponding terminals of all the others (connection via an additional component is not good enough). The resulting relationships are, for all parallel combinations: the potential differences across each element are the same, while the charge or current through the combination is the algebraic sum of that for each element. Thus, to jump-start a car, one connects a second battery in parallel with the car’s own; the circuits do not see too high a voltage, as they would with two batteries in series, but the second battery (presumably) has the energy to deliver the needed current.

A. Series formula for resistors

Resistors are reversible, but current can flow in either direction through them so signs are still relevant. Since they obey Ohm’s Law, V = IR, a rule for resistors in series is straightforward. Consider Figure 1.

Since VA - VC = (VA - VB) + (VB - VC), while IA→C = IA→B = IB→C = I, provided nothing else is connected at B, and

VA - VB = IA→BR1 = IR1, and VB - VC = IB→CR2 = IR2, we obtain

VA - VC = (VA - VB) + (VB - VC) = IR1 + IR2 = I (R1 + R2), from which Rseries = R1 + R2.

Note: Since capacitors and inductors do not obey Ohm’s Law, they do not necessarily obey the same rule.

B. Parallel formula for resistors

Again, the parallel rule for resistors is straightforward. Consider Figure 2.

Since VA - VB = I1R1 and VA - VB = I2R2, while IA→B = I1 + I2, we have

IA→B = I1 + I2 = (VA - VB)/R1 + (VA - VB)/R2 = (VA - VB)(1/R1 + 1/R2)

or VA - VB = IA→B/(1/R1 + 1/R2), so Rparallel = (R1-1 + R2-1)-1.

Again, since capacitors and inductors do not obey Ohm’s Law, they need not obey the same rule.

C. Other combinations.

When a circuit being analyzed contains similar elements in series, or in parallel, an equivalent, simpler, circuit can be obtained by replacing the series or parallel group with the equivalent single element. This circuit in turn may now have similar elements in parallel or series, and the process can continue until the circuit is simplified to something solvable. Unless the problem is merely to determine the equivalent element, one returns to the original circuit by reversing the replacements while using the information gained from the simpler circuits to fill in information for the previous version.

An example developed by Professor Kruse will illustrate the technique. Consider the circuit of Figure 1, on the next page. You are to find the overall equivalent resistance, the current I through the battery, the voltage across each resistor, and the current through each resistor. Since the resistors are all different, it will be sufficient identification if we label the voltages and currents for a single resistor by using a subscript corresponding to the resistance, and values for combinations by using multiple subscripts.

Choosing to begin at the lower right, there are 7 Ω and 1 Ω in series, which give an equivalent R71 of 8 Ω. These 8 Ω are in parallel with another 8 Ω. The equivalent resistance for this combination, R718, is then obtained from

1/R718 = 1/(8 Ω) + 1/(8 Ω), or R718 = 4 Ω.

At the top are three resistors in parallel. Again, their equivalent, R623, is obtained from

1/R623 = 1/(6 Ω) + 1/(2 Ω) + 1/(3 Ω), or R623 = 1 Ω.

Replacing the two parallel branches by their equivalent resistances gives the circuit of Figure 2.

Now we see that the equivalent resistance for the entire circuit is R = 9 Ω + 1 Ω + 4 Ω + 4 Ω = 18 Ω.

So we continue: I = (36 V)/(18 Ω) = 2 A.

This is also I9, I623, I4, and I871, so I9 = I632 = I4 = I871 = 2 A. Then V9 = (2 A)(9 Ω) = 18 V.

Now V623 = I623 R623, or V623 = (2 A)(1 Ω) = 2 V. This is also V6, V2, and V3, so V6 = V2 = V3 = 2 V.

Similarly V4 = (2 A)(4 Ω) = 8 V, and V8 = V71 = V871 = (2 A)(4 Ω) = 8 V.

From these voltages and the resistances we obtain

I6 = V6/(6 Ω) = (2 V)/(6 Ω) = 1/3 A, I2 = V2/(2 Ω) = (2 V)/(2 Ω) = 1 A,

I3 = V3/(3 Ω) = (2 V)/(3 Ω) = 2/3 A, I8 = V8/(8 Ω) = (8 V)/(8 Ω) = 1 A,

I71 = V71/(8 Ω) = (8 V)/(8 Ω) = 1 A.

Note that I6 + I2 + I3 = (1/3 A) + (1 A) + (2/3 A) = 2 A = I, and that I8 + I71 = 1 A + 1 A = 2 A = I. If these relationships did not check, we would know we had a mistake.