PHY 207 Chapter 28 Direct-Current Circuits / Dr. Miner

EMF

The source of electric energy that causes charges to move in electric circuits is the emf. Historically such energy sourrces were called electromotive force, however, it is not a "force" but a potential energy per unit charge, ora voltage. Thus we strongly discourage the use of the term "electromotive force." The short form, emf, is the preferred name.

A good example of such a source of electric energy is a battery. For now we will consider only a battery with a constant source of voltage. This leads to a constant current, or a steady state current, or an equilibrium current. We also know this tern as direct-current, or DC. When a battery is placed in a circuit loop with other circuit elements, such as capacitors or resistors, a DC current flows. The circuit must be a closed loop.

Consider a flashlight. WE have a battery and a light bulb in a closed loop. The light bulb is just a resistor through which a current flows. It heats the filament (a very thin wire) of the bulb and it glows giving off light. This continues until we turn it off (break the closed loop), or until the battery becomes discharged. A similar situation exists with an automobile battery (12 volts) and the headlights. Here the current is much larger than in the flashlight, but the proceedure is the same. We refer to these light bulbs as the load resistance.

Either of the above batteries hace two terminals, two conductors to connect the battery to the external circuit. In a simple picture, one can consider the negative terminal to be a large source of electrons that can flow unter the proper circumstances. The positve terminal can similarly be considered to be a large sink for electrons. As electrons flow into and out of the battery there is some internal resistance to flow. The internal chemistry of the battery provides some internal voltage, or emf, for the battery. The actual terminal voltage of the battery will be somewhat less due to the voltage drop over the "internal resistance." The expression describing terminal voltge is:

V = emf - I r

Here V is the terminal voltage (measured between the twoterminals), emf is the true emf of the battery, I is the current being drawn from the battery, and r is the battery's internal resistance.

In order to examine the rate at which energy is drawn from the battery, we consider the potential energy per unit charge, V. If we multiply by the charge, q, we get energy. If we divide by the time, t, we get energy per unit time, or power, P. We find that:

P = V q/t = V I

For a resistor, this expression can be rewritten as:

P = (I R) I = I² R

Kirchhoff's Loop Rule

Consider a single-loop circuit, one with a single path for current. If we keep track of the voltage around the loop, and remember that voltage (or potential, or electric potential) is potential energy per unit charge, we note that the sum of the potential changes (or potential "drops") around the loop must be zero. Compare to gravitational potential energy around a looping path around a mountain. On the path one may increase and decrease elevation, but by the time one returns to the starting point the elevation is back to the initial value. It is the same with electrical potential energy, and with electrical potential energy per unit charge, or voltage.

If one goes around the loop monitoring the voltage drops (or gains), one finds that at a battery the change is V = emf - Ir. For a resistor the change is V = I R. For a capacitor the change is V = Q/C. Of course, one must pay closed attention to sign of each term. For resistors and capacitors (passive circuit elements), the side closer to the positive side of the battery will be at a higher potential. For the battery (energy source), the positive terminal is at the higher voltage. We can summarize Kirchhoff's Loop Law as:

The sum of the potential changes around a closed path is zero.

This above rule can be for a simple loop, or for any closed loop in a more complex circuit, say one that has two or three loops. Next we will explore that option.

Kirchhoff's Point Rule

Consider a circuit made of two loops side by side. One side of each is in common. Where the loops come together three wires join. This is called a junction. If charge approaches this juction along one of the wires and leaves along the other two, it is clear that the total leaving must be the same as the total entering. (The junction has no place to store charge.) One could compare this situation to traffic flow. The total number of cars entering an intersection must be equal to the total leaving the intersection. We can summarize Kirchhoff's Point Law as:

The algebraic sum of the currents that enter a junction is zero.

This rule along with the Loop Rule will enable us to analyze various electric circuits.

Using Kirchhoff's Laws

To analyze a circuit with Kirchhoff's Laws, one must write all of the indepencent Point and Loop equations possible. One finds that it a Point Law expression is written for each junction (a union of three or more wires), the final expression is not independent of the others, that is, it can be obtained by a combination of the others. In a similar vein, if one writes all of the Loop Law expressions possible (for all unique loops), one notes that the final expression is also a duplication. To solve n equations for n unknowns it is necessary that the n equations be independent. After these independent equations are established, one can then solve.

Resistors in Series

Consider again a simple loop circuit. There is only one possible path around the loop. Current travels through one circuit element and then the next. It will travel through each element exactly once in one complete loop around the circuit. We say that the elements are in series. In such a case, the current through each element is exactly the same. Compare to a traffic situation. Imagine a curcular race track with three bridges. The number of cars crossing each bridge is the same, the same "current." If one checks the voltage around this simple loop circuit, one notes that the sum of the changes is zero. If we have three resistors in series, the voltage drop across the combination is the sum of the individual voltage drops. For example:

V_total = V₁ + V₂ + V₃

V_total is the total voltage. Let us replace the three resistors with one equivalent resistor that keeps the circuit unchanged. If we rewrite the voltages as V = I R, then the expression becomes:

I R_total = I R₁ + I R₂ + I R₃

After cancelling the current I, which is the same in every term, we have:

R_total = R₁ + R₂ + R₃

This is the rule for adding resistors in series.

Resistors in Parallel

Now let us consider a circuit in which there are three resistors connected side by side, that is, here all the left hand sides and joined and all of the right hand sides are joined. We say these resistors are in parallel. Now all three have the same voltage. If a current, I_total, comes down the wire from the rest of circuit to the junction of the left hand wires on the three resistors, some of it can go through each subject to keeping their voltages the same. The conservation of current requires:

I_total = I₁ + I₂ + I₃

We now note that for each resistor we can write I = V/R. Consider a resistor R_total that is equivatent to the three resistors in parallel, and in replacement keeps the circuit unchanged. we reqrite the current expression as:

V/R_total = V/R₁ + V/R₂ + V/R₃

Since all have the same voltage V, we can cancel V and find:

1/R_total = 1/R₁ + 1/R₂ + 1/R₃

This is the rile for adding resistors in parallel. Note that the rules for adding resistors in series and parallel are just the opposite of the comparable rules for capacitors in series and parallel. Also, note that while we have examined groups of three resistors, the same rule apply to two or more that three resistors. There is just one term for each resistor.

Measuring Instruments

We can measure current, voltage and resistance with ammeters, voltmeters, and ohmmeters. These devices are often combined into one meter called a multimeter, or VOM (Volt-Ohm-Milliammeter).

These meters can be made from an analog (not digital) galvanometer. A galvanometer is a coil of wire that rotates in a magnetic field when a current passes through it. The deflection is proportional to the current. This coil of wire has a definite resistant, and a certain current for full scale deflection. Details will be given in Chapter 30 after magnetic fields have been examined.

The Ammeter. An ammeter is inserted in a circuit at some point and indicates the current passing that point. It, of course, should not change the circuit being measured, and thus should have very little resistance (negligible resistance). To make an ammeter from a galvanometer, we add a very small resistance in parallel with it. We select the value of that small resistance so as to have the current for full scale deflection of the galvanometer be the value for full scale deflection of the ammeter. Let the resistance of the galvanometer be R_g and the current for maximum deflection of the galvanometer be I_g. Let the current for maximum deflection of the ammeter be I_max and the resistance in parallel be R_A. Then, since the galvanometer and resistance R_A are in parallel, their voltages should be the same:

I_g R_g = (I_max - I_g) R_A

Thus we have an ammeter that deflects proportionally to the current with an appropriate maximum value. Note that we could make an ammeter with several different ranges if we arrange several circuit options with different parallel, or shunt resistors.

The Voltmeter. A voltmeter is connected to two points in a circuit and measures the potential difference (or, potential drop) between the two points. For example, it could measure the voltage across a resistor. It would be connected in parallel with the resistor. If it is not to change the circuit noticeably, it must have a very high resistance so that the current drawn through it is negligible relative to that drawn through the resistor across which the voltage is measured. To make an analog voltmeter from a galvanometer, we attach a verylarge resistance R_V in series with the galvanometer. This will produce a very large total resistor for the voltmeter and it will draw a very low (or, negligible) current. If we call the maximum voltage read on the voltmeter V_max, and use the above symbols, we can write the voltage across the voltmeter as:

V_max = I_g(R_g + R_V)

Note again that a device with various ranges can be made if there are circuit options with difference R_V's. The Ohmmeter. To make an ohmmeter we put an ammeter in series with a battery and the unknown resistance R. If we keep trach of the voltage changes around the loop we find:

V_battery = I R + I R_ammeter = 0

Solving for the current gives:

I = (V_battery)/(R + R_ammeter)

We know V_battery and R_ammeter, and we measure I on the ammeter. We can then determine the unknown R. The device is calibrated to read the resistance directly. Incidentally, the instrument is nor linear, as can beseen from the above equation. Again, various ranges can be provided.

RC Circuits

RC circuits are circuits that contain both resistors and capacitors. In a DC circuit (or, steady-state circuit, or constant current circuit), a capacitor acts like an open switch. It steady-state voltage will be V = Q/C and will be equal to the voltage of the battery. However, when the switch is first closed for the circuit, charge must flow until the capaitor is charged. This may only take a fraction of a second, ot a few seconds. This is called a transient current.

Consider a simple loop circuit which has a battery with terminal voltage V_battery, a resistor R, a capacitor C, and a switch. Just before the switch is closed the current is zero. Just after it is closed a current flows and the capacitor starts to charge. Its voltage will be V = Q/C. By the time Q grows until Q/C = V_battery, the current will be zero. This early current is the transient current. Let us write Kirchhoff's Loop Law for the circuit:

V_battery - I R - Q/C = 0

Let us note that the current I can be written as I = dQ/dt. The expression then becomes:

V_battery - (dQ/dt)R - Q/C = 0

The solution to this equation is:

Q = (C V_battery)(1 - exp(-t/RC))

Note that aftera long time exp(-t/RC) becomes equal to 0, and the charge is constant. If we take the derivative of Q with respect to t, we get:

I = dQ/dt = ((V_battery)/R)exp(-t/RC)

We see that after a long time, the current is zero. The quantity RC is called the time constant. It has units of time, and determines how fast a capacitor charges and discharges.

If we start with a charged capacitor, and have only a resistor and an open switch in the curcuit, we can then discharge the capacitor by closing the switch. The Kirchhoff's Loop Law equation becomes:

- I R - Q/C = 0

Again, using I = dQ/dt, we have :

- (dQ/dt)R - Q/C = 0

The solution is:

Q = Q_o exp(-t/RC)

Here Q_o is C V_battery. Again, using I = dQ/dt, we have:

I = (Q₀/RC)exp(-t/RC) = (V_battery)/R)exp(-t/RC)

Both the charge and the current become zero after a time as the capacitor is discharged. For a measure of the time we use the time constant t = R C. When t = R C, the exp(-t/RC) = exp(-1) = 0.37, or the charge on the capacitor is down to just over one-third of the original value.