DC Circuits

KEY TERMS: Direct current, alternating current, resistors connected in series and parallel, equivalent resistance, Kirchoff's junction rule and loop rule, ammeter, voltmeter, ohmmeter.


Direct current (DC) circuits basically consist of a loop of conducting wire (like copper) through which an electric current flows. An electric current consists of a flow of electric charges, analogous to the flow of water (water molecules) in a river. In addition to the copper wire in a circuit there usually are components such as resistors which restrict the flow of electric charge, similar to the way rocks and debris in a river restrict the flow of the river water.

For a clear picture of an electric circuit go to an informative website with interactive pictures that will show you quickly what is going on in an electric circuit:


then click on "Electricity and Magnetism" button,
then "electric circuits" button and
then click "water circuit."

You can click anyplace on the interactive "water circuit" page to get an explanation of the water circuit as an analogy to an electric circuit. You will find this very helpful in getting the idea of the voltage, current and resistance in an electric circuit.


Now that we have an idea about electric charge and how they experience a force when placed in an electric field (SEE THE PAGE ON ELECTRIC CHARGE) we can discuss DC circuits (covered in Chapter 25 Current, Resistance, and Electromotive Force and in Chapter 26 DC Circuits - see below).

Chapter 25 Current, Resistance, and
Electromotive Force

Electron trajectories in a conductor are shown in the diagrams below. If there is no electric field inside a conducting material electrons move randomly. If a field is present the electric force F=qE imposes a small drift on the electron's motion. Note that since the electron has a negative charge the force F is in a direction opposite to the field E.

Fig. 25.1 Random motions of electrons in a conductor without and with an electric field present.



The motion of a ball
rolling down an inclined plane and bouncing off pegs in its path is analogous to the motion of an electron in a metallic conductor with an electric field present.

If the pegs were
rearranged, or some were removed, the ball would experience less resistance to its movement. If more
pegs were added the ball might experience more resistance to its downhill motion.

Electric charges move through a material in a similar way. If the charge has its motion greatly restricted we say the material through which it is moving has a large electrical RESISTANCE. The resistance of a material depends on the arrangement, or spacing, of the atoms or molecules in the material. The motion of the atoms affects the electrical resistance of the material. If the atoms or molecules vibrate a lot, in general, they will increase the resistance of the material. And since we know that as the temperature rises molecules vibrate more and more, we conclude that the electrical resistance of a material is increased as the temperature rises.

The amount of electrical resistance caused by a temperature increase is described by factor (a) called the temperature coefficient of resistivity (resistivity is closely related to resistance).

Other factors that affect the resistance of an electrical conductor are its length and cross-sectional area. The longer the conductor the greater the resistance, and the greater the cross-section area through which the charges flow the smaller the resistance of the conductor. We can formulate these ideas in an equation relating the resistance, the resistivity, the length, and the area of a conductor. Since the resistance is proportional to the length and inversely proportional to the area we can say that R = r L / A, where we have inserted a constant of proportionality called the resistivity (r). The resistivity is dependent on the characteristics of the material and its temperature and is, in general, different for each material (in units of ohm-meters).

The dependence of resistance (R) or resistivity (r) on temperature has been found to be linear over limited temperature ranges and can be described as a straight line:

R(T) = Ro [1 + a (T - To)]     

where T is the Celsius temperature, To is the reference temperature (usually room temperature or 20 degrees C), Ro is the resistance at the reference temperature, and a is the temperature coefficient of resistivity in units of 1/degrees C. You will measure the variation of resistance with temperature in the lab and verify the accuracy of this equation.







Fig. 25.3 Current through the cross section area A of a conductor.

The current through a cross-section area A is the net rate (dQ/dt) at which charge passes through the area. If the moving charges are positive the drift velocity is in the same direction as the field, as shown.





Fig. 25.7 Current flows in a conductor from higher to lower electric potential.

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Current through a conductor flows from a higher electric potential (or voltage) to a lower electric potential; just as water flows from a higher level of potential energy to a lower level of potential energy, i.e., downhill!

The greater the electric potential between the ends of a conductor, the greater the current through the conductor. The difference in voltage levels is often referred to as the voltage. If the current in a conductor is proportional the the potential difference (or voltage) driving the current through the conductor we say that Ohm's Law is obeyed: V = I R, where R, the resistance of the conductor, is the constant of proportionality or the slope of the V vs. I graph. See the V vs. I graphs below. Not all obey Ohm's Law.




Current-voltage relations for

(a) a resistor that obeys Ohm's Law, 1 / R = I / V, or

           V = I R        Ohm's Law


(b) a vacuum tube diode




(c) a semiconductor diode.

Fig. 25.10 Current-Voltage relations.




Below is a simple circuit that has an external resistor R and a 12 Volt source with an internal resistance r. The diagram following shows the electric potential (or voltage) drops in this circuit.

Fig. 25.18 Simple circuit with an external resistor R and a 12 Volt source with an internal resistance r.




The circuit above is redrawn below to show the voltage (or electric potential) rises and drops:

Fig. 25.21 Voltage (or electric potential) rises and drops in a circuit.




Energy and Power in Electric Circuits

We will see later that the electric potential (V, a scalar quantity with units of Volts) is defined as the potential energy per charge: V = U / Q = dU / dQ. The work-energy theorem tells us that U = W and we can write, since the current I = dQ / dt, V = dU / dQ = dW / dQ or the work done on the charge is  dW = V dQ = V I dt. Recall that the power is defined to be the energy transferred per unit time or the work done per unit time:

P = dW / dt = V I (the rate of delivering energy to a circuit element having a potential difference across it of V).

Using Ohm's Law V = I R, we can get a very useful equation to calculate the power in a circuit component, like an automobile headlight:

P = Vab I = I^2 R = Vab^2 / R

If the battery emf = e and the internal
battery resistance = r: Vab = e - I r

P = Vab I = e I - I^2 r   where

-- the term e I = rate of conversion of non-electrical (chemical) energy to electrical energy;

-- the term I^2 r = rate of energy dissipation in the source (battery)

-- the term e I - I^2 r = power output of the source delivered to the load (headlamp).

Fig. 25.23 Car battery and headlight.



Chapter 26 DC Circuits begins here.


For a clear picture of an electric circuit go to the informative website with interactive pictures that will show you quickly what is going on in an electric circuit:


then click on "Electricity and Magnetism" button,
then "electric circuits" button and
then click "Resistors" for resistivity, resistance, and resistor combinations,
then click on "Ohm's Law" for some sample problems done for you,
then click on "Power relationships."


In the figures (26.1) below:

(a) Three resistors in series have the same amount of current gassing through each of the three resistors sequentially. Their total resistance is


(b) Three resistors in parallel have three currents each passing through one of the three resistors. There are three parallel paths for the current to flow through. The total resistance of the three resistors is


Figures (c) and (d) show combinations of series and parallel resistors.


One way to calculate the current in each resistor (to see whether the component will burn out from too much current, for example) is to first find the total resistance of the circuit by simplifying the resistance of parallel combinations, as indicated. Once the total current is know the current in the individual branches can be calculated. Figure 26.3














To calculate the current in each of the branches of more complicated circuits like those on the left and below, one must resort to Kirchoff's Rules:

S V = 0
The sum of all the voltage drops and rises around ANY loop of the circuit
must be zero.
(Conservation of energy)

S I= 0
The sum of all the currants entering a junction in the circuit must be zero, i.e., the current into a junction must equal to the current coming out of a junction.
No current is lost or created in a junction.
(Conservation of electric charge)

Fig. 26.6 Two networks that cannot be reduced to series and parallel R's.


Fig. 26.9 Junction rule reduces the number of unknown currents from three to two.








The difference between an ammeter (measures current passing through a circuit) and a voltmeter (measures voltage between two points in a circuit) is in the path of the current through the meter.

(a) Ammeter has a very small shunt resistor that allows most of the current being measured to bypass the meter coil and add very little resistance to the circuit being measured.

(b) Voltmeter has a very large resistor in series with the meter coil to reduce the amount of current coming from the circuit being measured.



We will defer until later Resistance-Capacitance (RC) Circuits.
(Section 26.4)


In household circuits (actually alternating current circuits) the appliances are connected in parallel between the hot and neutral power lines as shown below. Why not series connections?

Fig. 26.25 Diagram of household wiring system.






Household appliances and tools usually have a third wire called a "ground" wire to provide a current path, bypassing your body, in the event of a wiring failure as shown on the right above.


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©2009 J. F. Becker