Up to now we have considered electric charges at rest. A flow of charge from one point to another is known as an electric current, and this normally takes place in a wire. If we measure the total charge flowing by any point in an interval of time, we have a measure of the current. we define current, i, to be:

i = or, current =

The units for this new quantity are thus coulombs/second, or amperes (amps for short).

In many ways electric current is analogous to water flowing in pipes. It flows continuously without accumulating in any spot, and if its flow is reduced at any one point, the current in other parts of the circuits will be increased or decreased as needed to maintain continuity.

1. Current enters a junction as shown. What current leaves? (A is for ampere.)

2. What are the unknown currents?

3. Find the missing currents.

In this text I have chosen to use the direction of electron flow as the direction of the current. I do this because in wires and resistors the moving charges are electrons. In ionized solutions and semiconductors this is not always the case, and the convention for electronics is that current is a positive charge flow (a decision made by Ben Franklin two centuries ago). The choice is really arbitrary for almost all circumstances, but I like to start out with the physical reality of fixed positive charge and moving electrons in a wire.

A great deal of work has been done on the manipulation of current. By reducing its flow in one circuit element, current can be diverted to other elements and controlled to serve useful purposes. In the last century it was found that certain materials were more
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effective than others at reducing the flow of electricity that is, materials differ in their electrical resistivity.

Depending on its shape and the material or materials that compose it, every object has a measurable resistance to the flow of electric current. Look at some laboratory results using three different objects. We will hook them to a battery (a source of potential) and measure the resulting current that passes through each.

Here are the results, showing the current that results from different applied potentials. (We use E rather than V to represent potential. This is done in electronics, so we'll do it here. E should not be confused with E, the electric field.)

Note that in every case E i, so we may say that E = iR where R is a constant that depends on the particular object under study. We call R the resistance, and call the relationship Ohm's Law. The units of R are volts/amp or ohms, symbolized by the Greek letter capital omega: W.

4. A 12v battery is connected to a 3W resistor. What current will flow through the resistor?

5. A 3A current flows through two resistors in series as shown. What is the potential across the 6W resistor? What is it across the 8W? What is it across the combined resistance?

6. Find the current through each resistor:

7. The current through the circuit is 3A.

What is VAB, VBC, VAC, VCD, and the potential of the battery?

8. What potential does each battery have to produce the currents indicated?

9. Given EAB = -5v and ECD = -3v,

find EBE, EEF, EAF, EBC, and EAC.

10. The current through the 5W is 6A.

a) What is the current through the 2W?

b) What is the potential across the 8W?

11. The potential across the 7W is 21v.
a) What is the current through the circuit?
b) What is the potential across the 4W?
c) What is the potential of the battery?

12. Find the current through the 16W resistor.

13. Find i1, i2, and i3.

14. Find i1, i2, i3, and i4.

15. The current from the battery is 1.2 A. Find Eab, Ecd, and Eae.

16. 6 A flows through the 12 ohm resistor. What current flows through the 7 ohm and 14 ohm resistors?

17. Find the unknown currents:

18. A current i enters two resistors as shown.
a) What is the potential across each resistor?
b) What is the total potential of the battery?
c) What is the effective resistance of R1 and R2 together?

19. Find the equivalent resistances:

20. Find the current through the circuit:

21. Two resistors are connected in parallel as shown.
a) What is the current through each?
b) What total current does the battery supply?
c) What is the effective resistance in the circuit?

Now THAT is a real shocker! We have taken not one, but TWO resistors, hooked them together, and found that the total resistance is less than either one individually. Is there an error in reasoning?

Of course not. Your favorite physics teacher is looking out for you. Just for a moment, instead of thinking of resistors as impediments to the flow of electricity, think of them as pathways. Between the top and bottom wire of problem 14 there will be no flow of electrons until we begin to put pathways in place. The more pathways, the more current and the less the resistance.

Think of resistors as check-out registers at a supermarket. When only one register is open, the line crawls. As new check-out people are brought in, the flow of groceries increases, no matter how slow (how large a resistance) a particular checker is.

22. What is the current through each resistor?

What is the total current?

What is the effective resistance?

We now have two working equations:

When resistors are in series, R = R1 + R2

When in parallel, = +

Note that this is the reverse of the situation with capacitors, yet it is just as logical. (In fact, a bit more logical, I'd say.)

23. Find the equivalent resistances:

24. Again, find the equivalent resistances:

25. The current through the 12W is 0.3A.

What is the current through the 4W?

What is the potential of the battery?

26. The current through the 160W resistor is 0.3A.
a) What is the potential across the 100W?
b) What is the potential of the battery?

1 1 1

R R1 R
27. The current in the 8W is 2A. What is the current in the 4W?

28. What is the potential across the 5K resistor?

29. What is the potential across the 400W?

30. What is the potential across R1?

31. The potential across the 18W is 6v.

What is it across the whole circuit?

32. 6.0 amps flows into a branched circuit as shown.

What current will go through the 8 ohm resistor?

33. The current in the 15 ohm resistors is 0.37 A.

What is the current in the 11 ohm?

34. What is the current through the 16W?

If you find it natural and logical to think of the large current passing through the small resistor, you will find this shortcut helpful. If it just seems like one more equation, you'll do better to draw a potential diagram and use E = iR.

Another useful shortcut is the voltage divider. Here it is in all its glory.

35. What is the potential across the 24W resistor?

36. The current through the 50W resistor is 0.83 A.

What is the potential across the 22W resistor?

37. Find the current in the 7W resistor by three different methods.

38. The potential across the 8W resistor is 12 v.

Find the following: i8, i11, V24, i24, i4, V4, V7, i3, VT.

A battery is a source of energy, causing a current to flow. Where has this energy gone when the battery runs out? It can't simply disappear. Think about it it goes into heat. Wires get hot, resistors get very hot when current runs through them. The rate at which this energy is dissipated, the work per unit time, is called the power, just as it was in mechanics.

Recall: power = = = current · potential

Thus P = iE (just remember "pie")

39. A 6v battery delivers 2 amps. What is its power output?

40. A 3A current flows through a 6W resistor. What power is dissipated?

41. A 12v battery is wired to a 24W resistor.

What power is dissipated by the resistor?

work charge · potential

time time
42. A current i flows through resistor R. What power is dissipated by it?

43. A battery of potential E is connected to a resistor R. What power is dissipated by the resistor?

Note that we have two new expressions: P = i2R and P = E2/R as well as P = iE. Remember, these are not really new, they're just consequences of stuff we already know.

44*. The power output of the 3W is 12 watts.

What is the power output of the 5W? ....the 2W?

45*. A 30v source is connected to a 20W resistor submerged in 280 gm of water at 22°C. How long will it take to boil away all the water?

Electronic circuits often contain more than one circuit element, so we will finish the chapter by combining resistors with capacitors. Furthermore, circuitry is best mastered thorugh repeated practice, as these final problems will indicate.

Capacitors normally appear in alternating current (AC) circuits, but some of their properties can be learned by thinking about them in direct current (DC) circuits. As AC circuits quickly get pretty difficult, we will confine our work to DC.

46. A 200µF capacitor holds a charge of 6.0 x 10-3C. It is discharged across the 4W resistor by closing the switch. What is the current in the resistor immediately after the switch is closed?

47. Draw a graph showing how current in problem 46 changes as the capacitor discharges.

48. A 40 µF capacitor holding 350 µC of charge is discharged across a 150W resistor. What is the initial current?

49. A 30v potential is connected to the capacitor and resistor as shown.

What is the initial current through the resistor?

What is the current long after the circuit has been completed?

50. In the R-C circuit shown, what is the potential across the capacitor immediately as the switch is closed? What is it long after the switch was closed?

We see, then, that initially the uncharged capacitor in a circuit acts as a wire, transmitting current freely. When a little time has passed, the fully charged capacitor stops all current flow in its part of the circuit.

A study of the rate at which capacitors charge and discharge and their behavior in AC circuits requires calculus. This will be left to more advanced courses.

51. What is the initial resistance of this circuit?

What is its resistance after equilibrium is reached?

52. Find the current in the 12W resistor (a) intially and (b) after a long time.

53. Find the charge on each capacitor after equilibrium is reached.