Today’s agendum: Electric Current.

2. Today’s agendum: • Electric Current. • You must know the definition of current, and be able to use it in solving problems. • Current Density. • You must understand the difference between current and current density, and be able to use current density in solving problems. • Ohm’s “Law” and Resistance. • You must be able to use Ohm’s “Law” and electrical resistance in solving circuit problems. • Resistivity. • You must understand the relationship between resistance and resistivity, and be able to calculate resistivity and associated quantities. • Temperature Dependence of Resistivity. • You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.

3. Electric Current Definition of Electric Current The average current that passes any point in a conductor during a time t is defined as where Q is the amount of charge passing the point. The instantaneous current is One ampere of current is one coulomb per second:

4. - + Currents in battery-operated devices are often in the milliamp range: 1 mA = 10-3 A. “m” for milli—another abbreviation to remember! Here’s a really simple circuit: current Don’t try that at home! (Why not?) The current is in the direction of flow of positive charge… …opposite to the flow of electrons, which are usually the charge carriers.

5. - + current electrons An electron flowing from – to + gives rise to the same “conventional current” as a proton flowing from + to -. An electron flowing from – to + “Conventional” refers to our convention, which is always to consider the effect of + charges (for example, electric field direction is defined relative to + charges).

6. “Hey, that figure you just showed me is confusing. “Hey, that figure you just showed me is confusing. Why don’t electrons flow like this?” - + current electrons Good question.

7. - + current electrons Electrons “want” to get away from - and go to +. Chemical reactions (or whatever energy mechanism the battery uses) “force” electrons to the negative terminal. The battery won’t “let” electrons flow the wrong way inside it. So electrons pick the easiest path—through the external wires towards the + terminal. Of course, real electrons don’t “want” anything.

8. Note! Current is a scalar quantity, and it has a sign associated with it. In diagrams, assume that a current indicated by a symbol and an arrow is the conventional current. I1 If your calculation produces a negative value for the current, that means the conventional current actually flows opposite to the direction indicated by the arrow.

9. Example: 3.8x1021 electrons pass through a point in a wire in 4 minutes. What was the average current? “This is a piece of cake so far!” Don’t worry, it gets “better” later.

10. Today’s agendum: • Electric Current. • You must know the definition of current, and be able to use it in solving problems. • Current Density. • You must understand the difference between current and current density, and be able to use current density in solving problems. • Ohm’s “Law” and Resistance. • You must be able to use Ohm’s “Law” and electrical resistance in solving circuit problems. • Resistivity. • You must understand the relationship between resistance and resistivity, and be able to use calculate resistivity and associated quantities. • Temperature Dependence of Resistivity. • You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.

11. + + + + Current Density When we study details of charge transport, we use the concept of current density. Current density is the amount of charge that flows across a unit of area in a unit of time. Current density: charge per area per time.

12. A current density J flowing through an infinitesimal area dA produces an infinitesimal current dI. dA J Current density is a vector. Its direction is the direction of the velocity of positive charge carriers. The total current passing through A is just Current density: charge per area per time.

13. J A If J is constant and parallel to dA (like in a wire), then Now let’s take a “microscopic” view of current. If n is the number of charges per volume, then the number of charges that pass through a surface A in a time t is vt v A q

14. The total amount of charge passing through A is the number of charges times the charge of each. vt v A q Divide by t to get the current… …and by A to get J:

15. To account for the vector nature of the current density, and if the charge carriers are electrons, q=-e so that The – sign demonstrates that the velocity of the electrons is antiparallel to the conventional current direction.

16. Currents in Materials Metals are conductors because they have “free” electrons, which are not bound to metal atoms. In a cubic meter of a typical conductor there roughly 1028 free electrons, moving with typical speeds of 1,000,000 m/s. But the electrons move in random directions, and there is no net flow of charge, until you apply an electric field...

17. E electron “drift” velocity - inside a conductor just one electron shown, for simplicity The voltage accelerates the electron, but only until the electron collides with a “scattering center.” Then the electron’s velocity is randomized and the acceleration begins again. Some predictions made by this theory are off by a factor or 10 or so, but it was the best we could do before quantum mechanics. We will see later in this course that the electron would follow curved trajectories, but the idea here is still valid.

18. Even though the details of the model on the previous slide are wrong, it points us in the right direction, and works when you take quantum mechanics into account. In particular, the velocity that should be used in is not the charge carrier’s velocity (electrons in this example). Instead, we should the use net velocity of the collection of electrons, the net velocity caused by the electric field. This “net velocity” is like the terminal velocity of a parachutist; we call it the “drift velocity.”

19. It’s the drift velocity that we should use in our equations for current and current density in conductors:

20. Example: the 12-gauge copper wire in a home has a cross-sectional area of 3.31x10-6 m2 and carries a current of 10 A. The conduction electron density in copper is 8.49x1028 electrons/m3. Calculate the drift speed of the electrons.

21. Today’s agendum: • Electric Current. • You must know the definition of current, and be able to use it in solving problems. • Current Density. • You must understand the difference between current and current density, and be able to use current density in solving problems. • Ohm’s “Law” and Resistance. • You must be able to use Ohm’s “Law” and electrical resistance in solving circuit problems. • Resistivity. • You must understand the relationship between resistance and resistivity, and be able to use calculate resistivity and associated quantities. • Temperature Dependence of Resistivity. • You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.

22. Resistance The resistance of a material is a measure of how easily a charge flows through it. Resistance: how much “push” is needed to get a given current to flow. The unit of resistance is the ohm: Resistances of kilo-ohms and mega-ohms are common:

23. This is the symbol we use for a “resistor:” All wires have resistance. Obviously, for efficiency in carrying a current, we want a wire having a low resistance. In idealized problems, we will consider wire resistance to be zero. Lamps, batteries, and other devices in circuits have resistance. Every circuit component has resistance.

24. Resistors are often intentionally used in circuits. The picture shows a strip of five resistors (you tear off the paper and solder the resistors into circuits). The little bands of color on the resistors have meaning, namely the amount of resistance.

25. Ohm’s “Law” In some materials, the resistance is constant over a wide range of voltages. For such materials, we write and call the equation “Ohm’s Law.” In fact, Ohm’s “Law” is not a “Law” in the same sense as Newton’s “Laws”… … and in advanced classes you will write something other than V=IR when you write Ohm’s “Law.” Newton’s Laws demand; Ohm’s “Law” suggests.

26. Today’s agendum: • Electric Current. • You must know the definition of current, and be able to use it in solving problems. • Current Density. • You must understand the difference between current and current density, and be able to use current density in solving problems. • Ohm’s “Law” and Resistance. • You must be able to use Ohm’s “Law” and electrical resistance in solving circuit problems. • Resistivity. • You must understand the relationship between resistance and resistivity, and be able to use calculate resistivity and associated quantities. • Temperature Dependence of Resistivity. • You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.

27. Resistivity It is also experimentally observed (and justified by quantum mechanics) that the resistance of a metal wire is well-described by where  is a “constant” called the resistivity of the wire material, L is the wire length, and A its cross-sectional area.Notice that this relates R to a property of the material of which the wire is made. This makes sense: a longer wire or higher-resistivity wire should have a greater resistance. A larger area means more “space” for electrons to get through, hence lower resistance.

28. R = L / A, units of  are m  A L The longer a wire, the “harder” it is to push electrons through it. The greater the resistivity, the “harder” it is to push electrons through it. The greater the cross-sectional area, the “easier” it is to push electrons through it. Resistivity is a useful tool in physics because it depends on the properties of the wire material, and not the geometry.

29. A =  (d/2)2 geometry! Resistivities range from roughly 10-8·m for copper wire to 1015 ·m for hard rubber. That’s an incredible range of 23 orders of magnitude, and doesn’t even include superconductors (we might talk about them some time). Example: Suppose you want to connect your stereo to remote speakers. (a) If each wire must be 20 m long, what diameter copper wire should you use to make the resistance 0.10  per wire. R = L / A A = L / R  (d/2)2 = L / R

30. d/2= ( L / R )½ don’t skip steps! d = 2 ( L / R )½ (d/2)2 = L / R d = 2 [ (1.68x10-8) (20) /  (0.1) ]½ d = 0.0021 m = 2.1 mm (b) If the current to each speaker is 4.0 A, what is the voltage drop across each wire? V = I R V = (4.0) (0.10) V = 0.4 V

31. Ohm’s “Law” Revisited The equation for resistivity I introduced five slides back is a semi-empirical one. Here’s how we define resistivity: Our equation relating R and  follows from the above equation. We define conductivity as the inverse of the resistivity:

32. With the above definitions, The “official” Ohm’s “Law”, valid for non-ohmic materials. Cautions! In this context:  is not volume density!  is not surface density!

33. Example: the 12-gauge copper wire in a home has a cross-sectional area of 3.31x10-6 m2 and carries a current of 10 A. Calculate the magnitude of the electric field in the wire.

34. Today’s agendum: • Electric Current. • You must know the definition of current, and be able to use it in solving problems. • Current Density. • You must understand the difference between current and current density, and be able to use current density in solving problems. • Ohm’s “Law” and Resistance. • You must be able to use Ohm’s “Law” and electrical resistance in solving circuit problems. • Resistivity. • You must understand the relationship between resistance and resistivity, and be able to use calculate resistivity and associated quantities. • Temperature Dependence of Resistivity. • You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.

35. Temperature Dependence of Resistivity Many materials have resistivities that depend on temperature. We can model* this temperature dependence by an equation of the form where 0 is the resistivity at temperature T0, and  is the temperature coefficient of resistivity. *T0 is a reference temperature, often taken to be 0 °C or 20 °C. This approximation can be used if the temperature range is “not too great;” i.e. 100 °C or so.

36. Example: a carbon resistance thermometer in the shape of a cylinder 1 cm long and 4 mm in diameter is attached to a sample. The thermometer has a resistance of 0.030 . What is the temperature of the sample? Look up resistivity of carbon, use it to calculate resistance. This is the resistivity at 20 C. This is the resistance at 20 C.

37. The result is very sensitive to significant figures in resistivity and .

38. Today’s agendum: • Emf, Terminal Voltage, and Internal Resistance. • You must be able to incorporate all of these quantities in your circuit calculations. • Electric Power. • You be able to calculate the electric power dissipated in circuit components, and incorporate electric power in work-energy problems.

39. circuit components in series Conservation of energy to implies that the voltage drop across circuit components in series is the sum of the individual voltage drops. Vab C1 C2 C3 a b -Q V2 V1 + - V Vab = V = V1 + V2 + V3

40. circuit components in series Therefore, the voltage drop across resistors in series is the sum of the individual voltage drops. R1 R2 R3 a b V1 V2 V3 - + V Vab = V = V1 + V2 + V3 This is a consequence of conservation of energy. Use this in combination with Ohm’s “Law”, V=IR.

41. DC Currents In Physics 104, whenever you work with currents in circuits, you should assume (unless told otherwise) “direct current.” Current in a dc circuit flows in one direction, from + to -. We will not encounter ac circuits much in this course. • For any calculations involving household current, which is ac, assuming dc will be “close enough” to give you “a feel” for the physics. • If you need to learn about ac circuits, you’ll have courses devoted to them. • The mathematical analysis is more complex. We have other things to explore this semester.

42. emf, terminal voltage, and internal resistance We have been making calculations with voltages from batteries without asking detailed questions about the batteries. Now it’s time to look inside the batteries. We introduce a new term – emf – in this section. Any device which transforms a form of energy into electric energy is called a “source of emf.” “emf” is an abbreviation for “electromotive force,” but emf does not really refer to force! The emf of a source is the voltage it produces when no current is flowing.

43. The voltage you measure across the terminals of a battery (or any source of emf) is less than the emf because of internal resistance. Here’s a battery with an emf. All batteries have an “internal resistance:” emf is the zero-current potential difference - + The “battery” is everything inside the green box. a b Hook up a voltmeter to measure the emf: emf - + The “battery” is everything inside the green box. a b Getting ready to connect the voltmeter (it’s not hooked up yet).

44. Measuring the emf??? emf - + The “battery” is everything inside the green box. a b As soon as you connect the voltmeter, current flows. I You can’t measure voltage without some (however small) current flowing, so you can’t measure emf directly. You can only measure Vab.

45. We model a battery as producing an emf, , and having an internal resistance r: - + The “battery” is everything inside the green box. a b  r  Vab The terminal voltage, Vab, is the voltage you measure with current flowing. When a current I flows through the battery, Vab is related to the emf, , by

46. Your text writes and expects you to put the correct sign (+ or -) on the I. I’ll go along with your text, so our equation is Why the  sign? If the battery is delivering current, the V it delivers is less than the emf, so the – sign is necessary. If the battery is being charged, you have to “force” the current through the battery, and the V to “force” the current through is greater than the emf, so the + sign is necessary.

47. When current passes through a battery in the direction from the - terminal toward the + terminal, the terminal voltage Vab of the battery is Useful facts: • The sum of the potential changes around a circuit loop is zero. Potential decreases by IR when going through a resistor in the direction of the current and increases by  when passing through an emf in the direction from the - to + terminal.   + + V is + V is - - - I I

48. - +  r To model a battery, simply include an extra resistor to represent the internal resistance, and label the voltage source* as an emf instead of V (units are still volts): *Remember, all sources of emf—not just batteries—have an internal resistance.

49. Example: a battery is known to have an emf of 9 volts. If a 1 ohm resistor is connected to the battery, the terminal voltage is measured to be 3 volts. What is the internal resistance of the battery? R=1  Because the voltmeter draws “no” current, r and R are in series with a current I flowing through both. I emf - + a b IR, the potential drop across the resistor, is just the potential difference Vab. internal resistance r terminal voltage Vab the voltmeter’s resistance is so large that approximately zero current flows through the voltmeter

50. R=1  I emf - + a b A rather unrealistically large value for the internal resistance of a 9V battery.