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Chapter 17

Chapter 17. Current and Resistance. Volta discovered that electricity could be created if dissimilar metals were connected by a conductive solution called an electrolyte. This is a simple electric cell. A battery transforms chemical energy into electrical energy.

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Chapter 17

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  1. Chapter 17 Current and Resistance

  2. Volta discovered that electricity could be created if dissimilar metals were connected by a conductive solution called an electrolyte. This is a simple electric cell.

  3. A battery transforms chemical energy into electrical energy. Chemical reactions within the cell create a potential difference between the terminals by slowly dissolving them. This potential difference can be maintained even if a current is kept flowing, until one or the other terminal is completely dissolved.

  4. Several cells connected together make a battery, although now we refer to a single cell as a battery as well.

  5. Electric Current • The current is the rate at which the charge flows through this surface • Look at the charges flowing perpendicularly to a surface of area A • The SI unit of current is Ampere (A) • 1 A = 1 C/s

  6. Active Figure

  7. Electric Current, cont • The direction of the current is the direction positive charge would flow • This is known as conventional current direction • In a common conductor, such as copper, the current is due to the motion of the negatively charged electrons • It is common to refer to a moving charge as a mobile charge carrier • A charge carrier can be positive or negative

  8. By convention, current is defined as flowing from + to -. Electrons actually flow in the opposite direction, but not all currents consist of electrons.

  9. Quick Quiz Consider positive and negative charges moving horizontally through the four regions in Figure 17.2. Rank the magnitudes of the currents in these four regions from lowest to highest. (Ia is the current in Figure 17.2a, Ib the current in Figure 17.2b, etc.) (a) Id , Ia , Ic , Ib (b) Ia , Ic , Ib , Id (c) Ic , Ia , Id , Ib (d) Id , Ib , Ic , Ia (e) Ia , Ib , Ic , Id (f) none of these

  10. Answer (d). Negative charges moving in one direction are equivalent to positive charges moving in the opposite direction. Thus, are equivalent to the movement of 5, 3, 4, and 2 charges respectively, giving .

  11. Current and Drift Speed • Charged particles move through a conductor of cross-sectional area A • n is the number of charge carriers per unit volume • n A Δx is the total number of charge carriers

  12. Current and Drift Speed, cont • The total charge is the number of carriers times the charge per carrier, q • ΔQ = (n A Δx) q • The drift speed, vd, is the speed at which the carriers move • vd = Δx/ Δt • Rewritten: ΔQ = (n A vd Δt) q • Finally, current, I = ΔQ/Δt = nqvdA

  13. Current and Drift Speed, final • If the conductor is isolated, the electrons undergo random motion • When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current

  14. Active Figure

  15. Electrons in a conductor have large, random speeds just due to their temperature. When a potential difference is applied, the electrons also acquire an average drift velocity, which is generally considerably smaller than the thermal velocity.

  16. Charge Carrier Motion in a Conductor • The zig-zag black line represents the motion of charge carrier in a conductor • The net drift speed is small • The sharp changes in direction are due to collisions • The net motion of electrons is opposite the direction of the electric field

  17. Electrons in a Circuit • The drift speed is much smaller than the average speed between collisions • When a circuit is completed, the electric field travels with a speed close to the speed of light • Although the drift speed is on the order of 10-4 m/s the effect of the electric field is felt on the order of 108 m/s

  18. Quick Quiz Suppose a current-carrying wire has a cross-sectional area that gradually becomes smaller along the wire, so that the wire has the shape of a very long cone. How does the drift speed vary along the wire? (a) It slows down as the cross section becomes smaller. (b) It speeds up as the cross section becomes smaller. (c) It doesn’t change. (d) More information is needed.

  19. Answer (b). Under steady-state conditions, the current is the same in all parts of the wire. Thus, the drift velocity, given by ,is inversely proportional to the cross-sectional area.

  20. Circuits A circuit is a complete conducting path.

  21. Meters in a Circuit – Ammeter • An ammeter is used to measure current • In line with the bulb, all the charge passing through the bulb also must pass through the meter

  22. Meters in a Circuit – Voltmeter • A voltmeter is used to measure voltage (potential difference) • Connects to the two ends of the bulb

  23. Quick Quiz Look at the four “circuits” shown in Figure 17.6 and select those that will light the bulb.

  24. Answer (c), (d). Neither circuit (a) nor circuit (b) applies a difference in potential across the bulb. Circuit (a) has both lead wires connected to the same battery terminal. Circuit (b) has a low resistance path (a “short”) between the two battery terminals as well as between the bulb terminals.

  25. Quick Quiz Suppose an electrical wire is replaced with one having every linear dimension doubled (i.e. the length and radius have twice their original values). Does the wire now have (a) more resistance than before, (b) less resistance, or (c) the same resistance?

  26. Answer (b). Consider the expression for resistance: Doubling all linear dimensions increases the numerator of this expression by a factor of 2, but increases the denominator by a factor of 4. Thus, the net result is that the resistance will be reduced to one-half of its original value.

  27. Example 1 A certain conductor has 7.50 × 1028 free electrons per cubic meter, a cross-sectional area of 4.00 × 10−6 m2, and carries a current of 2.50 A. Find the drift speed of the electrons in the conductor.

  28. Example 2 In a particular television picture tube, the measured beam current is 60.0 μA. How many electrons strike the screen every second?

  29. Example 3 An aluminum wire with a cross-sectional area of 4.0 × 10−6 m2 carries a current of 5.0 A. Find the drift speed of the electrons in the wire. The density of aluminum is 2.7 g/cm3. (Assume that one electron is supplied by each atom.)

  30. Practice 1 A 200-km-long high-voltage transmission line 2.0 cm in diameter carries a steady current of 1 000 A. If the conductor is copper with a free charge density of 8.5 × 1028 electrons per cubic meter, how many years does it take one electron to travel the full length of the cable?

  31. Resistance • In a conductor, the voltage applied across the ends of the conductor is proportional to the current through the conductor • The constant of proportionality is the resistance of the conductor

  32. Active Figure

  33. Resistance, cont • Units of resistance are ohms (Ω) • 1 Ω = 1 V / A • Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor

  34. Georg Simon Ohm • 1787 – 1854 • Formulated the concept of resistance • Discovered the proportionality between current and voltages

  35. Ohm’s Law • Experiments show that for many materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents • This statement has become known as Ohm’s Law • ΔV = I R • Ohm’s Law is an empirical relationship that is valid only for certain materials • Materials that obey Ohm’s Law are said to be ohmic

  36. Ohm’s Law, cont • An ohmic device • The resistance is constant over a wide range of voltages • The relationship between current and voltage is linear • The slope is related to the resistance

  37. Ohm’s Law, final • Non-ohmic materials are those whose resistance changes with voltage or current • The current-voltage relationship is nonlinear • A diode is a common example of a non-ohmic device

  38. Resistivity • The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A • ρ is the constant of proportionality and is called the resistivity of the material • See table 17.1

  39. Temperature Variation of Resistivity • For most metals, resistivity increases with increasing temperature • With a higher temperature, the metal’s constituent atoms vibrate with increasing amplitude • The electrons find it more difficult to pass through the atoms

  40. Temperature Variation of Resistivity, cont • For most metals, resistivity increases approximately linearly with temperature over a limited temperature range • ρ is the resistivity at some temperature T • ρo is the resistivity at some reference temperature To • To is usually taken to be 20° C •  is the temperature coefficient of resistivity

  41. Temperature Variation of Resistance • Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, you can find the effect of temperature on resistance

  42. Quick Quiz Two wires, A and B, are made of the same metal and have equal length, but the resistance of wire A is four times the resistance of wire B. How do their diameters compare? 1) dA = 4 dB 2) dA = 2 dB 3) dA = dB 4) dA = 1/2 dB 5) dA = 1/4 dB

  43. Answer 1) dA = 4 dB 2) dA = 2 dB 3) dA = dB 4) dA = 1/2 dB 5) dA = 1/4 dB Two wires, A and B, are made of the same metal and have equal length, but the resistance of wire A is four times the resistance of wire B. How do their diameters compare? The resistance of wire A is greater because its area is less than wire B. Since area is related to radius (or diameter) squared, the diameter of A must be two times less than B.

  44. Quick Quiz 1) it decreasesby a factor 4 2) it decreasesby a factor 2 3) it stays the same 4) it increasesby a factor 2 5) it increasesby a factor 4 A wire of resistance R is stretched uniformly (keeping its volume constant) until it is twice its original length. What happens to the resistance?

  45. Answer 1) it decreasesby a factor 4 2) it decreasesby a factor 2 3) it stays the same 4) it increasesby a factor 2 5) it increasesby a factor 4 A wire of resistance R is stretched uniformly (keeping its volume constant) until it is twice its original length. What happens to the resistance? Keeping the volume (= area x length) constant means that if the length is doubled, the area is halved. Since R=L/A , this increases the resistance by four.

  46. Example 4 A lightbulb has a resistance of 240 Ω when operating at a voltage of 120 V. What is the current in the bulb?

  47. Example 5 Eighteen-gauge wire has a diameter of 1.024 mm. Calculate the resistance of 15 m of 18-gauge copper wire at 20°C.

  48. Example 6 A rectangular block of copper has sides of length 10 cm, 20 cm, and 40 cm. If the block is connected to a 6.0-V source across two of its opposite faces, what are (a) the maximum current and (b) the minimum current that the block can carry?

  49. Practice 2 Suppose that you wish to fabricate a uniform wire out of 1.00 g of copper. If the wire is to have a resistance R = 0.500 Ω, and if all of the copper is to be used, what will be (a) the length and (b) the diameter of the wire?

  50. Example 7 A wire 3.00 m long and 0.450 mm2 in cross-sectional area has a resistance of 41.0 Ω at 20°C. If its resistance increases to 41.4 Ω at 29.0°C, what is the temperature coefficient of resistivity?

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