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Current and Resistance The Starting Point: Elements, Atoms and Charge Electrons and protons have, in addition to their mass, a quantity called charge Charge (unlike mass) can be either positive (protons) or negative (electrons) Like charges repel, unlike charges attract Free Electrons An electron that is not bound to any particular atom Ions External force can cause an electron to leave its orbit -atom is referred to as a positive ion External force can cause an atom to gain an electron -atom is referred to as a negative ion Electric Current Whenever electric charges, an electric current is said to exist 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 Q I t The SI unit of current is Ampere (A) 1 A = 1 C/s Electrical Current Electron Flow Versus Conventional Current Insert Figure 1.10 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 Current and Drift Speed 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 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 Current and Drift Speed Charged particles move through a conductor of crosssectional area A n is the number of charge carriers per unit volume nAΔx is the total number of charge carriers Current and Drift Speed The total charge is the number of carriers times the charge per carrier, q The drift speed, vd, is the speed at which the carriers move ΔQ = (n A Δx) q vd = Δx/ Δt Rewritten: ΔQ = (n A vd Δt) q Finally, current, I = ΔQ/Δt = nqvdA Quiz 1 Consider positive and negative charges moving horizontally through the four regions in the following Figure. Rank the magnitudes of the currents in these four regions from lowest to highest. (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 Answer Quiz 1 (d). Negative charges moving in one direction are equivalent to positive charges moving in the opposite direction. Thus, Ia, Ib, Ic and Id are equivalent to the movement of 5, 3, 4, and 2 charges respectively, giving Id < Ib < Ic< Ia Example A copper wire of cross-sectional area 3.00x10-6 m2 carries a current of 10 A. Assuming that each copper atom contributes one free electron to the metal, find the drift speed of the electron in this wire. The density of copper is 8.95 g/cm3. I 10.0C / s vd nqA 8.48 1022 electrons m3 1.6 1019 C 3.00 106 m2 2.46 106 m / s 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 Quiz 2 Suppose a current-carrying wire has a crosssectional 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. Answer Quiz 2 (b). Under steady-state conditions, the current is the same in all parts of the wire. Thus, the drift velocity, given by vd = I/nqA, is inversely proportional to the cross-sectional area 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 Meters in a Circuit – Voltmeter A voltmeter is used to measure voltage (potential difference) Connects to the two ends of the bulb Voltage The electrical potential difference is what causes current to flow. The basic unit of voltage is the volts . ~ Current flow-Amperes (Amps) Ampere: unit of measurement for electrical current. One volt across a resistance of one ohm will produce a flow of one amp. ~ Exercise The capacity of a car battery is usually specified in ampere-hours. A battery rated at, say, 100 A-h should be able to supply 100 A for 1 h, 50 A for 2 h, 25 A for 4 h, 1 A for 100 h, or any other combination yielding a product of 100 A-h. a. How many coulombs of charge should we be able to draw from a fully charged 100 A-h battery? b. How many electrons does your answer to part a require? Answer Assumptions: Battery fully charged. C s a) 100 A 1hr 100 (1hr ) 3600 360000C s hr b) charge on electron: -1.602 ´ 10-19 C no. of electrons = 360 10 22 224.7 10 19 1.602 10 3 Ohm’s Law German Physicist – George Simon Ohm Found that current is inversely proportional to resistance for a given voltage Known as Ohm’s law The Relationship Between Current and Voltage The Relationship Between Current and Resistance Basic Circuit Calculations Using Ohm’s Law to Calculate Current I V R where R = the circuit resistance V = the applied voltage Basic Circuit Calculations Using Ohm’s Law to Calculate Voltage V IR where I = the circuit current R = the circuit resistance Basic Circuit Calculations Using Ohm’s Law to Calculate V Resistance R I where V = the circuit voltage I = the circuit current Resistance Element 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 Ohm’s Law Conductivity and resistivity A I V L V I L V IR R A A I J E A 1 is current density is resistivity Example A nichrome wire of radius 0.321 mm has resistivity of 1.5 x 10-6 Ωm. If a potential difference of 10.0 V is maintained across a 1 m length of the nichrome wire, what is the current? Cross section: 2 A r 0.32110 m 3.24 107 m2 2 Resistance/unit length: 3 R 1.5 10 m 4.6 m 7 2 l A 3.24 10 m 6 V 10.0V I 2.2 A R 4.6 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 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 nonohmic device Ohmic devices vs. non-Ohmic Quiz 4 In Figure, does the resistance of the diode (a) increase or (b) decrease as the positive voltage ΔV increases? Answer Quiz 4 (b). The slope of the line tangent to the curve at a point is the reciprocal of the resistance at that point. Note that as increases, the slope (and hence ) increases. Thus, the resistance decreases. Temperature Dependence of Resistance Define temp coefficient of resistivity 1 d dT If is small and constant o 1 T To Units of is (oC)-1 Example Platinum Resistance Thermometer A resistance thermometer, which measures temperature by measuring the change in the resistance of a conductor, is made of platinum and has a resistance of 50 Ω at 20oC. When the device is immersed in a vessel containing melting indium, its resistance increases to 76.8 Ω. Find the melting point of Indium. α=3.92x10-3(oC)-1 R Ro o T To 137 C Ro o o Since the To= 20 c then T 157 C Microscopic view of Ohm’s Law m 2 ne Free electron Theory Resistivity is a constant Why do old light bulbs give less light than when new? The filament of a light bulb, made of tungsten, is kept at high temperature when the light bulb is on. It tends to evaporate, i.e. to become thinner, thus decreasing in radius, and cross sectional area. Its resistance increases with time. The current going though the filament then decreases with time – and so does its luminosity. Tungsten atoms evaporate off the filament and end up on the inner surface of the bulb. Over time, the glass becomes less transparent and therefore less luminous. Low temperature behavior of resistance Semiconductor Superconducti vity Exercise 1 If a current of 80.0 mA exists in a metal wire, how many electrons flow past a given cross section of the wire in 10.0 min? Sketch the direction of the current and the direction of the electrons’ motion. Exercise 2 If 3.25 × 10−3 kg of gold is deposited on the negative electrode of an electrolytic cell in a period of 2.78 h, what is the current in the cell during that period? Assume that the gold ions carry one elementary unit of positive charge. Exercise 3 An aluminum wire with a crosssectional 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.) Exercise 4 If the current carried by a conductor is doubled, what happens to (a) the charge carrier density? (b) the electron drift velocity?