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Transcript
Current, Resistance and Circuits Key Contents Electric current Current density and drift velocity Resistance and resistivity Ohm’s law Power in electric circuits The emf device Single-loop circuits Multi-loop circuits RC circuits Electric Current An electric current is a stream of moving charges. However, not all moving charges constitute an electric current. To have that, here must be a net flow of charge through a surface. Consider a flow of water through a garden hose. The flow of water through a garden hose represents the directed flow of positive charge (the proton in the water molecules) at a rate of perhaps several million coulombs per second. There is however no net transport of charge because there is a parallel flow of negative charge (the electrons in the water molecules) of exactly the same amount moving in exactly the same direction! Electric Current: the most common example The free electrons (conduction electrons) in an isolated length of copper wire are in random motion at speeds of the order of 106 m/s. If you pass a hypothetical plane through such a wire, conduction electrons pass through it in both directions at the rate of many billions per second—but there is no net transport of charge and thus no current through the wire. However, if you connect the ends of the wire to a battery, you slightly bias the flow in one direction, with the result that there now is a net transport of charge and thus an electric current through the wire. Electric Current The figure shows a section of a conductor, part of a conducting loop in which current has been established. If charge dq passes through a hypothetical plane (such as aa’) in time dt, then the current i through that plane is defined as: The charge that passes through the plane in a time interval extending from 0 to t is: Electric Current Under steady-state conditions, the current is the same for planes aa’, bb’, and cc’ and for all planes that pass completely through the conductor, no matter what their location or orientation. The SI unit for current is the coulomb per second, or the ampere (A): Electric Current: Conservation of Charge, and Direction of Current Current Density The magnitude of current density, J, is equal to the current per unit area through any element of cross section. It has the same direction as the current. This is a local property. If the current is uniform across the surface and parallel to dA, then J is also uniform and parallel to dA, and then Here, A is the total area of the surface. The SI unit for current density is the ampere per square meter (A/m2). Current Density Figure 26-4 shows how current density can be represented with a similar set of lines, which we can call streamlines. The current, which is toward the right, makes a transition from the wider conductor at the left to the narrower conductor at the right. Since charge is conserved during the transition, the amount of charge and thus the amount of current cannot change. However, the current changes—it is greater narrower conductor. density in the Current Density, Drift Speed What really happens with an electric current flowing in a conducting wire? Current Density, Drift Speed When a conductor has a current passing through it, the electrons move randomly, but they tend to drift with a drift speed vd in the direction opposite that of the applied electric field that causes the current. The drift speed is tiny compared with the speeds in the random motion. In the figure, the equivalent drift of positive charge carriers is in the direction of the applied electric field, E. If we assume that these charge carriers all move with the same drift speed vd and that the current density J is uniform across the wire’s crosssectional area A, then the number of charge carriers in a length L of the wire is nAL. Here n is the number of carriers per unit volume. Current Density, Drift Speed The total charge of the carriers in the length L, each with charge e, is then The total charge moves through any cross section of the wire in the time interval is the current Resistance and Resistivity We determine the resistance between any two points of a conductor by applying a potential difference V between those points and measuring the current i that results. The resistance R is then The SI unit for resistance that follows from Eq. 26-8 is the volt per ampere. This has a special name, the ohm (symbol W): In a circuit diagram, we represent a resistor and a resistance with the symbol Resistance and Resistivity Resistance and Resistivity The resistivity, r, of a resistor is defined as: This is a local property. The SI unit for r is W m. The conductivity s of a material is the reciprocal of its resistivity: Resistance and Resistivity, Variation with Temperature The relation between temperature and resistivity for copper—and for metals in general—is fairly linear over a rather broad temperature range. For such linear relations we can write an empirical approximation that is good enough for most engineering purposes: r = r0 (1+ a (T -T0 )) Resistance and Resistivity Calculating Resistance from Resistivity Ohm’s Law Ohm’s Law Power in Electric Circuits Charge dq moves through a decrease in potential of magnitude V, and thus its electric potential energy decreases in magnitude by the amount The power P associated with that transfer is the rate of transfer dU/dt, given by The unit of power is the volt-ampere (V A). Example, Rate of Energy Dissipation in a Wire Carrying Current Pumping Charges In order to produce a steady flow of charge through a resistor, one needs a “charge pump”, a device that—by doing work on the charge carriers—maintains a potential difference between a pair of terminals. Such a device is called an emf device, which is said to provide an emf. (emf stands for electromotive force) Pumping Charges A common emf device is the battery. The emf device that most influences our daily lives is the electric generator, which, by means of electrical connections (wires) from a generating plant, creates a potential difference in our homes and workplaces. Some other emf devices known are solar cells, fuel cells. An emf device does not have to be an instrument—living systems, ranging from electric ceels and human beings to plants, have physiological emf devices. Work, Energy, and Emf In any time interval dt, a charge dq passes through any cross section of the circuit shown, such as aa’. This same amount of charge must enter the emf device at its low-potential end and leave at its high-potential end. The emf device must do an amount of work dW on the charge dq to force it to move in this way. We define the emf of the emf device in terms of this work: Work, Energy, and Emf An ideal emf device is one that has no internal resistance to the internal movement of charge from terminal to terminal. The potential difference between the terminals of an ideal emf device is exactly equal to the emf of the device. A real emf device, such as any real battery, has internal resistance to the internal movement of charge. When a real emf device is not connected to a circuit, and thus does not have current through it, the potential difference between its terminals is equal to its emf. However, when that device has current through it, the potential difference between its terminals differs from its emf. Calculating the Current in a Single-Loop Circuit P =i 2R (dissipation in the resistor) The work that the battery does on this charge, is Calculating the Current in a Single-Loop Circuit Potential Method Calculating the Current in a Single-Loop Circuit Potential Method Other Single-Loop Circuits Internal Resistance The figure above shows a real battery, with internal resistance r, wired to an external resistor of resistance R. According to the potential rule Other Single-Loop Circuits Resistances in Series Potential between two points Going counterclockwise from a: Going clockwise from a: Potential across a real battery If the internal resistance r of the battery in the previous case were zero, V would be equal to the emf of the battery—namely, 12 V. However, since r =2.0W , V is less than that emf. The result depends on the value of the current through the battery. If the same battery were in a different circuit and had a different current through it, V would have some other value. Grounding a Circuit This is the same example as in the previous slide, except that battery terminal a is grounded in Fig. 27-7a. Grounding a circuit usually means connecting the circuit to a conducting path to Earth’s surface, and such a connection means that the potential is defined to be zero at the grounding point in the circuit. In Fig. 27-7a, the potential at a is defined to be Va =0. Therefore, the potential at b is Vb =8.0 V. Power, Potential, and Emf The net rate P of energy transfer from the emf device to the charge carriers is given by: where V is the potential across the terminals of the emf device. But But Pr is the rate of energy transfer to thermal energy within the emf device: The rate Pemf at which the emf device transfers energy both to the charge carriers and to internal thermal energy is then , therefore Example, Single loop circuit with two real batteries: Example, Single loop circuit with two real batteries, cont.: Multi-loop Circuits At junction d in the circuit This rule is often called Kirchhoff’s junction rule (or Kirchhoff’s current law). For the left-hand loop, For the right-hand loop, And for the entire loop, Multi-loop Circuits, Resistors in Parallel: where V is the potential difference between a and b. From the junction rule, Multi-loop Circuits Example, Resistors in Parallel and in Series Example, Resistors in Parallel and in Series, cont.: Example, Multi-loop circuit and simultaneous loop equations: Ammeter and Voltmeter An instrument used to measure currents is called an ammeter. It is essential that the resistance RA of the ammeter be very much smaller than other resistances in the circuit. A meter used to measure potential differences is called a voltmeter. It is essential that the resistance RV of a voltmeter be very much larger than the resistance of any circuit element across which the voltmeter is connected. RC Circuits Charging a Capacitor (# q =0, @ t =0) RC Circuits Time Constant The product RC is called the capacitive time constant of the circuit and is represented with the symbol t: At time t= t =( RC), the charge on the initially uncharged capacitor increases from zero to: RC Circuits Discharging a Capacitor Assume that the capacitor of the figure is fully charged to a potential V0 equal to the emf of the battery. At a new time t =0, switch S is thrown from a to b so that the capacitor can discharge through resistance R. i= dq q Á = - 0 e-t/RC = - e-t/t dt RC R Fig. 27-16 (b) This shows the decline of the charging current in the circuit. The curves are plotted for R =2000 W, C =1 mF, and emf =10 V; the small triangles represent successive intervals of one time constant t.