Chapter 25 Current Resistance, and Electromotive Force 1 Current In previous chapters we investigated the properties of charges at rest. In this chapter we want to investigate the properties of charges in motion. An electric current consists of charges in motion from one region to another. If the charges follow a conducting path that forms a closed loop, the path is called an electric circuit. 1.1 The direction of current flow When a constant, steady electric field E is established inside a conductor, the free electrons collectively move in a direction due to a steady force F = q E. Figure 1: This figure shows the random motion of electrons in a conductor without an E field. The second figure shows the collective motion of electrons in a conductor when an E field is present. 1
The collective motion of electrons in a conductor is characterized by their drift velocity v d. While the random motion of electrons has a very fast average speed of 10 6 m/s, the drift speed is very slow, often on the order of 10 4 m/s. The electric field is set up throughout the conductor at velocities approaching the speed of light, v c. Figure 2: The first figure (a) shows the direction of the conventional current, the flow of positive charges. The second figure (b) shows the flow of electrons in a metallic conducting moving in the opposite direction with respect to the current. 2
1.2 Current, Drift Velocity, and Current Density I = Q T = dq dt (the definition of current) (1) Figure 3: The current I is the time rate of charge transfer through the cross-sectional area A. The current is in the same direction as E. dq = q(nav d dt) = nqv d A dt I = dq dt = nqv d A The current per unit cross-sectional area is called the current density J: J = I A = nqv d (the current density) (2) J = nq v d (the current density in vector form) (3) 3
2 Resistivity The current density J in a conductor depends on the electric field E and on the properties of the material. The dependence can be quite complex, but for some materials, especially metals at a given temperature, J is nearly directly proportional to E, and the ratio of the magnitudes of E and J is constant. When materials exhibit this kind of behavior, it is said that they follow Ohm s Law. J = σ E = 1 ρ E (4) where σ is the conductivity and ρ is the resistivity shown in the table. Figure 4: This table shows the resistivities of various materials at room temperature (20 o C) 4
2.1 Resistivity and Temperature Figure 5: These figures show how the resistivity depends with the absolute temperature T for (a) a normal metal, (b) a semiconductor, and (c) a superconductor. The temperature dependence of the resistivity ρ can be written as: ρ(t ) = ρ o [1 + α (T T o )] (5) where α is the temperature coefficient of resistivity. 5
Figure 6: This table shows the coefficient of resistivities for various materials near room temperature. 3 Resistance For a conductor with resistivity ρ, the current density J and the electric field is E are related by the following equation: E = ρ J (6) Figure 7: This figure shows a conductor with uniform cross sectional area A. The current density is uniform over any cross section, and the electric field is constant along the length. Let V bet the potential difference between higher-potential and the lower-potential ends of the conductor, so that V is positive. V is the voltage across the con- 6
ductor. The direction of the current is always from the higher-potential to the lower-potential end. We can also relate the value of the current I to the potential difference between the ends of the conductor. Using the definition of current density J, we can write the current as I = JA. Likewise, the potential difference V between the ends is V = EL. Rewriting Eq. 6 above, we find: V L = ρi A or V = ( ) ρl A The ratio of V to I for a particular conductor is called its resistance R: I R = V I or V = IR (Ohm s Law) (7) We identify the resistance R to be R = ρl/a. 7
3.1 Interpreting Resistance Figure 8: This resistor has a resistance of 57 kω with an accuracy (tolerance) of ±10%. This is assuming that the color of the 3rd band is orange. Figure 9: This table shows the color codes used for labeling the resistances of resistors. The Resistance is Dependent Upon Temperature R(T ) = R o [1 + α (T T o )] where T 0 is often taken to be 0 o C or 20 o C. The temperature coefficient of resistance is represented by α, and the change in resistance is R 0 α(t T 0 ). 8
Figure 10: This figure shows the current-voltage relationships for two devices. In figure (a) the resistor obeys Ohm s law and the current I is proportional to the voltage V. In figure (b), a semiconductor diode, the current rises steeply with a modest amount of voltage, and the current flows only in the positive direction. 4 Electromotive Force and Circuits In order for a conductor to have a steady current, it must be part of a path that forms a closed loop or complete circuit. Otherwise, opposite charges would eventually build up on the two ends and negate the E field that produced the initial separation of charges. 9
Figure 11: This figure demonstrates why a steady current cannot persist in a section of conductor with a constant external E-field unless it is part of a circuit. Charges eventually accumulate on the two ends of the wire ultimately reducing the E field to E 0. J = 1 ρ E. 10
4.1 Electromotive Force In order to produce a steady current in a circuit, there has to be a device somewhere that acts to raise the potential (measured in volts) from low potential to high potential. Although in the circuit the electrostatic force is trying to push positive charge from high potential to low potential, the device that raises the potential pushes positive charges in the opposite sense (i.e., from low potential to high potential). Thus, the device must provide a force not derived from electrostatic means, buy by some other process. The non-electrostatic force that raises the voltage in a circuit is called an electromotive force, or emf, and is many times written as E. Figure 12: This is a schematic diagram of a source of emf (E) in an open-circuit situation. The electric-field force F e = q E and the nonelectrostatic force F n are shown acting on a positive charge q. The voltage across the battery terminals not connected. The most common source of emf is called a battery which uses an electrochemical process to move charge from low potential to high potential. When the battery is not connected, the voltage measured across the terminals is V ab = E. V ab = E Voltage measured across the battery not connected 11
Figure 13: This is schematic diagram of an ideal emf source in a complete circuit. The electric-field force F e = q E and the nonelectrostatic force F n are shown for a positive charge q. The current is in the direction from a to b in the external circuit and from b to a within the source. 12
4.2 Internal Resistance When the battery is connected to a simple circuit as shown in Fig. 13, current begins to flow through the whole circuit, including the battery. Because there is a small internal resistance r inside the battery, there is a slight voltage drop measured at the battery terminal V ab = E Ir. V ab = E Ir Voltage measured across the battery when connected 4.3 Symbols for Circuit Diagrams Figure 14: This table shows the common symbols found in a a circuit diagram. 13
4.4 Potential Changes around a Circuit Figure 15: This figure shows the potential rises and drops in a circuit. 14
5 Energy and Power in Electric Circuits When a charge q passes through a circuit element, there is a change in potential energy equal to qv ab. The potential energy decreases as the charge falls from potential V a to a lower potential V b. However, the moving charge does not gain kinetic energy (because there is conservation of charge and flux current is constant). In electric circuits we are more interested in the rate at which energy is either delivered to, or extracted from, a circuit element. If the current through the element is I, then a charge dq = I dt passes through the element in a time interval dt. The change in potential energy for this amount of charge crossing a potential difference V ab is du = V ab dq = V ab I dt. The power delivered to the circuit element must be Power = du dt = V ab I (power delivered to a circuit element) (8) Figure 16: This figure shows the power input to the circuit element between a and b. P = (V a V b )I = V ab I. 5.1 Power Input to a Pure Resistance Power = V I = V 2 R = I 2 R (9) 15
5.2 Power Output of a Source Figure 17: This figure shows the energy conversion in a simple circuit. Power output of a source a battery Power = V ab I = (E Ir)I = EI I 2 r 16
5.3 Power Input to a Source Figure 18: This figure shows two sources connected in a simple loop circuit. The source with the large emf delivers energy to the other source. Power input to a source a battery Power = V ab I = (E + Ir)I = EI + I 2 r 17