Chapter 25 Current, Resistance, and Electromotive Force


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1 Chapter 25 Current, Resistance, and Electromotive Force Lecture by Dr. Hebin Li
2 Goals for Chapter 25 To understand current and how charges move in a conductor To understand resistivity and conductivity To calculate the resistance of a conductor To learn how an emf causes current in a circuit To calculate energy and power in circuits
3 Current Without electric field, free electrons move randomly in all directions. There is no net flow of charge in any direction. An electric field in a conductor causes charges to flow. A current is any motion of charge from one region to another. We define the current through a crosssection to be the net charge flowing through the area per unit time: I = dq dt The SI unit of current is the ampere (A). 1 A = 1 C/s
4 Direction of current flow A current can be produced by positive or negative charge flow. Conventional current is treated as a flow of positive charges. (We simply define it this way!) The moving charges in metals are electrons. The direction of electron flow is opposite to the direction of current!
5 Current, drift velocity, and current density Microscopic view of electric current: The net motion of an electron can be described by the drift velocity v d. Suppose there are n moving charged particles per unit volume and each particle has a charge q. The charge dq that flows our of the end of the cylinder during time dt is dq = q nav d dt = nqv d Adt So the current is I = dq dt = nqv da The current density is J = I A = nqv d I = dq dt = n q v da J = nqv d Current V.S. Current density
6 Resistivity The resistivity of a material is the ratio of the electric field in the material to the current density it causes: ρ = E J (definition of resistivity), unit: Ω m The conductivity is the reciprocal of the resistivity.
7 Example A 18gauge copper wire has a crosssectional area of m 2. It carries a current of 1.67 A. Find (a) the electricfield magnitude in the wire; (b) the potential difference between two points in the wire 50.0 m apart; (c) the resistance of a 50.0m length of wire.
8 Example A copper wire has a square cross section 2.3 mm on a side. The wire is 4.0 m long and carries a current of 3.6 A. The density of free electrons is /m 3. Find the magnitude of (a) the current density in the wire and (b) the electric field in the wire. (c) how much time is required for an electron to travel the length of the wire?
9 Resistivity: there is more The resistivity is temperature dependent. Over a small temperature range, the resistivity of a metal can be represented approximately by the equation For other materials, the dependence can be different. For some materials, at a given temperature, ρ is constant and does not depend on E. Such a material is call an ohmic conductor or a linear conductor. For some materials, at a given temperature, ρ depends on E. Such a material is call an nonohmic conductor or a nonlinear conductor. (such as semiconductors)
10 Resistance Start with E = ρj Considering a wire with uniform crosssectional area A and length L So we have, V = EL, I = JA V = ρi L A or V = ρl A I The resistance R is defined as the ratio of V to I R = V I
11 Example: A material of resistivity ρ is formed into a solid, truncated cone of height h and radii r 1 and r 2 at either end. (a)calculate the resistance of the cone between the two flat end surfaces. (b)what does the answer to (a) lead to when r 1 = r 2?
12 Electromotive force and circuits An electromotive force (emf) makes current flow. In spite of the name, an emf is not a force. It represents the energy per unit charge (like potential). It has a unit of V. The figures below show a source of emf in an open circuit (left) and in a complete circuit (right).
13 Internal resistance For an ideal emf source, the terminal voltage is V ab = E Real sources of emf actually contain some internal resistance r. The terminal voltage of an emf source is V ab = E Ir. For example, the terminal voltage of the 12V battery shown at the right is less than 12 V when it is connected to the light bulb. The exact voltage depends on the current and internal resistance.
14 Symbols for circuit diagrams Table 25.4 shows the usual symbols used in circuit diagrams.
15 A source in an open circuit What are the readings on idealized voltmeter V and ammeter A? An idealized voltmeter has an infinitely large internal resistance. An idealized ammeter has a zero resistance. For an open circuit: I = 0, so The reading on A is zero. The reading on V is V ab = E Ir = E = 12 V
16 Source in a complete circuit What are the readings on V and A? The circuit is complete, the current in the circuit is I = E R+r = 12 V 4+2 Ω = 2 A The reading on A is 2 A. The reading on V is V ab = E Ir = 12 V 2 A 2 Ω = 8 V
17 Using voltmeters and ammeters
18 Potential changes around a circuit Follow the direction of the current, the potential increases by E going through an emf source. The potential decreases by IR going through a resistor. The net change in potential must be zero for a round trip in a circuit. E Ir IR = 0
19 Example: The circuit shown in the figure below contains two batteries, each with an emf and an internal resistance, and two resistors. Find (a)the current in the circuit (magnitude and direction). (b)the terminal voltage V ab of the 16.0V battery. (c)the potential difference V ac of point a with respect to point c.
20 Energy and power in electric circuits The rate at which energy is delivered to (or extracted from) a circuit element is P = V ab I. The power delivered to a pure resistor is P = V ab I = I 2 R = V ab2 /R. Here we used I = V ab /R.
21 Power output of a source Consider the power output of a battery: The power delivered to the external circuit is P = V ab I The potential V ab = E Ir, plug it into the eqn. above P = E Ir I = EI I 2 r Net power output. Total power converted from chemical to electric energy. the power dissipated in battery.
22 Example: Power input and output In the following circuit, find (a) The rate of energy conversion from chemical to electrical energy. (b) The rate of energy dissipation in the battery. (c) The rate of energy dissipation in the 4Ω resistor. (d) The battery s net power output.
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