2/5/07184 Lecture 161 PHY 184 Spring 2007 Lecture 16 Title: Electric Current and Resistance

2/5/07184 Lecture 162AnnouncementsAnnouncements  Homework Set 4 is due tomorrow at 8:00 am.  Midterm 1 will take place in class Thursday, February 8 Will cover Chapters Homework Set You may bring one 8.5 x 11 inch sheet of equations, front and back, prepared any way you prefer Bring a calculator Bring a No. 2 pencil Bring your MSU student ID card  We will post Midterm 1 as Corrections Set 1 after the exam You can re-do all the problems in the Exam You will receive 30% credit for the problems you missed To get credit, you must do all the problems in Corrections Set 1, not just the ones you missed

2/5/07184 Lecture 163ReviewReview  Electric current i is the net charge passing a given point in a given time  The ampere is abbreviated as A and is given by  The current per unit area flowing through a conductor is the current density J  If the current is constant and perpendicular to a surface, then and we can write an expression for the magnitude of the current density

2/5/07184 Lecture 164 Electron Drift Velocity  In a conductor that is not carrying current, the conduction electrons move randomly. (thermal motion)  When current flows through the conductor, the electrons have an additional coherent motion. (drift velocity, v d )  The magnitude of the velocity of random thermal motion is on the order of 10 6 m/s while the magnitude of the drift velocity is on the order of m/s  We can relate the current density J to the drift velocity v d of the moving electrons.

2/5/07184 Lecture 165 Electron Drift Velocity (2)  Consider a conductor with cross sectional area A and electric field E.  Suppose that there are n electrons per unit volume.  The negatively charged electrons will drift in a direction opposite to the electric field.  We assume that all the electrons have the same drift velocity v d and that the current density J is uniform.  In a time interval dt, each electron moves a distance v d dt.  The volume that will pass through area A is then Av d dt; the number of electrons is dn = nAv d dt.

2/5/07184 Lecture 166 Electron Drift Velocity (3)  Each electron has charge e so that the charge dq that flows through the area A in time dt is  So the current is  … and the current density is  The current density and the drift velocity are parallel vectors, pointing in opposite directions. As vectors,

2/5/07184 Lecture 167 Electron Drift Velocity (4)  Consider a wire carrying a current  The physical current carriers are negatively charged electrons.  These electrons are moving to the left in this drawing.  However, the electric field, current density and current are directed to the right. Comments Electrons are negative charges! On top of the coherent motion the electrons have random (thermal) motion.

2/5/07184 Lecture 168 Clicker Question  The figure shows positive charge carriers that drift at a speed v d to the left. In what directions are J and E ? A) J and E point to the right B) J points to the left, E to the right C) J points to the right, E to the left D) J and E point to the left

2/5/07184 Lecture 169 Example - current through a wire (1)  The current density in a cylindrical wire of radius R=2.0 mm is uniform across a cross section of the wire and has the value A/m 2. What is the current i through the outer portion of the wire between radial distances R/2 and R?  J = current per unit area = di / dA R

2/5/07184 Lecture 1610 Example - current through a wire (1)  The current density in a cylindrical wire of radius R=2.0 mm is uniform across a cross section of the wire and has the value A/m 2. What is the current i through the outer portion of the wire between radial distances R/2 and R?  J = current per unit area = di / dA R Area A’ (outer portion) Current through A’

2/5/07184 Lecture 1611 Resistance and Resistivity  Some materials conduct electricity better than others.  If we apply a given voltage across a conductor, we get a large current.  If we apply the same voltage across an insulator, we get very little current (ideal: none).  The property of a material that describes its ability to conduct electric currents is called the resistivity,   The property of a particular device or object that describes it ability to conduct electric currents is called the resistance, R  Resistivity is a property of the material; resistance is a property of a particular object made from that material.

2/5/07184 Lecture 1612 Resistance and Resistivity (2)  If we apply an electric potential difference V across a conductor and measure the resulting current i in the conductor, we define the resistance R of that conductor as  The unit of resistance is volt per ampere.  In honor of George Simon Ohm ( ) resistance has been given the unit ohm, 

2/5/07184 Lecture 1613 Resistance and Resistivity (3)  We will assume that the resistance of the device is uniform for all directions of the current; e.g., uniform metals.  The resistance R of a conductor depends on the material from which the conductor is constructed as well as the geometry of the conductor  First we discuss the effects of the material and then we will discuss the effects of geometry on resistance.

2/5/07184 Lecture 1614ResistivityResistivity  The conducting properties of a material are characterized in terms of its resistivity.  We define the resistivity, , of a material by the ratio  The units of resistivity are E: magnitude of the applied field J: magnitude of the current density

2/5/07184 Lecture 1615 Typical Resistivities  The resistivities of some representative conductors at 20° C are listed in the table below  As you can see, typical values for the resistivity of metals used in wires are on the order of  m.  -cm)

2/5/07184 Lecture 1616ResistanceResistance  Knowing the resistivity of the material, we can then calculate the resistance of a conductor given its geometry. Derivation:  Consider a homogeneous wire of length L and constant cross sectional area A.  … the resistance is

2/5/07184 Lecture 1617 Resistance and resistivity  For a wire,

2/5/07184 Lecture 1618 Clicker Question  You have three cylindrical copper conductors. Rank them according to the current through them, the greatest first, when the same potential difference V is placed across their lengths. A: a, b, c B: a and c tie, then b C: b, a, c D: a and b tie, then c

2/5/07184 Lecture 1619 Clicker Question  You have three cylindrical copper conductors. Rank them according to the current through them, the greatest first, when the same potential difference V is placed across their lengths. B: a and c tie, then b D: a and b tie, then c

2/5/07184 Lecture 1620 Example: Resistance of a Copper Wire  Standard wires that electricians put into residential housing have fairly low resistance.  Question:  What is the resistance of a length of 100 m of standard 12- gauge copper wire, typically used in household wiring for electrical outlets?  Answer:  The American Wire Gauge (AWG) size convention specifies wire cross sectional area on a logarithmic scale.  A lower gauge number corresponds to a thicker wire.  Every reduction by 3 gauges doubles the cross-sectional area.

2/5/07184 Lecture 1621 Example: Resistance of a Copper Wire (2)  The formula to convert from the AWG size to the wire diameter is  So a 12-gauge copper wire has a diameter of 2.05 mm  Its cross sectional area is then  Look up the resistivity of copper in the table …

2/5/07184 Lecture 1622 Clicker Question  A rectangular block of iron has dimensions 2.0cm x 2.0 cm x 10cm. A potential difference is to be applied to the block between parallel sides. What is the ratio of the resistances R(1)/R(2) of the block for the two arrangements (1) and (2). A) B) C) 10 cm 2.0 cm (1) (2) -

2/5/07184 Lecture 1623 Clicker Question  A rectangular block of iron has dimensions 2.0cm x 2.0 cm x 10cm. A potential difference is to be applied to the block between parallel sides. What is the ratio of the resistances R(1)/R(2) of the block for the two arrangements (1) and (2). A) 10 cm 2.0 cm (1) (2) -

2/5/07184 Lecture 1624ResistorsResistors  In many electronics applications one needs a range of resistances in various parts of the circuits.  For this purpose one can use commercially available resistors.  Resistors are commonly made from carbon, inside a plastic cover with two wires sticking out at the two ends for electrical connection.  The value of the resistance is indicated by four color- bands on the plastic capsule.  The first two bands are numbers for the mantissa, the third is a power of ten, and the fourth is a tolerance for the range of values.

2/5/07184 Lecture 1625 Resistors (2)  The number associated with the colors are: black = 0 brown = 1 red = 2 orange = 3 yellow = 4 green = 5 blue = 6 purple = 7 gray = 8 white = 9  In the tolerance band gold means 5% silver means 10% no tolerance band means 20% For example, the single resistor shown here has colors (top to bottom) brown, green, brown and gold Using our table, we can see that the resistance is 15×10 1  = 150  with a tolerance of 5%

2/5/07184 Lecture 1626SummarySummary.. speed of an electron.. resistance to current