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e 1. R. I 1. e 2. I 2. R. I 3. e 3. R. Lecture 11 Current & Resistance. +. +. +. +. +. A. I. Electric Current. Definition: the current is the rate at which charge flows through this surface.

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  1. e1 R I1 e2 I2 R I3 e3 R Lecture 11 Current & Resistance

  2. + + + + + A I Electric Current Definition: the current is the rate at which charge flows through this surface. Given an amount of charge, DQ, passing through the area A in a time interval Dt, the current is the ratio of the charge to the time interval. The SI units of current is the ampere (A). • 1 A = 1 C/s • 1 A of current is equivalent to 1 C of charge passing through the area in a time interval of 1 s.

  3. Example: The amount of charge that passes through the filament of a certain light bulb in 2.00 s is 1.67 c. Find the current in the light bulb. Find no. of electrons? The current density The current density is electric current per unit area

  4. vd q A vdDt Current and Drift Speed • Consider the current on a conductor of cross-sectional area A.

  5. Volume of an element of length Dx is : DV = A Dx. • Let n be the number of carriers per unit of volume. • The total number of carriers in DV is: n A Dx. • The charge in this volume is: DQ = (n A Dx)q. • Distance traveled at drift speed vdby carrier in time Dt: Dx = vd Dt. • Hence: DQ = (n A vd Dt)q. • The current through the conductor: I = DQ/ Dt = n A vd q. • The current density : J = I/A = n vd q.

  6. 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. A = 3.00x10-6 m2 ; I = 10 A, q = 1.6 x 10-19 C. n = 8.48 x 1022 electrons/ m3. [Q] If the current density in a copper wire is equal to 5.8*106A/m2, calculate the drift velocity of the free electrons in this wire.

  7. Drift speeds are usually very small. • Drift speed much smaller than the average speed between collisions. • Electrons traveling at 2.46x10-6 m/s would take 68 min to travel 1m. • So why does light turn on so quickly when one flips a switch? • The info (electric field) travels at roughly 108 m/s… [Q] A silver wire 1 mm in diameter transfers a charge of 65 C in 1 hr, 15 min. Silver contains 5.80 x 1028 free electrons per cubic meter. a) What is the current in the wire? b) What is the magnitude of the drift velocity of the electrons in the wire? Ans. a) 0.0144 A; b) 1.98 x 10-6 m/s

  8. Resistance and Ohm’s Law • When a voltage (potential difference) is applied across the ends of a metallic conductor, the current is found to be proportional to the applied voltage. • In situations where the proportionality is exact, one can write. The proportionality constant R is called resistance of the conductor.

  9. The resistance is defined as the ratio. In SI, resistance is expressed in volts per ampere. A special name is given: ohms Example: if a potential difference of 10 V applied across a conductor produces a 0.2 A current, then one concludes the conductors has a resistance of 10 V/0.2 a = 50 W.

  10. Ohm’s Law • Resistance in a conductor arises because of collisions between electrons and fixed charges within the material. • In many materials, including most metals, the resistance is constant over a wide range of applied voltages. • This is a statement of Ohm’s law. Ohm’s Law

  11. Non-Linear or Non-Ohmic Material Linear or Ohmic Material I I DV DV Most metals, ceramics Semiconductors e.g. devices called diodes

  12. Resistivity • Electrons moving inside a conductor subject to an external potential constantly collide with atoms of the conductor. • They lose energy and are repeated re-accelerated by the electric field produced by the external potential. • The collision process is equivalent to an internal friction. • This is the origin of a material’s resistance.

  13. The resistance of an ohmic conductor is proportional to the its length, l, and inversely proportional to the cross section area, A, of the conductor. • The constant of proportionality r is called the resistivity of the material. • Every material has a characteristic resistivity that depends on its electronic structure, and the temperature. • Good conductors have low resistivity. • Insulators have high resistivity.

  14. Resistivity - Units • Resistance expressed in Ohms, • Length in meter. • Area are m2, • Resistivity thus has units of W m.

  15. Material Resistivity (10-8Wm) Material Resistivity (10-8Wm) Silver 1.61 Bismuth 106.8 Copper 1.70 Plutonium 141.4 Gold 2.20 Graphite 1375 Aluminum 2.65 Germanium 4.6x107 Pure Silicon 3.5 Diamond 2.7x109 Calcium 3.91 Deionized water 1.8x1013 Sodium 4.75 Iodine 1.3x1015 Tungsten 5.3 Phosphorus 1x1017 Brass 7.0 Quartz 1x1021 Uranium 30.0 Alumina 1x1022 Mercury 98.4 Sulfur 2x1023 Resistivity of various materials

  16. Example (a) Calculate the resistance per unit length of a nichrome wire of radius 0.321 m. Cross section: Resistivity (Table): 1.5 x 10-6Wm. Resistance/unit length: (b) If a potential difference of 10.0 V is maintained across a 1.0-m length of the nichrome wire, what is the current?

  17. The reciprocal of the resistivity is called the conductivity, [Q] Stretching changes resistance: A wire of resistance R is stretched uniformly until it is twice its original length. What happens to its resistance? The resistance of the wire increases by a factor of four if the length increases twice [Q] Speaker wires: Suppose you want to connect your stereo to remote speakers. (a) If each wire must be 20m long, what diameter copper wire should you use to keep the resistance less than 0.1-W per wire? (b) If the current on each speaker is 4.0A, what is the voltage drop across each wire? [Q] A 2.4m length of wire that is 0.031cm2 in cross section has a measured resistance of 0.24W.  Calculate the conductivity of the material.

  18. Temperature Variation of Resistance • The resistivity of a metal depends on many (environmental) factors. • The most important factor is the temperature. • For most metals, the resistivity increases with increasing temperature. • The increased resistivity arises because of larger friction caused by the more violent motion of the atoms of the metal.

  19. For most metals, resistivity increases approx. linearly with temperature. r T Metallic Conductor • ris the resistivity at temperature T (measured in Celsius). • rois the reference resistivity at the reference temperature To (usually taken to be 20 oC). • ais a parameter called temperature coefficient of resistivity. For a conductor with fixed cross section. r T Superconductor

  20. Example: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 50.0 W at 20oC. When the device is immersed in a vessel containing melting indium, its resistance increases to 76.8 W. Find the melting point of Indium. Solution: Using a=3.92x10-3(oC)-1 from table. Ro=50.0 W. To=20oC. R=76.8 W.

  21. A resistance thermometer using a platinum wire is used to measure the temperature of a liquid. The resistance is 2.42 ohms at 0oC, and when immersed in the liquid it is 2.98 ohms. The temperature coefficient of resistivity of platinum is 0.0038 . What is the temperature of the liquid? • Solution:

  22. Superconductivity • 1911: H. K. Onnes, who had figured out how to make liquid helium, used it to cool mercury to 4.2 K and looked at its resistance: • At low temperatures the resistance of some metals0, measured to be less than 10-16•ρconductor (i.e., ρ<10-24 Ωm)!

  23. V = IR + - I Electrical energy and power • In any circuit, battery is used to induce electrical current • chemical energy of the battery is transformed into kinetic energy of mobile charge carriers (electrical energy gain) • Any device that possesses resistance (resistor) present in the circuit will transform electrical energy into heat • kinetic energy of charge carriers is transformed into heat via collisions with atoms in a conductor (electrical energy loss) D C A B

  24. Electrical energy • Consider circuit on the right in detail • AB: charge gains electrical energy form the battery (battery looses chemical energy) • CD: electrical energy lost (transferred into heat) • Back to A: same potential energy (zero) as before • Gained electrical energy = lost electrical energy on the resistor C B A D

  25. Power • Compute rate of energy loss (power dissipated on the resistor) • Use Ohm’s law • Units of power: SI: watt delivered energy: kilowatt-hours

  26. [Q] Calculate Determine the total current drawn by all the devices in the circuit in the figure.

  27. Example A high-voltage transmission line with resistance of 0.31 W/km carries 1000A , starting at 700 kV, for a distance of 160 km. What is the power loss due to resistance in the wire? • Observations: • Given resistance/length, compute total resistance • Given resistance and current, compute power loss Now compute power

  28. (1) An aluminum wire carrying a current has a diameter 0.800 mm. The electric field in the wire is 0.640 V/m. What is: a) the current carried by the wire? b) the potential difference between two points in the wire 12.0 m apart? C) the resistance of a 12.0 m length of the wire? Ans. a) 12.2 A; b) 7.68 V; c) 0.628 Ω (2) A copper wire has resistance 5 Ohms. Given that the resistivity of silver is 85 percent of the resistivity of copper, what is the resistance of a silver wire three times as long with twice the diameter? (3) A current of 5A exists in a 10 W resistor for 4min. (a) How many coulombs, and (b) how many electrons pass through any cross section of the resistor in this time? (4) What is the resistance of a device that operates with a current of 7A when the applied voltage is 110V?

  29. (5) Thermal energy is developed in a resistor at a rate of 100W when the current is 3.0A. What is the resistance in ohms? (6) A 1250W radiant heater is constructed to operate at 115V. (a) What will be the current in the heater? (b) What is the resistance of the heating coil?

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