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Lecture 12 Current & Resistance

Lecture 12 Current & Resistance. Electric Current. Definition : the current is the rate at which charge flows through this surface. Given an amount of charge, D Q, passing through the area A in a time interval D t, the current is the ratio of the charge to the time interval.

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Lecture 12 Current & Resistance

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  1. Lecture 12 Current & Resistance General Physics II, Lec 12, By/ T.A. Eleyan

  2. 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. General Physics II, Lec 12, By/ T.A. Eleyan

  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? General Physics II, Lec 12, By/ T.A. Eleyan

  4. High current density Small current density Current Density • When we care only about the total current I in a conductor, we do not have to worry about its shape. • However, sometimes we want to look in more detail at the current flow inside the conductor. Similar to what we did with Gauss’ Law (electric flux through a surface), we can consider the flow of charge through a surface. To do this, we consider (charge per unit time) per unit area, i.e. current per unit area, or current density. The units are amps/square meter (A/m2). • Current density is a vector (since it has a flow magnitude and direction). We use the symbol . The relationship between current and current density is General Physics II, Lec 12, By/ T.A. Eleyan

  5. Current and Drift Speed • Consider the current on a conductor of cross-sectional area A. General Physics II, Lec 12, By/ T.A. Eleyan

  6. 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. General Physics II, Lec 12, By/ T.A. Eleyan

  7. 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. General Physics II, Lec 12, By/ T.A. Eleyan

  8. 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 General Physics II, Lec 12, By/ T.A. Eleyan

  9. 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. General Physics II, Lec 12, By/ T.A. Eleyan

  10. 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. General Physics II, Lec 12, By/ T.A. Eleyan

  11. 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 General Physics II, Lec 12, By/ T.A. Eleyan

  12. The current–potential difference curve for an ohmic material. The curve is linear, and the slope is equal to the inverse of the resistance of the conductor. • A nonlinear current–potential difference curve for a junction diode. This device does not obey Ohm’s law. General Physics II, Lec 12, By/ T.A. Eleyan

  13. 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. General Physics II, Lec 12, By/ T.A. Eleyan

  14. 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. General Physics II, Lec 12, By/ T.A. Eleyan

  15. Resistivity - Units • Resistance expressed in Ohms, • Length in meter. • Area are m2, • Resistivity thus has units of W m. General Physics II, Lec 12, By/ T.A. Eleyan

  16. 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 General Physics II, Lec 12, By/ T.A. Eleyan

  17. 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? General Physics II, Lec 12, By/ T.A. Eleyan

  18. 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 0.1Ω 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 0.24Ω.  Calculate the conductivity of the material. General Physics II, Lec 12, By/ T.A. Eleyan

  19. 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. General Physics II, Lec 12, By/ T.A. Eleyan

  20. For most metals, resistivity increases approx. linearly with temperature. • 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. General Physics II, Lec 12, By/ T.A. Eleyan

  21. 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. Using a=3.92x10-3(oC)-1 from table. Ro=50.0 W. To=20oC. R=76.8 W. General Physics II, Lec 12, By/ T.A. Eleyan

  22. [Q] 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? General Physics II, Lec 12, By/ T.A. Eleyan

  23. 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)! Resistance versus temperature for a sample of mercury (Hg). The graph follows that of a normal metal above the critical temperature Tc. The resistance drops to zero at Tc, which is 4.2 K for mercury. General Physics II, Lec 12, By/ T.A. Eleyan

  24. 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) General Physics II, Lec 12, By/ T.A. Eleyan

  25. 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 General Physics II, Lec 12, By/ T.A. Eleyan

  26. Power • Compute rate of energy loss (power dissipated on the resistor) • Use Ohm’s law • Units of power : watt delivered energy: kilowatt-hours General Physics II, Lec 12, By/ T.A. Eleyan

  27. [Q] Calculate Determine the total current drawn by all the devices in the circuit in the figure. General Physics II, Lec 12, By/ T.A. Eleyan

  28. 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 General Physics II, Lec 12, By/ T.A. Eleyan

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