10/25/2005 Section 5_2 Conductors empty.doc 1/3 Jim Stiles The Univ. of Kansas Dept. of EECS 5-2 Conductors Reading Assignment: pp. 122-132 We have been studying the electrostatics of free-space (i.e., a vacuum). But, the universe is full of stuff! Q: Does stuff (material) affect our electrostatics knowledge? A: HO: Dielectrics and Conductors A. Ohm’s Law So, in conductors, applying an electric field ()rE will cause current ()rJ to form. Q: A: HO: Ohm’s Law10/25/2005 Section 5_2 Conductors empty.doc 2/3 Jim Stiles The Univ. of Kansas Dept. of EECS B. Resistance Q: I thought Ohm’s Law was R=V/I ? A: HO: Resistors C. Perfect Conductors Consider now a perfect conductor (i.e., σ=∞). Q: Does this mean that current density ()rJ is likewise infinite? A: HO: Perfect Conductors D. Kirchoff’s Voltage Law Recall since a static field ()rE is conservative: ()0Crd⋅=∫E Av10/25/2005 Section 5_2 Conductors empty.doc 3/3 Jim Stiles The Univ. of Kansas Dept. of EECS 0nvV=∑ HO: Kirchoff’s Voltage Law E. Joule’s Law Conducting material will absorb energy—it will heat up! HO: Joule’s Law10/25/2005 Dielectrics and Conductors.doc 1/3 Jim Stiles The Univ. of Kansas Dept. of EECS Dielectrics and Conductors Consider a very simple model of an atom: Say an electric field is applied to this atom. Note the field will apply a force on both the positively charged nucleus and the negatively charged electron. However, these forces will move these particles in opposite directions! Two things may occur. In the first case, the atom may stretch, but the electron will remain bound to the atom: + - - + = electron (negative charge) = nucleus (positive charge)10/25/2005 Dielectrics and Conductors.doc 2/3 Jim Stiles The Univ. of Kansas Dept. of EECS Note, an electric dipole has been created ! For the second case, the electron may be break free from the atom, creating a positive ion and a free electron. We call these free charges, and the electric field will cause them to move in opposite directions. Moving charge! We know what moving charge is. Moving charge is electric current ()rJ . These two examples provide a simple demonstration of what occurs when an electric field is applied to some material (e.g., plastic, copper, water, oxygen). + - ()rE+u−u+ - ()rEp10/25/2005 Dielectrics and Conductors.doc 3/3 Jim Stiles The Univ. of Kansas Dept. of EECS 1) Materials where the charges remain bound (and thus dipoles are created) are called insulator (or dielectric) materials. 2) Materials where the electrons are free to move are called conductors. Of course, materials consists of molecules with many electrons, and in general some electrons are bound and some are free. As a result, there are no perfect conductors or perfect insulators, although some materials are very close! Additionally, some materials are lie between being a good conductor or a good insulator. We call these materials semi-conductors (e.g., Silicon).10/25/2005 Ohms Law.doc 1/3 Jim Stiles The Univ. of Kansas Dept. of EECS Ohm’s Law Recall that a positively charge particle will move in the direction of an electric field, whereas a negative chare will move in the opposite direction. Both types of charge, however, result in current moving in the same direction as the electric field: Q: So, the direction of current density ()rJand electric field ()rE are the same. The question then is, how are their magnitudes related ? A: They are related by Ohm’s Law: ()()()rrrσ=JE The scalar value ()rσ is called the material’s conductivity. Note the units of conductivity are: ()rE+ - +u−u()rJ10/25/2005 Ohms Law.doc 2/3 Jim Stiles The Univ. of Kansas Dept. of EECS ()()()()()()()()()122rAmpsVoltsr rmmrAmpsm rmVoltsrAmps rVoltsmr1 rOhmmσ−⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞=⎜⎟⋅⎝⎠⎛⎞=⎜⎟⋅⎝⎠JEJEJEJE In other words, the unit of conductivity is conductance/unit length. We emphasize that conductivity ()rσ is a material parameter. For example, the conductivity of copper is: 7158x10m.copperσ⎡⎤=⎢⎥Ω⎣⎦ and the conductivity of polyethylene (a plastic) is: -12115x10m.polyethyleneσ⎡⎤=⎢⎥Ω⎣⎦ Note the vast difference in conductivity between these two materials. Copper is a conductor and polyethylene is an insulator.10/25/2005 Ohms Law.doc 3/3 Jim Stiles The Univ. of Kansas Dept. of EECS Georg Simon Ohm (1789-1854) was the German physicist who in 1827 discovered the law that the current flow through a conductor is proportional to the voltage and inversely proportional to the resistance. Ohm was then a professor of mathematics in Cologne. His work was coldly received! The Prussian minister of education announced that "a professor who preached such heresies was unworthyto teach science." Ohm resigned his post, went into academic exile for several years, and then left Prussia and became a professor in Bavaria. From: www.ee.umd.edu/~taylor/frame2.htm10/25/2005 Resistors.doc 1/7 Jim Stiles The Univ. of Kansas Dept. of EECS Resistors Consider a uniform cylinder of material with mediocre to poor to pathetic conductivity ()rσσ=. This cylinder is centered on the z-axis, and has length A . The surface area of the ends of the cylinder is S. Say the cylinder has current I flowing into it (and thus out of it), producing a current density ()rJ . By the way, this cylinder is commonly referred to as a resistor! Q: What is its resistance R of this resistor, given length A, cross-section area S, and conductivity σ? A: Let’s first begin with the circuit form of Ohm’s Law: VRI= ()rˆzJa=JIA S a b z10/25/2005 Resistors.doc 2/7 Jim Stiles The Univ. of Kansas Dept. of EECS where V is the potential difference between the two ends of the resistor (i.e., the voltage across the resistor), and I is the current through the resistor. From electromagnetics, we know that the potential difference V is: ()rbabaVVd==⋅∫E A and the current I is: ()rSIds=⋅∫∫J Thus, we can combine these expressions and find resistance R, expressed in terms of electric field ()rE within the resistor, and the current density ()rJ within the resistor: ()()rrbaSdVRIds⋅==⋅∫∫∫EJA Lets evaluate each integral in this expression to determine the resistance R of the device described