10/25/2005 Resistors.doc 1/7 Resistors Consider uniform cylinder of mteril with mediocre to poor to r. pthetic conductivity ( ) ˆ This cylinder is centered on the -xis, nd hs length. The surfce re of the ends of the cylinder is. y the cylinder hs current flowing into it (nd thus out of r. it), producing current density ( ) By the wy, this cylinder is commonly referred to s resistor! Q: Wht is its resistnce R of this resistor, given length, cross-section re, nd conductivity? A: Let s first egin with the circuit form of Ohm s Lw: R im tiles The Univ. of Knss Dept. of EEC
10/25/2005 Resistors.doc 2/7 where is the potentil difference etween the two ends of the resistor (i.e., the voltge cross the resistor), nd is the current through the resistor. From electromgnetics, we know tht the potentil difference is: nd the current is: ( r E ) d ds Thus, we cn comine these expressions nd find resistnce R, E r within the resistor, expressed in terms of electric field ( ) nd the current density within the resistor: R E d ds Lets evlute ech integrl in this expression to determine the resistnce R of the device descried erlier! im tiles The Univ. of Knss Dept. of EEC
10/25/2005 Resistors.doc 3/7 1) The voltge is the potentil difference etween point nd point : ( r E ) d Q: But, wht is the electric field E? A: The electric field within the resistor cn e determined from Ohm s Lw: E We cn ssume tht the current density is pproximtely constnt cross the cross section of the cylinder: ˆ Likewise, we know tht the conductivity of the resistor mteril is constnt: As result, the electric field within the resistor is: E ˆ im tiles The Univ. of Knss Dept. of EEC
10/25/2005 Resistors.doc 4/7 Therefore, integrting in stright line long the -xis from point to point, we find the potentil difference to e: 1 E d ˆ ˆ d d 2) We likewise know tht the current through the resistor is found y evluting the surfce integrl: ds ˆ ˆ ds ds Therefore, the resistnce R of this prticulr resistor is: im tiles The Univ. of Knss Dept. of EEC
10/25/2005 Resistors.doc 5/7 R 1 An interesting result! Consider resistor s sort of clogged pipe. ncresing the cross-sectionl re mkes the pipe igger, llowing for more current flow. n other words, the resistnce of the pipe decreses, s predicted y the ove eqution. Likewise, incresing the length simply increses the length of the clog. The current encounters resistnce for longer distnce, thus the vlue of R increses with incresing length. Agin, this ehvior is predicted y the eqution shown ove. For exmple, consider the cse where we dd two resistors together: R 1 1 R 2 2 1 2 im tiles The Univ. of Knss Dept. of EEC
10/25/2005 Resistors.doc 6/7 We cn view this cse s single resistor with length 1 + 2, resulting in totl resistnce of: R totl 1 + 2 1 2 + R + R 1 2 But, this result is not the lest it surprising, s the two resistors re connected in series! Now let s consider the cse where two resistors re connected in different mnner: 1 R1 1 2 R2 2 im tiles The Univ. of Knss Dept. of EEC
10/25/2005 Resistors.doc 7/7 We cn view this s single resistor with totl cross sectionl re of 1 + 2. Thus, its totl resistnce is: R totl ( 1 + 2) ( + ) 1 1 2 1 2 + 1 1 + R1 R2 Agin, this should e no surprise, s these two resistors re connected in prllel. MPORTANT NOTE: The result R is vlid only for the resistor descried in this hndout. Most importntly, it is vlid r ). only for resistor whose conductivity is constnt ( ( ) 1 1 f the conductivity is not constnt, then we must evlute the potentil difference cross the resistor with the more generl expression: E d d im tiles The Univ. of Knss Dept. of EEC