Measurement of Electrical Resistance and Ohm's Law

Transcription

1 Physics Labratry Manual _ Lyd LABORATORY Measurement f Electrical Resistance and Ohm's Law OBJECTIVES Define the cncept f electrical resistance using measurements f the vltage acrss and current in a wire cil. Investigate the dependence f the resistance n the length, crss-sectinal area, and resistivity f the wire. Investigate the equivalent resistance f series and parallel resistrs. EQUIPMENT LIST Resistance cils (standard set available frm Sargent-Welch r Central Scientific cnsisting f 10 m and 20 m length f cpper and German silver wire) Direct current ammeter (0-2 A), direct current vltmeter (0-30 V, preferably digital readut) Direct current pwer supply (0-20 V at 1 A) THEORY If a vltage V is applied acrss an element in an electrical circuit, the current I in the element is determined by a quantity knwn as the resistance R. The relatinship between these three quantities serves as a definitin f resistance. V R= I (Eq. 1) The units f resistance are vlt/ampere, which are given the name hm. The symbl fr hm is Q. Sme circuit elements bey a relatinship knwn as Ohm's Law. Fr these elements the quantity R is a cnstant fr different values f V. If a circuit element beys Ohm's Law, when the vltage V is varied the current I will als vary, but the rati VII shuld remain cnstant. In this labratry we will perfrm measurements n five cils f wire t investigate if they bey Ohm's Law. We als will determine the resistance f the cils. The resistance f any bject t electrical current is a functin f the material frm which it is cnstructed, the length, the crss-sectinal area, and the temperature f the bject. At cnstant temperature the resistance R is given by THOMSON 2008 Thmsn Brks/Cle, a part fthe Thmsn Crpratin. Thmsn, the Star lg, and Brks/Cle are trademarks used herein under license. ALL RIGHTS RESERVED. N part f Ihis wrk * _cvered by the cpyright heren maybe reprcuced r used In any frm r by any means--graphic, eiectrnic, r mechanicaf, including phtcp0ng, recrding, taping, web distributin, infrmalln BROOKS/COLE strage and retrieval systems, r in any ther manner-withut the written permissin f the publisher. 279

2 280 Physics Labratry Manual Lyd Figure 28-1 Resistrs in series. Figure 28-2 Resistrs in parallel. L R=p"A (Eq.2) where R is the resistance (Q), L is the length (m), A is the crss-sectinal area (mz), and p is a cnstant dependent upn the material called the resistivity (Q-m). Actually p is a functin f temperature, and if the temperature f the cils f wire rises as a result f the current in them, this may be a surce f errr. Circuit elements in an electrical circuit can be cnnected in series r parallel. Three resistrs (R1, Rz, and R3) are cnnected in series as shwn in Figure Fr resistrs in series the current is the same fr all the resistrs, but the vltage drp acrss each resistr is different. Fr resistrs in series the equivalent resistance Re f the three resistrs is given by (Eq.3) The same three resistrs are shwn cnnected in parallel in Figure Fr resistrs in parallel the current is different in each resistr, but the vltage acrss each resistr is the same. In this case the equivalent resistance Re f the three resistrs in terms f the individual resistrs is given by (Eq.4) One f the bjectives f this labratry will be t bserve the behavir f resistrs in series and parallel. EXPERIMENTAL PROCEDURE 1. Cnnect the ammeter A, the vltmeter V, and the pwer supply PS t the first resistr as shwn in Figure The basic circuit is the pwer supply in series with a resistr. T measure the current in the resistr, the ammeter is placed in series. T measure the vltage acrss the resistr, the vltmeter is placed in parallel. 2. Vary the current thrugh resistr Rl in steps f A up t A. Fr each specified value f the current, measure the vltage acrss the resistr and recrd the values in Data Table 1. The resistrs will heat up and may be damaged by allwing current in them fr lng perids f time. Measurements shuld be made quickly at each value f the current. APPLY VOLTAGE ONLY WHEN DATA ARE BEING TAKEN. 3. Repeat Step 2 fr each f the five resistrs. Fr each resistr the ammeter must be in series with that resistr and the pwer supply, and the vltmeter must be in parallel with the resistr. Recrd all values in Data Table Cnnect the first fur resistrs in series t measure the equivalent resistance f the cmbinatin. Use tw values f current, A and A, and measure the value f the vltage fr each f these values f current. Recrd the vltage in Data Table 2.

3 Labratry 28 Measurement f Electrical Resistance and Ohm's Law 281 Ammeter Pwer Supply Figure 28-3 Measurement f current and vltage fr resistr RI' Pwer Supply Ammeter Figure 28-4 Resistrs RI and R2 in parallel. 5. Measure the vltage acrss the cmbinatin f R2, R3, and R4 in series fr currents f 0.500A and 1.000A and recrd the values in Data Table Cnnect RI and R2 in parallel as shwn in Figure 28-4 and measure the vltage acrss the cmbinatin fr current values f A and A and recrd in Data Table Cnnect RI and R3 in parallel as shwn in Figure 28-5 and measure the vltage fr current values f A and A and recrd in Data Table Cnnect R2 and R3 in parallel and perfrm the same measurements as described in Steps 6 and 7. Recrd the results in Data Table 2. Pwer Supply Figure 28-5 Resistrs RI and R3 in parallel. Ammeter

4 282 Physics Labratry Manual Lyd CALCULATIONS 1. The first fur cils are made f cpper with resistivity f p = 1.72 X 10-8 Q-m. The fifth cil is made f an ally called German silver with resistivity f p =28.0 X 10-8 Q-m. The first, secnd, and fifth cils are 10.0 m lng, and the third and furth cils are 20.0 m lng. The diameters f the first, third, and fifth cils are m, and the diameters f the secnd and furth cils are m. Use these values in Equatin 2 t calculate the value f the resistance fr each f the five cils and recrd the results in Calculatins Table 1 as the theretical values fr the resistance Rthe. 2. If Equatin 1 is slved fr V, the result is V =IR. There is a linear relatinship between the vltage and the current, and the slpe f V versus I will be the resistance R. Perfrm a linear least squares fit t the data in Data Table 1 with Vas the vertical axis and I as the hrizntal axis. Recrd in Calculatins Table 1 the slpe f the fit fr each resistr as the experimental value fr the resistance Rexp' recrd the value f the crrelatin cefficient r fr each f the fits. 3. Calculate the percentage errr in the values f Rexp cmpared t the values f Rthe fr the five resistrs and recrd the results in Calculatins Table Fr the data f Data Table 2 calculate the values f the equivalent resistance fr the varius series and parallel cmbinatins listed in the table as the value f the measured vltage divided by the apprpriate current. Calculate and recrd the mean f the tw trials as (Re )exp in Calculatins Table Equatins 3 and 4 give the theretical expressins fr equivalent resistance fr series and parallel cmbinatins f resistance. Calculate a theretical value fr the equivalent resistance fr each series and parallel cmbinatin measured in Data Table 2. Fr the values f the individual resistances R1, Rz, and R3 in Equatin 3 and 4, use the experimental values determined frm the fit t the data n the individual resistrs. Recrd this theretical value fr the equivalent resistance in each case as (Re)the in Calculatins Table Calculate the percentage difference between the values f (Re )exp and (Re)the fr each f the series and parallel cmbinatins measured and recrd the results in Calculatins Table 2. Als GRAPHS 1. Cnstruct graphs f the data in Data Table 1 with Vas the vertical axis and I as the hrizntal axis. Use nly ne piece f graph paper fr all five resistrs, making five small graphs n that ne sheet. Chse different scales fr each graph if needed, but make the five graphs as large as pssible while still fitting n ne page. Als shw n each small graph the straight line fr the linear least squares fit.

5 Name Sectin Date LAB 0 RAT 0 R Y 2 8 Measurement f Electrical Resistance and Ohm's Law PRE-LABORATORY ASSIGNMENT 1. Three resistrs RI, Rz, and R3 are cnnected in series with RI <Rz <R3' Chse all crrect answers belw. The ttal resistance f the cmbinatin is (a) less than Rv (b) less than R3, (c) greater than R3, (d) greater than 3 R3, (e) equal t RI + Rz + R3. 2. Tw resistrs RI and Rz are cnnected in parallel with RI < Rz. Chse all crrect answers belw. The ttal resistance f the cmbinatin is (a) less than Rv (b) less than Rz, (c) greater than Rz, (d) greater than 2 Rz, (e) equal t (RIRz)/(RI +Rz). 3. A wire f length LI and diameter dl has resistance RI' A secnd wire f the same material has length Lz = 2 LI and diameter dz = 2 dl The resistance f wire tw is Rz. Chse the crrect value fr Rz. (a) Rz = RI, (b) Rz = 2RI, (c) Rz = YzRI, (d) Rz = 4RI' 4. If a circuit element carries a current f 3.71 A, and the vltage drp acrss the element is 8.69V, what is the resistance f the circuit element? Shw yur wrk. R= Q " Q. '" -'" 5. A resistr is knwn t bey Ohm's Law. When there is a current f 1.72 A in the resistr, it has a vltage drp acrss its terminals f 7.35V. If a vltage f 12.0 V is applied acrss the resistr, what is the current in the resistr? Shw yur wrk. 1= A W c '" E.r: I- N l I (!J a: > 0.. ) 283

6 284 Physics Labratry Manual Lyd 6. The resistivity f cpper is 1.72 x 10-8 O-m. A cpper wire is 15.0m lng, and the wire diameter is em. What is the resistance f the wire? Shw yur wrk. R= 7. A wire f crss-sectinal area 5.00 x 10-6 m2 has a resistance f 1.75 O. What is the resistance f a wire f the same material and length as the first wire, but with a crss-sectinal area f 8.75 x 10-6 m2? Shw yur wrk. R= 8. Three resistrs f resistance 20.00, 30.00, and are cnnected in series. What is their equivalent resistance? Shw yur wrk. R= 9. Three resistrs f resistance 15.00, 25.00, and are cnnected in parallel. What is their equivalent resistance? Shw yur wrk. R=

7 Name Sectin Date Lab Partners LAB 0 RAT 0 R Y 2 8 Measurement f Electrical Resistance and Ohm's Law LABORATORY REPORT Data Table 1 I (A) Calculatins Table 1 Rs R4 R3 Rz Rl Rexp r% Errr Rexp 285

8 I 286 Physics I V (V) ILabratry V IManual I (V) Lyd I Rl R2 R2 R3 R4 Parallel Series Data TableCmbinatin 2 I! I (A) I I (A) Calculatins Table 2 Cmbinatin (Re)the (Re)ex/ % Diff (m )exp (0) (Re)exp 1 (0) Rl R2 R2 R3 R4 Parallel Series SAMPLE 1. Rthe = pl/a = 2. (Re)exp = VII = 3. % Errr = CALCULATIONS 4. (Series) (Re)the = Rl + R2 + R3 + R4 = 5. (Parallel) (Re)the = (1)/(1/R 1 + l/r2) = 6. % Difference =

9 Labratry 28 Measurement f Electrical Resistance and Ohm's Law 287 QUESTIONS 1. D the individual resistrs yu have measured bey Ohm's Law? In answering this questin, cnsider the least squares fits and the graphs yu have made fr each resistr. Remember that linear behavir f V versus I is the prf f hmic behavir. 2. Evaluate the agreement between the theretical values fr the individual resistances and the experimental values. 3. Des yur agreement between the experimental and theretical values f the series cmbinatins f resistrs supprt Equatin 3 as the mdel fr series cmbinatin f resistrs? The agreement is nt expected t be perfect, but yu are t determine if the agreement is reasnable within the expected experimental uncertainty. 4. Evaluate the agreement between the experimental and theretical values f the parallel cmbinatins f resistrs. D the results supprt Equatin 4 as the mdel fr the parallel cmbinatin f resistrs within the expected experimental uncertainty? 5. The first and secnd cils have the same length, and the third and furth cils have the same length. They differ nly in the crss-sectinal area. Accrding t thery, what shuld be the rati f the resistance f the secnd cil t the first and the furth cil t the third? Calculate these ratis fr yur experimental results and cmpare the agreement with the expected rati.

10 288 Physics Labratry Manual Lyd 6. The first and third cils have the same crss-sectinal area, and the secnd and furth cils have the same crss-sectinal area. They differ nly in length. Accrding t thery, what shuld be the rati f the resistance f the third cil t the first and the furth cil t the secnd? Calculate these ratis fr yur experimental results and cmpare the agreement with the expected rati.