P Physics Relatinships between f,, and. Relatinships Between Frequency, apacitance, nductance and Reactance. Purpse: T experimentally verify the relatinships between f, and. The data cllected will lead t the cnclusin the is inversely prprtinal t and f. This activity will als verify the capacitive reactance frmula. Equipment: Patch rds, Alligatr lips 2 DMMs apacitance Bx Functin Generatr Decade Resistance Bx Tridal inductrs (5H) R meter Thery: A battery cnnected in series with a resistr frms a simple circuit where Ohm's law can be applied. Using Ohms law, the current in the circuit is given by Where ε ο is the battery vltage, and R is the resistance. ε = Eq. 1 f a capacitr is placed a series circuit with a dc vltage surce such as a battery, the current thugh the circuit will be exactly zer. f yu replace the battery with an ac (sinusidal) vltage surce, then the current will nt be zer. nstead it will be a sinusidal functin with amplitude given by: ε R 1, where = 2πf = Eq. 2 n sme ways, then the capacitr is a type f "A resistr" with effective resistance given by. Prf: The vltage drp acrss a capacitr is given by q V =. Since q = idt and i is a sinusidal f the frm: i( t) = sin 2πft then ( ) 1 f 8
P Physics Relatinships between f,, and. q( t) = sin( 2 π ft) dt = cs( πft) 2πf 2 thus Vc can be written as: V = cs( 2πft) = V cs( 2πft). max 2πf We define the reactance f the capacitr as: Thus, this implies that we can calculate c by Eq. 3. Vmax = Eq. 3 max 1 = = Eq. 4 2πf 2πf f an inductr is placed in a series circuit with a sinusidal vltage supply, then the current is given by: ε = where = 2πf n the same way, an inductr can be cnsidered an A resistr, with effective resistance. Prf: The vltage drp acrss an inductr is given by di V =. dt f i( t) = sin( 2πft), then V = 2πf cs 2πft and we can define the inductive resistance by: thus, max ( ) V max = Eq. 5 2πf = = 2πf. Eq. 6 This implies that we can calculate c using Eq. 5 by measuring the vltage drp acrss the inductr and the current thrugh the circuit. We can als calculate frm Eq. 6 with knwledge f f and. 2 f 8
P Physics Relatinships between f,, and. Experiment and Analysis: Part A 1. a. nstruct the circuit shwn in Figure 1 belw using a 1-µF capacitr fr 1. Use the R meter t determine a "gd" value fr the capacitance. Next, set the generatr frequency t apprximately 100 Hz. Adjust the utput cntrl fr apprximately 1-3 V acrss the capacitr. Recrd the current flwing thrugh the capacitr. Nte: yu may have difficulty using the DMM meter t recrd the current! t is likely t be rather small. An alternative strategy is t set up the circuit as shwn in Figure 2. The resistance shuld be n the rder f 1 / 2πf. Yu can then measure the vltage acrss the resistance bx and divide by the resistance ( = V/R) t determine the current. Keep in mind that yu must use figure (1) t measure the vltage acrss the capacitance, and figure (1b) t measure the vltage acrss the resistr. Als remember nt t change the resistance when yu measure vltage acrss the capacitr. Nte that the DMM measures RMS values fr current: rms = (Essentially, this means that that = the DMM reading multiplied by 2 ). V DMM A Vltage readings are als rms values, s that V =. rms 2 2 R Resistance Bx Functin generatr + _ ~ Vc DMM set n A vlts. Figure 1 Measuring the Vltage with the DMM 3 f 8
P Physics Relatinships between f,, and. Functin generatr + _ ~ R Resistance Bx Vr DMM set n A vlts. Figure 2 Measuring the current using the resistance bx. b. Using yur measured current and vltage, cmpute the reactance f the capacitr using = V /. Yu can use the DMM readings fr V and since the factr 2 cancels ut. c. alculate the reactance using the reactance frmula 1 = 2πf d. Are the values cmputed in Steps 1b and 1c within 10 percent f each ther? f nt, use the R meter t get gd values fr the capacitance, and resistance. f they check ut, then use the scillscpe t determine the perid and frequency f the functin generatr. Als be sure that yu have a "gd" sine wave. When yu are satisfied, enter yur results in Table 1. e. D yur results cnfirm the reactance frmula (within the tlerances f the meter, the generatr and the capacitr )? That is, des = V? 2. a. Suppse the generatr frequency is changed t 50 Hz while the vltage remains at 3 V. What will happen t the reactance? b. What will happen t the circuit current? c. hange the generatr frequency t 50 Hz and recrd the new current in Table 1. d. Des yur answer t Step 2c supprt yur answer t Step 2b? e. Hw much current wuld yu expect if the frequency were changed t 200 Hz and V remained at 3 V? f. hange the frequency t 200 Hz and adjust the generatr fr 3 V acrss the capacitr. Recrd the current. 4 f 8
P Physics Relatinships between f,, and. 3. hange the capacitr in Fig. 1 t 0.15 µf. Repeat Steps 1 and 2. Enter yur results in Table. 4. D yur values fr V and agree in each case? f nt, is the agreement better at high r lw frequencies? Why d yu think this is? Part B 1. Repeat the prcedure frm Part 1 using the 5H inductr in place f the capacitr. Use the R meter t get a "gd" value fr inductance. mplete Table 2 using Eqs. 5 and 6. Yur values fr shuld agree within 10 r 15% (due t uncertainties in measuring A current accurately). f yu are using the resistance bx t determine current, yu will have t change the setting s that enugh vltage is drpped acrss it t be measured. n general, set R bx apprximately equal t 2πf fr whatever frequency yu are using. A DMM n A current setting Functin generatr + ~ - Vc DMM set n A vlts. Figure 3 2. heck t see if the reactance determined abve agrees with the value calculated by the reactance frmula. alculate and recrd the reactance using the frmula = 2πf. The reactance calculated here shuld be within 15% f the reactance yu listed in Table 2. f it is nt, recheck yur measurements and yur calculatins. 3. hange the frequency f the generatr t 500 Hz. Measure the utput vltage f the generatr. Nte that the utput vltage is likely t drp. Make the measurements and calculatin necessary t cmplete the secnd rw f the table. Des yur data table supprt yur predictins abut what happens t the reactance and the current? 5 f 8
P Physics Relatinships between f,, and. 4. hange the generatr frequency t 1000 Hz. Frm the reactance yu determined in the first tw rws f Table 2, predict and recrd the reactance yu will have at 1000 Hz. mplete the third rw f the table t check yur predictin. Results: Write at least ne paragraph describing the fllwing: what yu expected t learn abut the lab (i.e. what was the reasn fr cnducting the experiment?) yur results, and what yu learned frm them Think f at least ne ther experiment might yu perfrm t verify these results Think f at least ne new questin r prblem that culd be answered with the physics yu have learned in this labratry, r be extraplated frm the ideas in this labratry. 6 f 8
P Physics Relatinships between f,, and. lean-up: Befre yu can leave the classrm, yu must clean up yur equipment, and have yur instructr sign belw. Hw yu divide clean-up duties between lab members is up t yu. lean-up invlves: mpletely dismantling the experimental setup Remving tape frm anything yu put tape n Drying-ff any wet equipment Putting away equipment in prper bxes (if applicable) Returning equipment t prper cabinets, r t the cart at the frnt f the rm Thrwing away pieces f string, paper, and ther detritus (i.e. yur water bttles) Shutting dwn the cmputer Anything else that needs t be dne t return the rm t its pristine, pre lab frm. certify that the equipment used by has been cleaned up. (student s name),. (instructr s name) (date) 7 f 8
P Physics Relatinships between f,, and. TABE 1 V f c (thery) c (exp) %diff 50 100 200 400 Frequency f urrent Vltage acrss 1, V 1 (measured) TABE 2 nductive Reactance = 2πf nductive reactance % V = difference 100 Hz 500 Hz 1000 Hz 8 f 8