physical practices

Introduction: A capacitor is an electrical component used to store energy in an electrical field. It contains at least two electrical conductors separated by a dielectric. When there is a potential difference (voltage) across the conductors, a static electric field develops through the insulator, causing positive charge which is collected on one plate and negative charge which is collected on the other plate. Energy is stored in the electrostatic field. An ideal capacitor is characterized by a constant value, the capacitance, measured in Farads. This is the ratio of the electric charge on each conductor to the potential difference between them. Capacitors are widely used as parts of electrical circuits in many common electrical devices: in electronic circuits for blocking direct current while allowing alternating current to pass; for smoothing the output of power supplies in filter networks; in the resonant circuits that tune radios to particular frequencies, in the field of electric power in the transmission of systems for stabilizing and powering flow…

1. Equipment: -Plate capacitors

-Connecting cords

-Spacer plates

-Optical profile bench

-Electric field meter

-Slide mount

-Potential probe -Power supply -Fan

-Support rod -Rule

2. Procedure and data collected: The objective of our practice is to understand the how the capacitor runs. With this propose, first of ell we will measure the relationship between voltage and electric field strength with constant place spacing, and the other target will consist in studying the relationship of electric field strength with different place spacing at constant voltage.

2.1

Constant distance: The experimental set up consists in measure the electric field at various voltages keeping constant different plate distances. To measure the voltage we have needed to change the scal, so when we worked with potential differences under 100V we used a scale with two units of accuracy, and when it was over 100V we changed to a new scale with 10 units of precision.

d=2 cm V±1(v) 0 52 74 130 145 175 190 250

E(kV/m) 0 2,69±0,01 3,7±0,1 6,4±0,1 7,2±0,1 8,6±0,1 9,4±0,1 12,1±0,1

d=8 cm V± 1(v) 52 75 100 120 145 170 190 250

E(mAkV/m) 0,82±0,01 1,09±0,01 1,43±0,01 1,7±0,1 2,02±0,01 2,34±0,01 2,6±0,1 3,3±0,1

d=4 cm V± 1(v) 52 74 100 120 145 175 195 250

E(kV/m) 1,28±0,01 1,75±0,01 2,33±0,01 2,7±0,1 3,2±0,1 3,9±0,1 4,3±0,1 5,4±0,1

d=10cm V± 1(v) 52 75 100 125 145 175 195 250

1

E(mAkV/m) 0,7±0,1 0,94±0,01 1,19±0,01 1,48±0,01 1,66±0,01 1,92±0,01 2,13±0,01 2,7±0,1

d= 6cm V± 1(v) 52 74 100 120 145 175 195 250

E(kV/m) 1,06±0,01 1,39±0,01 1,89±0,01 2,18±0,01 2,63±0,01 3,1±0,1 3,4±0,1 4,3±0,1

d= 12cm V± 1(v) 52 75 100 125 145 175 195 250

E(mAkV/m) 0,63±0,01 0,81±0,01 1,03±0,01 1,28±0,01 1,44±0,01 1,62±0,01 1,85±0,01 2,33±0,01

2.2

Constant voltage:

The experimental set up consists in measuring the electric field at various plate distances keeping constant the voltage:

v=75 V d(m) ±0,01 0,02 0,04 0,05 0,06 0,07 0,08 0,1 0,12

v=150 V d(m)±0,01 0,02 0,04 0,05 0,06 0,07 0,08 0,1 0,12

E(kV/m) 4,6±0,1 2,3±0,1 1,92±0,01 1,46±0,01 1,14±0,01 1,05±0,01 0,91±0,01 0,77±0,01

E(kV/m) 9,1±0,1 4,3±0,1 3,7±0,1 2,9±0,1 2,5±0,1 2,25±0,01 1,83±0,01 1,47±0,01

3. Analysis: In every experiment we will use these formulas to analyze on the one hand the data collected and on the other hand the error and the mistakes:

∑

s=√

∑

s(b)=

r= √∑

2

∑ √ ∑

∑

3.1 d=0,02m V(v)±1 0 52 74 130 145 175 190 250 1016

E(kV/m) 0 2,69±0,01 3,7±0,1 6,4±0,1 7,2±0,1 8,6±0,1 9,4±0,1 12,1±0,1 50,09

Constant distance: X 0 52 74 130 145 175 190 250 1016

Y 0 2,69 3,7 6,4 7,2 8,6 9,4 12,1 50,09

X·Y 0 139,88 273,8 832 1044 1505 1786 3025 8605,68

b s s(b) R R2

X2 0 2704 5476 16900 21025 30625 36100 62500 175330

Y2 0 7,2361 13,69 40,96 51,84 73,96 88,36 146,41 422,46

b·x 0 2,55 3,63 6,38 7,12 8,59 9,33 12,27 49,87

490,8·10-4 0,011 2,6 0,999 0,999

d=0,02m

E(kV/m) 14 12 10

y = 0,0491x R² = 0,9994

8 6 4 2

V(v)

0 0

Slope of the graph=

50

=0,0491

100

150

d=

3

200

=

250

300

= =20,37·10-3 m= 0,02 m

(y-b·x)2 0 0,018 0,005 0,0003 0,006 0,0001 0,006 0,029 0,066

d=4cm V(v) 52 74 100 120 145 175 190 250 1106

E(kV/m) 1,28 1,75 2,33 2,7 3,2 3,9 4,3 5,4 24,86

X 52 74 100 120 145 175 190 250 1106

Y 1,28 1,75 2,33 2,7 3,2 3,9 4,3 5,4 24,86

b s s(b) R R2

X·Y 66,56 129,5 233 324 464 682,5 817 1350 4066,56

X2 2704 5476 10000 14400 21025 30625 36100 62500 182830

Y2 1,63 3,06 5,42 7,29 10,24 15,21 18,49 29,16 90,52

b·x 2,55 3,63 4,91 5,89 7,12 8,59 9,33 12,27 54,29

222,423·10-4 12,59 0,029 0,999 0,999

d=0,04m

E(kV/m) 6 5 4

y = 0,0222x R² = 0,9947

3 2 1

V(v) 0 0

50

Slope of the graph=

100

= 0,0222

150

d=

4

200

=

250

300

= =45,04·10-3 m= 0,04 m

(y-b·x)2 1,51 2,89 5,20 7,03 9,93 14,83 18,07 28,63 88,10

d=6cm V(v) 52 74 100 120 145 175 190 250 1106

E(kV/m) 1,06 1,39 1,89 2,18 2,63 3,1 3,4 4,3 19,95

X 52 74 100 120 145 175 190 250 1106

Y 1,06 1,39 1,89 2,18 2,63 3,1 3,4 4,3 19,95

X·Y 55,12 102,86 189 261,6 381,35 542,5 646 1075 3253,43

b s s(b) R R2

E(kV/m)

X2 2704 5476 10000 14400 21025 30625 36100 62500 182830

Y2 1,12 1,93 3,57 4,75 6,92 9,61 11,56 18,49 57,96

b·x 2,55 3,63 4,91 5,89 7,12 8,59 9,33 12,27 54,29

177,94·10-4 8,01 0,02 0,999 0,999

d=0,06m

5 4 3 2

y = 0,0178x R² = 0,9923

1 V(v) 0 0

50

Slope of the graph=

100

= 0,0178

150

d=

5

200

=

250

300

= =56,18 10-3 m= 0,06 m

(y-b·x)2 1,02 1,79 3,39 4,54 6,66 9,31 11,23 18,07 56,02

d=8cm V(v) 52 75 100 120 145 170 190 250 1102

E(kV/m) 0,82 1,09 1,43 1,7 2,02 2,34 2,6 3,3 15,3

X 52 75 100 120 145 170 190 250 1102

Y 0,82 1,09 1,43 1,7 2,02 2,34 2,6 3,3 15,3

X·Y 42,64 81,75 143 204 292,9 397,8 494 825 2481,09

b s s(b) R R2

E(kV/m)

4

X2 2704 5625 10000 14400 21025 28900 36100 62500 181254

Y2 0,67 1,19 2,04 2,89 4,08 5,48 6,76 10,89 34,01

b·x 2,55 3,68 4,91 5,89 7,12 8,34 9,32 12,27 54,09

136,885·10-4 4,645 0,011 0,999 0,999

d=0,08m

3,5 3 2,5 2 y = 0,0137x R² = 0,9918

1,5 1 0,5

V(v)

0 0

50

Slope of the graph=

100

= 0,0137

150

d=

6

200

=

250

300

=72,99 10-3 m= 0,072 m

(y-b·x)2 0,59 1,08 1,91 2,73 3,88 5,25 6,51 10,57 32,52

d=10cm V(v) 52 75 100 125 145 175 195 250 1117

E(kV/m) 0,7 0,94 1,19 1,48 1,66 1,92 2,13 2,7 12,72

X 52 75 100 120 145 170 190 250 1102

Y 0,82 1,09 1,43 1,7 2,02 2,34 2,6 3,3 15,3

X·Y 42,64 81,75 143 204 292,9 397,8 494 825 2481,09

b s s(b) R R2

X2 2704 5625 10000 14400 21025 28900 36100 62500 181254

Y2 0,67 1,19 2,04 2,89 4,08 5,48 6,76 10,89 34,00

b·x 2,55 3,68 4,91 5,89 7,12 8,34 9,33 12,27 54,09

136,885·10-4 4,646 0,011 0,999 0,999

E(kV/m) 3

d=0,1 m

2,5 2 1,5

y = 0,0112x R² = 0,9832

1 0,5 V(v) 0 0

50

Slope of the graph=

100

= 0,0112

150

d=

7

200

=

250

300

=89,28 10-3 m= 0,09 m

(y-b·x)2 0,59 1,08 1,91 2,73 3,88 5,25 6,51 10,57 32,52

d=12cm V(v) 52 75 100 125 145 175 195 250 1117

E(kV/m) 0,63 0,81 1,03 1,28 1,44 1,62 1,85 2,33 10,99

X 52 75 100 120 145 170 190 250 1102

Y 0,82 1,09 1,43 1,7 2,02 2,34 2,6 3,3 15,3

X·Y 42,64 81,75 143 204 292,9 397,8 494 825 2481,09

b s s(b) R R2

3

E(kV/m)

X2 2704 5625 10000 14400 21025 28900 36100 62500 181254

Y2 0,67 1,19 2,04 2,89 4,08 5,48 6,76 10,89 34,00

b·x 2,55 3,68 4,91 5,89 7,12 8,34 9,32 12,27 54,09

136,885·10-4 4,645 0,01 0,999 0,999

d=0,12 m

2,5 2 1,5 y = 0,0096x R² = 0,9784

1 0,5

V(v) 0 0

50

Slope of the graph=

100

= 0,0096

150

d=

8

200

=

250

300

= 104,17 10-3 m= 0,1 m

(y-b·x)2 0,59 1,08 1,91 2,73 3,88 5,25 6,51 10,57 32,52

3.2

Constant voltage:

3.2.1 75 V d(m)±0,01

E(kV/m) ±0,01

0,02

4,6

0,04

2,3

0,05

1,92

0,06

1,46

0,07

1,14

0,08

1,05

1

0,1

0,91

0

0,12

0,77

1/d(m) 50 25 20 16,67 14,29 12,5 10 8,33 156,79

E(kV/m) 4,6 2,3 1,92 1,46 1,14 1,05 0,91 0,77 14,15

b

912,07·10-4

s

403,74

s(b)

6,134

R2

0,999

5

E(kV/m)

v=75 V

4 y = 0,0828x-1,026 R² = 0,9901

3 2

d(m) 0

X2 2500 625 400 277,78 204,08 156,25 100 69,44 4332,55

0,02

Y2 21,16 5,29 3,69 2,13 1,3 1,1 0,83 0,59 36,09

0,04

0,06

X·Y 230 57,5 38,4 24,33 16,29 13,13 9,1 6,427 395,16

E(kV/m) 5

0,08

0,1

b·x 815,98 407,99 326,39 271,99 233,14 203,99 163,19 135,99 2558,69

0,12

0,14

(y-b·x)2 658340,32 164585,08 105282,53 73188,60 53822,90 41186,82 26336,86 18286,33 1141029,47

v=75 V

4 y = 0,0912x R² = 0,9955

3 2

Slope=E(kV/m): (m)= 1

E(kV/m)·d(m)=0,091kV= 91 V

1/d(m)

0 0

10

9

20

30

40

50

60

4.2.2 150 V

d(m)±0,01 E(kV/m)±0,01 9,1

0,04

4,3

0,05

3,7

0,06

2,9

6

0,07

2,5

4

0,08

2,25

2

0,1

1,83

0,12

1,47

1/d(m) 50 25 20 16,67 14,29 12,5 10 8,33 156,79

b s s(b) R^2

E(kV/m) 9,1 4,3 3,7 2,9 2,5 2,25 1,83 1,47 28,05

v=150 V

E(kV/m) 10

0,02

y = 0,1764x-1,004 R² = 0,9979

8

d(m) 0 0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

X2

Y2

X·Y

b·x

(y-b·x)2

2500 625 400 277,78 204,08 156,25 100 69,44 4332,56

82,81 18,49 13,69 8,41 6,25 5,06 3,35 2,16 140,22

455 107,5 74 48,33 35,71 28,13 18,3 12,25 779,22

8,99 4,5 3,6 2,99 2,57 2,25 1,8 1,5 28,2

0,0115245 0,0385431 0,0105968 0,0095159 0,0048064 3,378E-06 0,0009904 0,000828 0,0768084

179,853·10-3 0,105 0,001 0,999

Slope=E(kV/m): (m)=

10

E(kV/m)

v=150 V

8 6 4

y = 0,1799x R² = 0,9982

2

E(kV/m)·d(m)=0,1799 kV= 180 V

1/d(m)

0 0

10

10

20

30

40

50

60

4. Conclusions: Our results are quite close to the expected ones, because if we compare the relationships associated by the theoretical formula with the relationships obtained experimentally, both results are quite close. However, we don’t obtain perfect results and the source of mistakes in the measurements is quite big. First of all, the plastic rule we use to measure distances was quite old and broken, so it was really difficult to get accurate measures using it. The mistakes are more obvious when we measure higher distances, but this problem can be also caused by the polarization of the air. The capacitor we have used in the lab included a fan to avoid air’s polarization, but with bigger distances the fan didn’t work so well so we have more problems and we don’t have exact results. This can be also the cause of our mistakes in the second part, where we took constant the voltage while measuring a wide range of distances. In conclusion, I think we could improve this practice by improving the ventilation system. Another improvement could be to install directly in the spacer plates the rule, to get exact measures of distances. On top of that, we can’t forget that we are supposing that our capacitor is composed of plane-parallel plates with infinite surface, and this last part isn’t true, so our calculations can never be completely correct.

Antía Núñez Martínez 1st year Degree in Chemical Engineering

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