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Object: Determination of Planck’s Constant ‘h’ by measuring radiation in a fixed spectral range. Apparatus used: The complete experimental arrangement as shown in figure (1) consists of filament bulb, its power supply (0-12 V d.c.) , A solar cell fitted on mini optical bench, one Ammeter ( 0- 2 A) and one voltmeter (0-10 V) to read filament current and voltage, one digital D.C. Microammeter (0-200 μA.).

Planck’s Constant Set-Up

Filament Bulb Solar Cell

Figure 1(a)

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Theory: For a black body at temperature ‘T’ the total radiations as well as the spectral distribution for this radiation are functions of temperature T alone. The spectral distribution involves Planck’s constant ‘h’.

dkT

hchcdE

1

51)exp(

8

………………….1

Working with visible light and temperature upto 2500 oK , we have, kT

hc

>>1.

Hence equation (1) simplifies to,

dkT

hchcdE )exp(

85

………………….2

If the radiation is received through a filter on a photocell and galvanometer response θ is measured, we get:

5

8

BhcA

dkT

hc)exp( ………………….3

Where A is a factor depending on geometry of the arrangement and sensitivity of the galvanometer and Bλ is a function of λ which includes (i) transmission characteristics of the filter and (ii) wavelength wise response of the photocell. The integral has to cover all range of λ for which Bλ is non zero. If we have filter, which has, transmission characteristics schematically represented by figure 2 (a) and a photovoltaic cell with response schematically represented in figure 2 (b) then the Bλ function is given by products of ordinates T and R plotted against respective values figure 2 (c).

Figure 1(b)

Current Control

Filament Power

Supply

A

Filament Current

V Filament Voltage

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However, if the filter has a narrow transmission band, one may reduce equation (3) to ,

08 hcAC 0)exp(

kT

hc ………………….4

Where λ0 is some effective mean wavelength, Δ λ0 is the effective band width and C λ0 is a constant depending on λ0. At two different temperatures, we get

)

11(exp

2101

2

TTk

hc

………………….5

Where the constant A, B λ and Δ λ0 all cancel out due to the simplifying assumptions made to go from (3) to (4) We note that unless the transmission band is narrow, the theory would not hold, further effective λ0 is not just the center of transmission band of the filter, but it will be seriously effected by response curve of the photocell. In the photo voltaic cell the shift usually will

be towards longer wavelength. Even more seriously the term )exp(kT

hc

increases

rapidly with increasing λ. For T = 2000 0K, as we pass from λ = 5000 0A to λ = 7000 0A. This exponential term increases from exp(-3) to exp.(-2). This fact will also place effective λ0 on the higher side of the mean transmission wavelength of the filter. With in these limitation equation (5) gives

2.303 10log =Tk

hc 1

0 ………………….6

5000 6000 7000 λ (Α˚)

T

5000 6000 7000 λ (Α˚)

R

5000 6000 λ (Α˚)

B λ

(a) (b) (c) Figure 2

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Thus log θ vs 1/T graph should be a straight line from whose slope we may deduce h.

)1

(

)(log.

303.2 100

Tc

kh

………………….7

The experimental task reduces to measure θ for different temperatures of the filament keeping the geometry and λ0 constant. METHOD:- (a) To determine the value of Rg and Rt/Rg with the V-I characteristics of Bulb.

1. To find Rg the resistance of the filament bulb when it just starts glowing: Connect

the set-up with the mains and switched it ON. Apply filament current by the power supply control knob marked CONTROL on the panel such that the filament just starts glowing. Note the corresponding filament current and voltage and record these readings in table-I as shown below. Take at least three readings to the get the better value of Rg. Refer fig. 3

TABLE: - I For Rg, the resistance of the filament when it just starts glowing.

S.No. Filament Current

I Amp. Filament Voltage

V Volt Rg Ohm Average Rg

1. 2. 3.

2. To find Rt/Rg from the V-I characteristics of bulb: Further increase the filament

current in steps of 0.1 amp. and note the corresponding values of filament voltage for each value of filament current. Record all these readings in table-II as shown below.

3. Calculate Rt and Rt/ Rg for each value of filament current and voltage.

TABLE: - II For V-I Characteristics of the Filament bulb

S.No. Filament Current I Amp.

Filament Voltage V Volt

Rt Ohm Rt / Rg

1. 2. 3. 4.

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(b) Observations of θ and to determine the value of Planck’s constant ‘h’ 4. Adjust the filament bulb at some distance ‘d’ from the solar cell on mini optical

bench. 5. Increase the filament current to such extend that the digital micro ammeter reads

some photo-current say mini 2 µA to 3 µA. Note this value of filament current and photocurrent and record it in table- II as shown below.

6. Increase the filament current in steps of 0.1 amp. As in previous case and note the

corresponding photo-current for each value of filament current. Record these readings in table- III.

7. Note the value of Rt/ Rg for each value of filament current from table-II and record

these readings in table -III. 8. Note the corresponding temperatures for each value of Rt/ Rg from graph no I or

table-IV for Rt/ Rg Vs Temp T oK for tungsten. 9. Calculate log and 1/T oK. 10. Plot graph between log Vs 1/ToK and find the slope of the curve. 11. Repeat the experiment at different value of ‘d’ between the bulb and solar cell.

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Table-III For θ For distance between the filament bulb and solar cell d = cm Wavelength = 6000 A0

S.No. Filament Current I Amp.

Filament voltage V volt

Corresponding Rt/Rg from table or graph.

Photo-current µA

log ToK from graph Rt/Rg Vs ToK

4101 xKT o

1.

2.

3.

4.

Calculations:

From equation (7)

40 10log

303.2 xKT

xc

Kh

o

Where o= 6000 A0 mean wavelength of light

k= 1.38 x 10-16 erg/deg. Boltzmann’s constant

c= 3x10 10 meter/sec. velocity of light

T

1log

= Slope of the curve.

Results: 1. The experimentally observed values of Planck’s constant ’ h ‘= …………erg/sec. 2. The standard value of Planck’s constant ‘ h’ =

The experimentally observed value of h is always with in 10 to 15% of the standard value.

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Table –IV Relation Between Rt/Rg and Temperature K for Tungsten Filament.

S. No. Temperature (K) Rt/R 01.

02.

03.

04.

05.

06.

07.

08.

09.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

0273

0300

0400

0500

0600

0700

0800

0900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400

0.25

0.29

0.43

0.55

0.71

0.85

0.99

1.15

1.29

1.45

1.61

1.79

1.95

2.11

2.30

2.46

2.65

2.85

3.05

3.26

3.46

3.66

3.85