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Determination of Planck’s Constant Using the Photoelectric ...

Determination of Planck s Constant Using the Photoelectric EffectLulu Liu (Partner: Pablo Solis) MIT Undergraduate(Dated: September 25, 2007)Together with my partner, Pablo Solis, we demonstrate the particle-like nature of light as charac-terized by photons with quantized and distinct energies each dependent only on its frequency ( ) andscaled by Planck s Constant (h). Using data obtained through the bombardment of an alkali metalsurface (in our case, potassium) by light of varying frequencies, we calculate Planck s Constant ,hto be 10 15 10 15eV THEORY AND MOTIVATIONIt was discovered by Heinrich Hertz that light incidentupon a matter target caused the emission of electronsfrom the target.

Sep 25, 2007 · of electron-Volts (eV). Kmax[in eV] = Vs = (h/e)ν −W0[in eV] (4) Figure 5 shows a plot of the maximum determined ki-netic energy (in electron-Volts) of photoelectrons as a function of the frequency of light (in Hz). The linear regression shows the best fit line through this set of re-duced data points. The slope of this line corresponds to

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Transcription of Determination of Planck’s Constant Using the Photoelectric ...

1 Determination of Planck s Constant Using the Photoelectric EffectLulu Liu (Partner: Pablo Solis) MIT Undergraduate(Dated: September 25, 2007)Together with my partner, Pablo Solis, we demonstrate the particle-like nature of light as charac-terized by photons with quantized and distinct energies each dependent only on its frequency ( ) andscaled by Planck s Constant (h). Using data obtained through the bombardment of an alkali metalsurface (in our case, potassium) by light of varying frequencies, we calculate Planck s Constant ,hto be 10 15 10 15eV THEORY AND MOTIVATIONIt was discovered by Heinrich Hertz that light incidentupon a matter target caused the emission of electronsfrom the target.

2 The effect was termed the Hertz Effect(and later the Photoelectric Effect) and the electrons re-ferred to as photoelectrons. It was understood that theelectrons were able to absorb the energy of the incidentlight and escape from the coulomb potential that boundit to the to classical wave theory, the energy of alight wave is proportional to the intensity of the lightbeam only. Therefore, varying the frequency of the lightshould have no effect on the number and energy of resul-tant photoelectrons. We hope to disprove this classicalhypothesis through experimentation, by demonstratingthat the energy of light does indeed depend on the fre-quency of light, and that this dependence is linear withPlanck s constanthas the Constant of HYPOTHESISL ight comes in discrete packets, calledphotons, eachwith an energy proportional to its (1)For each metal, there exists a minimum binding energyfor an electron characteristic of the element, also calledthe work function (W0).

3 When a photon strikes a boundelectron, it transfers its energy to the electron . If thisenergy is less than the metal s work function, the photonis re-emitted and no electrons are liberated. If this energyis greater than an electron s binding energy, the electronescapes from the metal with a kinetic energy equal tothe difference between the photon s original energy andthe electron s binding energy (by conservation of energy).Therefore, the maximum kinetic energy of any liberatedelectron is equal to the energy of the photon less theminimum binding energy (the work function).

4 Expressedconcisely the relationship is as such:Kmax=h W0(2) Electronic maximum kinetic energy can be determined byapplying a retarding potential (Vr) across a vacuum gapin a circuit with an amp meter. On one side of this gapis the photoelectron emitter, a metal with work functionW0. We let light of different frequencies strike this emit-ter. WheneVr=Kmaxwe will cease to see any currentthrough the circuit. By finding this voltage we calculatethe maximum kinetic energy of the electrons emitted as afunction of radiation frequency. This relationship (withsome variation due to error) should model EXPERIMENTAL SchematicFIG.

5 1: Schematic of experimental setup with retarding volt-age applied. Photons are incident on the photocathode andtravel toward the anode to complete the circuit unless stoppedby a high enough applied ApparatusOur monochromatic light source was an Oriel 65130 Mercury Lamp in combination with a narrow band passfilter wheel with four different wavelength wavelengths are listed in (nm) I: Spectrum from Oriel mercury Leybold photocell served as our target, containing apotassium (W0= ) photosurface as the cathodeand a platinum ring (Wa= ) as the anode sepa-rated by a vacuum.

6 It was enclosed in a black box witha small circular opening to allow for incoming light. Pre-cautions were taken to shield the setup from ambientlight, to protect the filters and photocell from overheat-ing, and to minimize illumination of the DATA AND ANALYSISAn example of our tabulated raw data is shown in Ta-bleIIfor the nm wavelength over five normalized currents for each wavelength are plot-ted against their respective retarding voltages in Figure2with the standard deviations as error bars. The nor-malization removes the scaling effects of the non-uniformdistribution of intensities across the spectrum of our lightsource.

7 With intensity normalized away, it is already ev-ident from this figure that the cut-off voltages have somedependence on 50510152025303540 Retarding Voltage Applied (V)Photocurrent Detected (pA)Plot of Photocurrents Induced by Four Wavelengths of Light (Normalized to nm Curve) nmFIG. 2: Plot of photocurrent as a function of the magnitude ofretarding voltage applied for each wavelength of incident to the zero-voltage point of the lowest back current was a noticeable effect for several wave-lengths. Back current is caused by photoelectrons liber-ated from the platinum anode as a result of scatteredlight.

8 Its effects become most prominent when the re-tarding voltage is high. The retarding voltage is seen asan accelerating voltage by these electrons and they traveluninhibited from the anode to the cathode, opposite thedirection of the expected feature that should not be neglected is thenon-linear nature of the photocurrent vs. voltage curvenear the stopping voltage. Theoretically, at the stoppingvoltage, we expect the current to be zero. However, aswe decrease the stopping voltage, the current rises onlyvery slowly until it begins to take on a familiarI= Vrlinear form.

9 This is to be expected since the numberof states with energy near the minimum binding energyN(E W0) may in fact be very small, and increasesas the binding energy effects such as the two represented abovecompromise the reliability of zero-current crossings fordetermination of stopping voltages. Instead, we look totwo different methods for extrapolating the data pointsof interest. Any differences in results will be used incalculation of a lower bound on our systematic Method One for Voltage Cut-OffDetermination: Linear Fit 50510152025303540 Magnitude of Retarding Voltage Applied (V)Photocurrent Detected (pA)Demonstration of the Linear Fit Method of Stopping Voltage Approximationy = mx + b m = + = + X2 = y = mx + b m = + b = + = 0 (Vr = ,I = )Measured Values (I vs.)

10 Vr) for PhotonsLow Voltage Best Fit LineHigh Voltage Best Fit LineExtrapolated Zero PointFIG. 3: Graphical demonstration of the High and Low VoltageLinear Fit Method of stopping voltage Determination . Calcu-lations based on the normalized data points with error nm wavelength the asymptotic behavior of each curve at bothlow and high values of retarding voltages (discountingsaturation) is linear, both sections can be fit to sepa-rate linear regressions. The criteria for determining howmany data points to fit on each end was simple: mini-mum number of points required for a meaningful fit whilemaintaining a reasonable 2.


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