ph110 physics laboratory instruction manual

PH110 PHYSICS LABORATORY

Instruction Manual January – April 2013

Course Coordinator: Dr. S. B. Santra Co-coordinators: Dr. S. K. Khijwania, Dr. P. K. Padmanabhan, Dr. B. Bhuyan, Dr. U. Raha

Department of Physics

Indian Institute of Technology Guwahati

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CONTENTS Topic Page No Instructions to Students 03 Introduction to Error Analysis 05 Expt.1: Determination of g by compound pendulum 09 Expt.2: Time constant of a capacitive circuit 11 Expt.3: Magnetic field along the axis of a coil 14 Expt.4: Resonance and Q factor of a LCR circuit 16 Expt.5: Study of Hall Effect in an extrinsic semiconductor 19 Expt.6: Speed of light in glass 22 Expt.7: Fraunhofer diffraction: Single Slit 25 Expt.8: Newton’s ring 27

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INSTRUCTIONS TO STUDENTS

(1) Class timing: 09:00 – 12:00 Hrs. No student will be allowed after 09:10 Hrs. (2) You must bring with you the following material to the lab: report sheets (A4 size

papers), pen, pencil, small scale, instruction manual, graph sheets, calculator and any other stationary item required. PENCIL is to be used only for graph plotting. Data should be recorded with PEN.

(3) Submit a cover file to the technical assistant on the first day of Laboratory class. Write down your Name, Roll No., and Group on the file cover. You should file your laboratory record regularly in your file. You are NOT allowed to take your reports back. You will be given your file and records back before the end-semester examination of the laboratory course.

(4) At least one set of observation should be signed by the instructor. An unsigned observation data will be awarded ZERO.

(5) You are expected to perform the experiment, complete the calculation and data analysis during the laboratory hours. Submit the report before 12:00 hrs, late submission may cause subtraction of marks.

(6) Laboratory manual will be available in Xerox shop, Core IV. (7) Read the instruction manual carefully before coming to the laboratory class.

Report must be prepared in the following format: 1) Name of the experiment, Your Name, Roll No., lab group, and date of experiment. 2) Working formula, meaning of symbols and the schematic diagram of the

experimental setup. (DO NOT write the theory of the experiment.) 3) Experimental specifications: apparatus/instruments used in the experiment 4) Least count of all the equipment, constants if any to be used and the well tabulated

observations. Observation tables should be neat and self-explanatory. (Typical tabular columns have been given for some of the experiments. You may make your own format). All physical quantities in the table should have suitable units.

5) Graph(s), if applicable, with title of the graph, scaling used for the two axes, suitable units for each axis, equation of linear fit, if any.

6) Relevant substitutions, calculations, data analysis and results. 7) An error analysis should be performed and the results should be reported with

maximum possible error.

Your lab report should be ready with information (1) and (2) before coming to the lab. Marks are allocated for this initial preparation.

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Scheme of evaluation Laboratory reports will have 30% marks and the end semester will have 70% marks. Each laboratory report will be evaluated for 30 marks and final laboratory report mark will be the average of 8 experiments. Performing 8 experiments is must. If a student misses an experiment, marks for that experiment will be taken as zero for averaging.

LABORATORY REPORT END SEMESTER EXAMINATION

(a) Initial Preparation 5 (a) Working formula, diagrams, etc. 10 (b) Data collection 10 (b) Data collection 15 (c) Calculation 10 (c) Calculation 15 (d) Error analysis and result 5 (d) Error analysis and result 10 (e) Viva ---- (e) Viva 20

Total 30 Total 70

1. Though there is no mark for viva during the lab, there will be discussion on the experiment at the time of singing the data and initial preparation.

2. End semester viva will be conducted individually during the examination. Viva will cover questions from all eight experiments.

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INTRODUCTION TO ERROR ANALYSIS All physical measurements are subject to various types of error. It is important to reduce the effect of errors to a minimum. In order to know the uncertainty in measurement or to know the deviation from the true value of a measured quantity, it is important to have an idea of the sources of error as well as estimates. Error involved in any measurement may be broadly classified as (a) systematic error and (b) random error. (a) Systematic Error: Errors that are not revealed through an entire set of measurements

are termed systematic errors. Systematic errors may arise because of instrumental defect or experimental bias. (i) Instrumental errors: Zero offset (instrument does not read zero when input is

zero) or incorrect calibration of the instrument or changes of calibration conditions (due change in temperature, pressure or any other environmental changes) are the example of instrumental errors. Zero error can be detected beforehand and all the observations are corrected accordingly.

(ii) Experimenter’s bias: This is a common source of error arising from some bias

of the experimenter and is difficult to eliminate. For example, parallax error in reading an analog meter is often encountered if suitable care is not taken to view the indicator needle perpendicular to the meter face.

Systematic errors are hard to handle. They are best identified and eliminated.

(b) Random Error: Fluctuations in the recording of data or in the instrumental measuring process result in random errors. The effect of random errors can be minimized by appropriate data processing techniques.

Probable error: Most of the experiments involve measurement of several different quantities which are combined to arrive at the final deduced quantity y. Measurement of each of these quantities is limited in accuracy by the least count of the instrument. These errors give rise to a maximum possible error. It can be estimated in the following manner. Suppose the physical quantity, y, is given by the relation = (7)

where C, m and n are known constants. Experimental determination of y involves measurement of x1 and x2. The overall maximum uncertainty or maximum possible error in y is given in terms of errors ∆x1 and ∆x2 (uncertainties in the measurement) in the quantities x1 and x2 respectively, by

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∆ = .∆ + ∆ (8)

Note that both contributions add up to give the maximum possible error in y, irrespective of whether m or n is positive or negative. This can be illustrated with the help of the following example. The electrical resistivity of a wire of circular cross section is given by = (9)

where r is the radius and l is the length of the wire, V is the voltage and I is the current flowing through the wire. The maximum possible error in the measurement of resistivity depends on the fractional uncertainties in the voltage, current, etc. and is given by ∆ = 2∆ + ∆ + ∆ + ∆ (10)

Least Squares Fit: When the data (xi ,yi) are linearly related by = + (11)

the best estimates for the slope a and intercept b of the straight line are obtained as follows: If y is the true value as defined by the eqn.(11), then one should minimize the quantity ∑ ( − ) = ∑ ( − − ) with respect to a and b. By differentiating this expression with respect to a and b, setting them to zero and solving the two simultaneous equations, we get the best estimates of a and b as = ∑ − ∑ ∑ ∑ − (∑ ) (12)

= ∑ ∑ −∑ ∑ ∑ − (∑ ) (13)

After obtaining the values of a and b, plot the straight line y = ax + b using those values. Plot the observed points too on the same graph. The errors in and obtained in this way are given below. ∆ = ± ∑ (∑ ) ∆ = ± ∑ ∑ (∑ ) (14)

See how well the data are clustered around this straight line. Quite often you may be able to reduce the equation to the linear form by a suitable rearrangement. For example if y = cex, then ln(y) = lnc+ x, so a plot of ln(y) Vsx would be a straight line. References [1]. J. R. Taylor, An Introduction to Error Analysis, University Science Books (1997).

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Experiment: 1

Determination of the value of acceleration due to gravity with the help of a

compound pendulum Theory: The time period of oscillation of a body constrained to rotate about a horizontal axis for small amplitudes is given by = 2 /

(1)

where m is the mass of the body, d is the distance between centre of mass (CM) and the axis of oscillations and I is the moment of inertia (MI) about the axis of oscillations. If Io is the MI of a body about a parallel axis through CM, then by parallel axis theorem, = + (2)

Let K be the radius of gyration (i.e., I0 = mK2), then = 4 ( + ) (3)

By observing period of oscillation T by varying d we can obtain the values of gravitational acceleration g as well as moment of inertia Io of the body. Note that the time period will be minimum at d = K. Experimental Setup: In this experiment the rigid body is a long rectangular bar with a series of holes drilled at regular interval to facilitate suspension at various points along its length. A screw type knife-edge can be fitted to the bar at these points. The knife-edge can be rested on a wall-mount so that the bar is free to move in a vertical plane. The radius of gyration for this bar is = + 12 (4)

Procedure:

1) Determine the centre of mass of the bar by balancing it on a knife-edge. 2) Measure length l and breadth b of the rectangular bar at least three times to calculate

K using eqn.(4). 3) Suspend the bar by means of knife-edge. 4) Measure d from the sharp end of the knife-edge (and not from the centre of the hole),

which is the point of suspension, to the centre of mass of the bar. 5) Measure time for around 20 oscillations for different d (only on one side of CM).

Repeat each observation three times for a given d.

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Suggested Observation Table:

1. Length of the bar = 2. Breath of the bar = 3. Mass of the bar =

Sl. d (m)

d2

(m2) t1

(sec) t2

(sec) t3

(sec) t=

(t1+t2+t3)/3 T=t/20 (sec)

T2d (m2-Sec)

Analysis: 1. Plot T2d versus d2. 2. Calculate K and g least square fitting. 3. Calculate the maximum possible error in the measurement of g.

References:

1. D. Kleppner and R. J. Kolenkow, An Introduction to Mechanics, McGraw-Hill (1999).

2. C. Kittel, W. D. Knight, M. A. Ruderman, C. A. Helmholz and B. J. Moyer, Mechanics (Berkley Physics Course - Volume 1), Tata McGraw-Hill (2008).

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Experiment: 2

Determination of time constant of a RC Circuit

Theory: A capacitor can be used to store the energy by accumulation of charges and a charged capacitor can be used to deliver the energy by discharging it through a load. Both charging and discharging are not instantaneous. It can be shown that the charging and discharging times depend on the capacitance of the capacitor and total resistance of the circuit.

Fig. 2.1

Charging: For the circuit (Fig.2.1), (with switch K1 closed and K2 open) containing a capacitance C, a resistance R and a source of constant voltage V0, the equation for the potential (neglecting the source resistance and resistance of the ammeter) is = + (1)

where is the current in the circuit and q is the charge accumulated on the capacitor. Both i and q may be functions of time. Since i = dq/dt we get the equation for q as = + (2)

The solution of this equation is = 1 − (3)

whereτ is the time constant of the circuit and is equal to RC. The current iis given by = = / (4)

The voltage across C is given by = 1 − / (5)

Equations (4) and (5) describe the decay of current in the circuit and growth of voltage on the capacitor, respectively, during the charging of a capacitor.

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Discharging: When key K1 is open and K2 is closed, with the capacitor charged to a voltage V0,discharging of the capacitor through the resistance R is described by the following relations: = / (6)

= = − / (7)

= / (8)

Equations (7) and (8) describe the decay of current in the circuit and decay of voltage across C, respectively, during the discharging process.

Procedure:

1. Assemble the circuit as shown in Fig. 2.1. The experiment is to be performed for only one value of R.

2. Set the source voltage V0 (from the power supply) between 2 to 4 V such that the largest (initial) current (i0=V0/R) has a reasonable value.

3. Set the ammeter (shown as µA in Fig. 2.1) in an appropriate range (estimate the range from V0 and R).

4. Decay of current during charging: Complete the charging circuit by closing the key K1 (keeping K2 open) and simultaneously start a stopwatch. Measure the current i as a function of time at convenient intervals (say 10 seconds).

5. Decay of current during discharging: After charging C to the full voltage, discharge it through R. For this purpose disconnect the power supply by opening key K1. Close the Key K2 and start the stopwatch. Again measure the current as a function of time at convenient intervals (say, 10 seconds).

6. Study decay of current either during charging or during discharging the capacitor. Observation Tables: Value of the capacitance C=2200µF. Resistance R=

Sl. Time(in sec) Current(in µA) ln(i/i0) 1 10 2 20 . . . .

30 300

1. Plot ln(i/i0) as a function of time t. 2. Using the linear regression technique to fit the data and draw the best-fit line on the

graph.

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3. Determine the time constant of the circuit and compare it with the theoretically calculated value.

4. Estimate the error in for = .

References:

1. E. M. Purcell, Electricity and Magnetism (Berkley Physics Course - Volume 2), Tata McGraw-Hill (2008).

2. D. Halliday and R. Resnick, Physics II, Wiley Eastern Limited (1966).

Experiment: 3

Determination of the magnetic field intensity along the axis of a circular current carrying coil

Theory: For a circular coil of n turns, carrying a current I, the magnetic field along its axis is given by ( ) = 2( + ) / (1)

where R is the radius of the coil.

Fig. 3.1.

In this experiment, the coil is oriented such that the plane of the coil is vertical and parallel to the north-south direction. The axis of the coil and the field produced by the coil are parallel to the east-west direction (Fig. 3.1). The net field at any point x along the axis is the vector sum of the fields due the coil B(x) and that due to the earth BE. = ( ) (2)

Fig.3.2

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Procedure: 1. The apparatus consists of a coil mounted perpendicular to the base. A sliding compass

box is mounted on aluminium rails so that the compass is always on the axis of the coil. Orient the apparatus such that the coil is in the north-south plane. (You may use the red dot on the bar magnet as reference). Adjust the leveling screws to make the base horizontal. Make sure that the compass is moving freely. Connect the circuit as shown in the Fig. 3.2. Place the compass box at the centre of the coil and rotate it so that the pointers indicate 0-0. The experiment is to be performed only along one side of the coil.

2. Close the keys K and KR (make sure that you are not shorting the power supply) and adjust the current with the potentiometer, Rh so that the deflection of the pointer is between 50o and 60o. The current is kept fixed at this value for the rest of the experiment.

3. Note down the readings θ1 and θ2. Reverse the current by suitably connecting the keys of KR and note down θ3 and θ4.

4. Repeat the experiment by moving the compass box at intervals of 2cm along the axis upto 20cm (10 data points).

Observation Tables: No of turns of the coil, n =......... Radius of the coil, R = 10 cm Current in the coil, I =......... Permeability of air, µ0 = 4π× 10 −7 N/A 2 Earth’s magnetic field, BE = 0.39 × 10-4 T

All θs should be in suitable units.

1. Plot ( ) = tan against x. 2. Identify the values of for = 0, 10 & 20cms from the graph and compare with the

theoretical values. 3. Estimate the maximum possible errors in B at these positions.

References:

1. D. J. Griffiths, Introduction to Electrodynamics, Prentice-Hall (1999).

x (cm) θ1 θ2 θ3 θ4

4

θavg tan4

θ

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Experiment: 4

Determination of resonant frequency and quality factor of a LCR circuit

Theory: Consider the following LCR series circuit.

Fig.4.1 If the applied voltage is = ( ) = ( ), the output voltage is given by = + + = + + = + −

The magnitude of is given by | | = + −

The phase of with respect to is defined as = | | ( ), where the phase angle is defined as tan = Imaginary Real = − Depending on , and , the phase angle could be positive or negative.

Fig.4.2

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The current amplitude = | | ⁄ is plotted against the frequency = /2 in Fig.4.2 for different values of . The current becomes maximum at the resonant frequency for all . The maximum current amplitude decreases with increasing . For a given , when is small the reactance = is small and the reactance = 1 ⁄ is large. Thus the current is mainly capacitive and the impedance is dominated by , which keeps the current low. As increases, decreases and increases. The decrease in reduces the impedance and allowing the current to increase. The current becomes maximum when = corresponding to the resonant frequency . If the frequency is further increased beyond , will dominate over and cause increase in the impedance which will keep the current low again. There is an exact analogy between a LCR circuit and a forced damped harmonic oscillator. The equation for the charge oscillation in the LCR circuit considered here is given by + + = cos ( )

with natural frequency = 1 √ ⁄ . On the other hand, if a particle of mass is subject to an external periodic force ( ) = , the equation of motion is given by + + = cos ( ) where is the damping force constant, is the restoring force constant, with natural frequency = ⁄ . One finds both the equations are exactly same with the equivalent quantities: ≡ , ≡ , ≡ and ≡ 1 ⁄ . In both the cases, resonance occurs when the driving frequency is equal to the natural frequency . Resonant frequency : At resonance, the imaginary part of the impedance vanishes. For the series LCR circuit considered here the impedance is given by = + + = + ( − 1 ⁄ ) Hence, at resonance one has = 1 ⁄ or, = 1√

Quality factor Q of a circuit: Q determines how well the LCR circuit stores energy and it is defined by = 2

per cycle. The maximum energy stored in the inductor is 2⁄ with = . There is no energy stored in the capacitor at this instant because and are 90 out of phase. The energy lost in one cycle is (Power)x(time for cycle), i.e., × = × . Hence

= 2 2⁄ = = 1

There is another equivalent expression for and it is given by = ∆

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where ∆ω is the bandwidth of the resonance curve. corresponds to the width where VR drops to 1/√2 of its maximum value as shown in Fig.4.3. Note that, can be measured from the resonance curve itself, one does not need to know , , or to find .

Procedure:

1. Assemble the circuit as shown in Fig 4.1. 2. Set the function generator for sinusoidal signal and adjust the peak to peak amplitude

of the signal to some suitable value (around 1 to 2V) and keep it constant throughout the experiment.

3. Record the voltage drop across the resistance R as a function of frequency in a suitable step. Make sure that you have sufficient data point on either side of the resonance frequency (so as to measure the value of τ).

Observation Table: L=4mH, C=0.1µF. R=

Sl. Frequency (f, in kHz) ω = 2πf

Input voltage (V)

Output voltage (across R, in mV)

1 2.5 2 3.0 3 3.5 .

1. Plot the amplitude of voltage VR against the frequency . 2. Obtain the resonant frequency and measure the bandwidth τ from the plot.

Estimate Q factor. 3. Compare the values of and Q with theoretical values. 4. Estimate the maximum possible error in the measurement of Q.

References:D. Halliday and R. Resnick, Physics II, Wiley Eastern Limited (1966).

Fig.4.3

∆

Vm/√2

Vm

VR

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Experiment: 5

Determination of type and density of majority charge carriers in an extrinsic semiconductor by Hall Effect

Theory: Consider a rectangular slab of semiconductor with thickness d kept in XY plane (see Fig.4.l). An electric field is applied in x-direction so that a current I flows through the sample. If w is width of the sample and d is the thickness, the current density is given by Jx= I/wd

Fig.5.1

Now a magnetic field B is applied along positive Z-axis (Fig.5.1). The moving charges are under the influence of magnetic force ⃗ × ⃗ , which results in accumulation of majority charge carriers towards one side of the material (along Y direction in the present case). This process continues until the electric force due to accumulated charges (qE) balances the magnetic force. So, in a steady state the net Lorentz force experienced by charge carriers will be zero. The potential thus developed across Y direction is known as Hall voltage VH (perpendicular to both current and the field directions) and this effect is called Hall effect. Thus under steady state condition ⃗ = ⃗ + ⃗ × ⃗ = 0 (1)

where ⃗ is the drift velocity of charge carriers. In the present case Eq.(1) can be written as, = = (2)

where n is the charge density and q is the charge of each carrier. The ratio (Ey/JxBz) is called the Hall co-efficient RH. Thus = = (3)

From Eqs.(2) and (3), the Hall co-efficient can also be written as

Y

X

Z Jx

q

B

d

w VH

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= 1 (4)

From Eq.(4) it is clear that the type of charge carrier and its density can be estimated from the sign and the value of Hall co-efficient RH. It can be obtained by studying the variation of VH

as a function of I for a given B.

Experimental Setup: Sample is mounted on an insulating sheet with four spring type pressure contacts. A pair of green colour leads is provided for current and that of red colour for Hall voltage. Note the direction of current and voltage measurement carefully. Do not exceed current beyond 8 mA. The unit marked "Hall Effect Set-up" consists of a constant current source (CCS) for supplying current to the sample and a digital milli-voltmeter to measure the Hall voltage. The unit has a single digital display used for both current and Hall voltage measurement. For applying the magnetic field an electromagnet with a constant current supply is provided. It is capable of generating a magnetic field of 0.75 Tesla for 10mm gap between its pole pieces. The magnetic field can be measured using gauss meter along with the given Hall probe.

Procedure:

1. Connect the leads from the sample to the "Hall effect Set-up" unit. Connect the electromagnet to constant current supply.

2. Switch on the electromagnet and set suitable magnetic field density (< 0.3 Tesla) by varying the current supplied to the electromagnet. You can measure this magnetic field density using the Hall probe. Find out the direction of magnetic field using the given bar magnet. Do not change the magnetic field and perform steps 3 to 6 with this value only.

3. Insert the sample between the pole pieces of the electromagnet such that the direction of magnetic field is perpendicular to the direction of current and the line connecting the Hall voltage probes (Fig.5.1).

4. From the direction of current and magnetic field estimate the direction of accumulation of majority carriers. Connect the one of the Hall voltage probes into which charge carriers are expected to accumulate to the positive side of the milli-voltmeter. Connect the other Hall voltage probe to the negative side of the milli-voltmeter. Don’t change this voltmeter connection throughout the experiment.

5. Record the Hall voltage as a function of sample current. Collect four sets of readings: V1(B,I), V2(B,-I), V3(-B,I) and V4(-B,-I) for each current; V1 is for positive (initial) current and field, V2 is for reverse current, V3 is for reverse field, V4 is for reverse field and current. Note that field direction can be changed by changing the direction of current through the electromagnet.

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Suggested Observation Table: Sample number: Thickness d=0.5mm

Sample current I (mA)

V1(B, I) (mV)

V2(B, -I) (mV)

V3(-B, I) (mV)

V4(-B, -I) (mV)

Hall voltage VH(mV)

1. The Hall voltage VH is obtained by, = [ ( , )− ( ,− )− (− , ) + (− ,− )]4

Thus the stray voltage due to thermo-emf and misalignment of Hall voltage probe is eliminated.

2. Calculate , determine charge carrier density and the type of majority charge carrier.

3. Estimate maximum possible errors in the measurement of . References:

1. C. Kittel, Introduction to Solid State Physics, John Wiley & Sons (1996). 2. D. J. Griffiths, Introduction to Electrodynamics, Prentice-Hall (1999).

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Experiment: 6

Determination of speed of light in glass by dispersion of light through a glass prism

Theory: Light travels with the speed c (2.998 × 108m/s) in vacuum. In a material medium its speed (v) is less. As a result, light waves undergo refraction at the interface of two media. In this experiment, we take the material of the medium in the form of a glass prism. A parallel stream of waves travelling from a medium 1 (here air) is incident on the interface of air and glass (of the prism), at the angle incidence θ1. The angle of refraction is θ2 Snell’s law connects the two by the relation, sin = sin (1)

where n1 and n2 are the refractive indices of the two media 1 and 2 respectively. Since the medium 1 here is air (n1≅ 1.000), the speed of light in the second medium is given by = sin sin (2)

We know that for a certain direction of incidence, the ray travels parallel to the base of the prism and the angular displacement of the final ray that emerges from the second interface of the prism has the lowest possible value. For this minimum angular deviation, δm, and the corresponding incidence angle θ1, the geometry of symmetric propagation inside the medium leads to the equation for ν = sin sin [ ] (3)

where α is the angle of the prism. Thus, from a measurement of the angle of the prism and the value of the minimum angular displacement δm, the speed of light in the material can be determined.

Procedure:

1. A spectrometer is used to measure the necessary angles. The spectrometer consists of three units: (a) collimator, (b) telescope, and (c) prism table

2. The prism table, its base and the telescope can be moved independently around their common vertical axis. A circular angular scale mounted with the telescope enables one to read angular displacements with the help of two verniers mounted on the prism table and located diametrically opposite to each other.

3. In this experiment, we need to produce a parallel beam of rays to be incident on the prism. This is done with the help of the collimator. The collimator has an adjustable rectangular slit at one end and a convex lens at the other end. When the illuminated slit is located at the focus of the lens (Fig. 6.1), a parallel beam of ray emerges from

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the collimator. We can test this point with the help of the telescope adjusted to receive the parallel rays. We first prepare the telescope towards this purpose as follows:

Fig. 6.1

4. Setting the eyepiece: Focus the eyepiece of the telescope on its cross-wires (for

viewing the cross wires against a white background) such that a distinct image of the cross-wires is seen by you. Henceforth do not disturb the eyepiece.

5. Setting the telescope: Focus the telescope onto a distant (infinity!) object. Test for the absence of a parallax between the image of the distant object and the vertical crosswire. Parallax effect (i.e. separation of two things when you move your head across horizontally) exits, if the cross-wire and the image of the distant object are not at the same distance from your eyes. Now the telescope is adjusted for receiving parallel rays. Henceforth do not disturb the telescope focusing adjustment.

6. Setting the collimator: Use the telescope for viewing the illuminated slit through the collimator and adjust the collimator (changing the separation between its lens and slit) till the image of the slit is brought to the plane of cross-wires as judged by the absence of parallax between the image of the slit and cross-wires.

7. Optical leveling of the prism table: Check whether the prism table is horizontally leveled (use a spirit level).

8. Finding the angle of minimum deviation (δm): Unlock the prism table for the measurement of the angle of minimum deviation. Locate the image of the slit after refraction through the prism as shown in Fig.6.2. Keeping the image always in the field of view, rotate the prism table till the position where the deviation of the image of the slit is smallest. At this position, the image will go backward, even when you keep rotating the prism table in the same direction. Lock both the telescope and the prism table and to use the fine adjustment screw for finer settings. Note the angular position of the prism. In this position the prism is set for minimum deviation. Without disturbing the prism table, remove the prism. Unlock the telescope (not the prism table) and turn the telescope towards the direct rays from the collimator. Note the reading of this position. The angle of the minimum angular deviation, viz, δm is the difference between the readings for these two settings.

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Observation Tables: Spectrometer No: ____, Angle of the prism = 60°. LC=Least count of main scale, VC (Vernier constant) =1 main scale division-1 vernier scale division. If m VSD=n MSD, VC=(1-n/m)LC. Angle of minimum deviation: Vernier A Vernier B Mini. deviation

position (units)

Direct position (units)

δm

(units) Mini. deviation

position (units)

Direct position (units)

δm (units)

MSR

V S C

TR ( ) MS R

V S C

TR ( ) | − | M

S R

V S C

TR ( ) MS R

V S C

TR ( ) | − |

DLF

DRT

MSR – Main Scale Reading; VSC – Vernier scale coincident; LC-Least Count; TR – Total Reading = MSR + VSC * VC. DLF: Deviation to the left, DRT: Deviation to the right. The angle should be taken in degree, minute and seconds.

1. Knowing and , calculate . 2. Estimate error in .

References:

1. F. A. Jenkins and H. E. White, Fundamentals of Optics, McGraw-Hill (1981). 2. E. Hecht, Optics, Pearson Education (2002).

Telescope Minimum deviation position

Fig.6.2 Minimum deviation geometry

Telescope Direct position

Prism table

Collimator Prism

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Experiment : 7

Determination slit width by Fraunhofer diffraction through a single slit Theory: Fraunhofer diffraction pattern (far field) is obtained when the source of light and the observation screen on which diffraction pattern observed are far away (effectively at infinite distance) from the aperture. The diffraction pattern for a single slit S is shown in Fig. 7.1. The variation of the intensity I at the observation plane P is given by = (1)

where I0 is the intensity of the incident light and 2βm is the phase difference between the waves coming from edges of the slit given by = 12 (2)

where d is the width of the slit and k = 2π / λ, is the magnitude of the wave vector. The minima of the fringes, according to eqn.(7.1) are located at = , = ±1, ±2. ±3 …

(3)

Hence, 12 2πλ d sinθ = mπ , or, sinθ = mλd

For small angle, sin ≈ = ∆ ⁄ where ∆ = | − |, is the position of the mth minima and is the position of the principal maximum, and is the distance between the slit and the photodiode. Therefore, = ∆ = mλd

∆

Fig.7.1

Source

Screen Slit x

-3

-2

-1

3

0

1

2

D

d

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Experimental Setup: The experimental set up consists of a coherent source (laser), a slit and a photodiode (with a slit on to it for better resolution) as shown in Fig. 7.2. All these components are mounted on an optical bench along with a micrometer stage for each of them for alignment and measurement. Intensity will be measured by measuring the photo voltage developed across the photodiode.

Fig. 7.2 Procedure:

1. Align the laser and the slit to get a clear diffraction pattern. You can observe the diffraction pattern by placing a white screen at certain distance from the slit, Fig.7.2.

2. Align the photodiode so that diffraction pattern falls on to it (ensure that you should be able to record the full diffraction pattern up to few orders on both side of principal maxima by moving micrometers of photodiode only.)

3. Record the intensity (photovoltage) as a function of x (Fig. 7.1) by moving the photodiode along the line perpendicular to the laser beam in a convenient step.

Observation Tables: Distance between the slit and the screen D= Wavelength of the laser beam = 650nm.

Sl. Micrometer reading

(units) Photovoltage (units)

1. Plot the photovoltage as a function of x. 2. Note down the location of the minima’s (m) from your graph (at least three), calculate

the corresponding diffraction angle θm for various minima’s.

3. Find the slit width = ⁄ . 4. Estimate the maximum possible error in the measurement of d.

References:

1. A. Ghatak, Optics, Tata McGraw-Hill (2005). 2. F. A. Jenkins and H. E. White, Fundamentals of Optics, McGraw-Hill (1981). 3. E. Hecht, Optics, Pearson Education (2002).

Sl. Position of minima from principal maxima (units)

m

θm d

Laser

Slit Photodiode

DMM

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Experiment: 8

Determination of radius of curvature of a plano convex lens using the

method of Newton’s ring Theory: Interference effects are observed in a region of space where two or more coherent waves are superimposed. Depending on the phase difference, the effect of superposition is to produce varying intensities - varying from a maximum of 2A to a minimum of zero where A is the amplitude of the waves. For the interference effect to be observed, the two waves should be coherent. One way of realizing two coherent waves is to derive them from a single wave front. In this experiment, a single wave front is split into two portions by amplitude division. Fig.8.1(a) shows the Newton’s ring assembly with a plano convex lens (L) placed over a plane glass plate (G). The system is illuminated from above by light from a sodium vapour lamp. Interference fringes are formed due to the superimposition of light reflected by the upper and lower boundaries of the air gap formed between the lower face of the lens L and the glass plate G. The thickness of the air gap is zero at the point of contact and increases as we go radially outward from the centre. Loci of constant air gap thickness d will be concentric circles around the point of contact. A fringe of minimum intensity is obtained if the separation (air gap) satisfies the relation, 2 cos = 2 2 where n is the refractive index of the medium (air in this case), λ is the wavelength of the light used, is the angle of refraction at the glass to air interface. For normal incidence = 0 and 2d = mλ.

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Fig.2

The centre of the ring corresponds to d = 0, and this condition is satisfied with m = 0. At a small distance away from the centre, where the air thickness d = λ/2, we have the first dark ring. When d = mλ/2, we obtain the mth dark ring. Midway between successive dark rings, we have rings of maximum intensity corresponding to = (2 + 1) . We can relate the separation d to the distance ρ (radius of the ring) from the centre (Fig.1b). If R is radius of curvature of the convex surface of the lens L, then = − ( − ) . For small d, we may write ≈ . This gives the radius of the mth dark ring as = and for right rings, = (2 + 1) 2 Procedure:

1. As the Plano-convex lens (L) is placed over a plane glass plate (G) as shown in Fig. 1a and a glass plate is mounted above the centre of the lens at about 45° to the horizontal, a number of alternate bright and dark concentric rings appear and can be viewed through the microscope. The microscope must be focused so that there is no parallax between the cross wires and the image of the rings. The cross wires can be rotated to make one of the cross wires to pass through the central spot of the ring system and the other tangential to one of the rings. Move the microscope along the horizontal scale till the tangential cross wire coincides with different successive rings.

2. Measure Dm=2ρ, the diameter of a Newton’s ring by placing the cross wire tangential to a particular ring at the two extreme positions and noting the main and vernier scale readings. Find ρ for many rings. Note that the centre ring is the 0th order fringe. As one moves radially outward from the centre, higher order interference fringes (rings) are encountered.

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Observational table: Vernier constant of the main scale: VC= Wavelength of the light used = 589.3 . Take data either from left to right or from right to left. Consider at least 10 dark rings. Table-1: Measurement of the diameter of the ring

Ring No (m)

Microscope readings Diameter Dm(2ρm) =x2-x1

Dm2

Left (x1) Right (x2) Main Scale

Vernier Total Main Scale

Vernier Total

10 1

1. Plot Dm2 versus m.

2. Estimate = by least squares fitting to data points. 3. Calculate error in R.

References:

1. A. Ghatak, Optics, Tata McGraw-Hill (2005). 2. F. A. Jenkins and H. E. White, Fundamentals of Optics, McGraw-Hill (1981). 3. E. Hecht, Optics, Pearson Education (2002).

ph110 physics laboratory instruction manual

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