To Find the Focal Length of a Convex Lens by Plotting Graphs Between U and V or Between 1/u and 1/v

 

To Find the Focal Length of a Convex Lens by Plotting Graphs Between U and V or Between 1/u and 1/v

Aim
To find, the focal length of a convex lens by plotting graphs between u and v or between 1/u and 1/v.

Apparatus
An optical bench with three uprights (central upright fixed, two outer uprights with lateral movement), a convex lens with lens holder, two optical needles, (one thin, one thick) a knitting needle and a half metre scale.

Theory
The relation between u, v and f for a convex lens is

where,
f = focal length of convex lens
u = distance of object needle from optical centre of the lens
v = distance of image needle from optical centre of the lens.

Note. According to sign-convention, u has negative value and v has positive value. Hence, f comes positive.

Ray diagram

Procedure
To determine rough focal length

1. Mount the concave mirror in mirror holder.
2. Go out in the open and face the mirror towards distant tree or building.
3. Obtain the image of the tree or the building on a white painted wall (screen) and move the mirror forward and backward to get a sharp image on the wall.
4. Measure the distance between the mirror and the wall (screen). This will be equal to the rough focal length of the mirror.
   To set the lens
5. Clamp the holder with lens in a fixed upright and keep the upright at 50 cm mark.
6. Adjust the lens such that its surface is vertical and perpendicular to the length of the optical bench.
7. Keep the upright fixed in this position throughout.
   To set the object needle
8. Take the thin optical needle as object needle (O). Mount it in outer laterally move¬able upright near zero end.
9. Move the object needle upright and clamp it at a distance (in full cms) nearly 1.5 times the obtained rough focal length of the lens.
10. Adjust height of the object needle to make its tip lie on horizontal line through the optical centre of the lens.
11. Note the position of the index mark on the base of the object needle upright.
    To set the image needle
12. With left eye closed, see with the right open eye from the other end of the optical bench. An inverted and enlarged image of the object needle will be seen. Tip of the image must lie in the middle of the lens.
13. Mount the thick optical needle (image needle) in the fourth upright near the other end of the optical bench.
14. Adjust the height of the image needle so that its tip is seen in line with the tip of the image when seen with right open eye.
15. Move the eye towards right. The tips will get separated. The image tip and the image needle tip have parallax.
16. Remove the parallax tip to tip.
17. Note the position of the index mark on base of the image needle upright.
18. Record the position of the index marks on the base of upright of the lens, the object needle and the image needle in the table against observation 2.
    To determine index correction
19. Find the index correction for distance between optical centre of lens and tip of the object needle and also for distance between optical centre of lens and tip of the image needle as described.
    To get more observations
20. Move object needle upright towards mirror in steps of 1 cm to get observation 2 and 1. Repeat the experiment.
21. Move object needle upright away from mirror (from position of observation 2) in steps of 1 cm to get observations 4, 5 and 6. Repeat the experiment.
22. Record all the observations as given ahead.
    (Note. Same as in Experiment 1).

Observations
Rough focal length of the given convex lens = …….cm
Actual length of the knitting needle x=…….cm
Observed distance between the object needle and the lens
when knitting needle is placed between them y =…….cm
Observed distance between the image needle and the
lens when knitting needle is placed between them z =…….cm
Index correction for the object distance u, x -y =…….cm

Index correction for the image distance v, x-z =…….cm

Calculations
Calculations of focal length by graphical methods:
(i) u-v Graph. Select a suitable but the same scale to represent u along X’-axis and v along Y-axis. According to sign conventions, in this case, u is negative and v is positive. Plot the various points for different sets of values of u and v from observation table second quadrant. The graph comes out to be a rectangular hyperbola as shown in graph between u and v.
Draw a line OA making an angle of 45° with either axis (i.e., bisecting ∠YOX’) and meeting the curve at point A. Draw AB and AC perpendicular on X’- and Y-axes, respectively.

The values of u and v will be same for point A. So the coordinates of point A must

Explanation
Same as for concave mirror:
(iii) Another u-v Graph. Select a suitable but the same scale to represent u along
X’-axis and v along Y-axis. Mark the points at distances u1, u2, u3,…… etc. along the OX’-axis
and the corresponding points at distances v1, v2, v3,…… etc. along OY- axis for different sets of observations from the table.
Draw straight lines joining u1 with v1; u2 with v2; u3 with v3;……. etc. These lines will intersect at point K as shown in the following graph.
Draw KL and KM perpendiculars on X’- and Y-axes, respectively

Explanation
Same as for concave mirror:
Note. It will be better to choose any four suitable sets of (a, v) from the observation table. All the six sets of observations may complicate the graph.

Precautions

1. Tips of the object and image needles should lie at the same height as the centre of the lens.
2. Parallax should be removed from tip to tip by keeping eye at a distance at least 30 cm away from the needle.
3. The object needle should be placed at such a distance that only real, inverted image of it is formed.
4. Index correction for u and v should be applied.

Sources of error

1. The uprights may not be the vertical.
2. Parallax removal may not be perfect.

Result
The focal length of the given convex lens as determined from

 

1.Focal length from f =uv/(u-v) =

2.From u-v graph  focal length =

3.From 1/u -1/v graph focal length =

To determine refractive index of a glass slab using a travelling microscope

 

To determine refractive index of a glass slab using a travelling microscope

Aim
To determine refractive index of a glass slab using a travelling microscope.

Apparatus
Three “glass slabs of different thickness but same material, a travelling microscope, lycopodium powder. A slab is a piece of transparent material with rectangular faces. All faces are transparent and opposite faces are parallel. The dimension along with the light travels inside the slab is called its thickness.
A Short Description of a Travelling Microscope
It is a compound microscope fitted vertically on a vertical scale. It can be moved up and down, carrying a Vernier scale moving along the main scale.
In any position, the reading is taken by combining main scale and vernier scale reading.

Theory

Diagram

Procedure
Adjustment of travelling microscope

1. Place the travelling microscope (M) on the table near a window so that sufficient light falls on it.
2. Adjust the levelling screws so that the base of the microscope becomes horizontal.
3. Make microscope horizontal. Adjust the position of the eye piece so that the cross wires are clearly visible.
4. Determine the vernier constant of the vertical scale of the microscope.
   Other steps
5. Make a black-ink cross-mark on the base of the microscope. The mark will serve as
   point P. ,
6. Make the microscope vertical and focus it on the cross at P, so that there is no parallax between the cross-wires and the image of the mark P.
7. Note the main scale and the vernier scale readings (R1) on the vertical scale.
8. Place the glass slab of least thickness over the mark P.
9. Raise the microscope upwards and focus it on the image P1 of the cross-mark.
10. Note the reading (R2) on the vertical scale as before (Step 7).
11. Sprinkle a few particles of lycopodium powder on the surface of the slab.
12. Raise the microscope further upward and focus it on the particle near S.
13. Note the reading (R3) on the vertical scale again (Step 7).
14. Repeat above steps with other glass slab of more thicknesses.
15. Record observations in tabular form as given below.

Observations and calculations
Vernier constant (least count) for vertical scale of microscope = 0.001cm.

S  No	Reading on vertical scale while Focused microscope in			Real thickness (  R3 – R1) cm	Apparent thickness (  R3 – R2) cm	Refractive index n = (  R3 – R1)/ (  R3 – R2)
	Cross mark without glass slab R1  cm	Cross mark with glass slab R2 cm	Lycopodium Powder R3  cm
1 2 3	3.4 3.425 3.35	3.845 4.052 3.95	4.6 5.226 5.1	1.2 1.8 1.75	0.775 1.173 1.15	1.54 1.53 1.52

 

Precautions

1. In microscope, the parallax should be properly removed.
2. The microscope should be moved in upper direction only to avoid back lash error.

Sources of error
The microscope scale may not be properly calibrated.

Result

The ratio (  R₃ – R₁)/(  R₃ – R₂) = constant

It gives refractive index n= (1.54+1.53+1.52)/3 = 1.53

To draw the I-V characteristic curve of a p-n junction in forward bias and reverse bias

 

To draw the I-V characteristic curve of a 

p-n junction in forward bias and reverse bias

Aim
To draw the I-V characteristic curve of a p-n junction in forward bias and reverse
bias.
Apparatus
A p-n junction (semi-conductor) diode, a 3 volt battery, a 50 volt battery, a high resistance rheostat, one 0-3 volt voltmeter, one 0-50 volt voltmeter, one 0-100 mA ammeter, one 0-100 μA ammeter, one way key, connecting wires and pieces of sand paper.

Theory
Forward bias characteristics. When the p -section of the diode is connected to positive terminal of a battery and n-section is connected to negative terminal of the battery then junction is said to be forward biased. With increase in bias voltage, the forward current increases slowly in the beginning and then rapidly. At about 0.7 V for Si diode (0.2 V for Ge), the current increases suddenly. The value of forward bias voltage, at which the forward current increases rapidly, is called cut in voltage or threshold voltage.
Reverse bias characteristics. When the p -section of the diode is connected to negative terminal of high voltage battery and n-section of the diode is connected to positive terminal of the same battery, then junction is said to be reverse biased.

When reverse bias voltage increases, initially there is a very small reverse current flow, which remains almost constant with bias. But when reverse bias voltage increases to sufficiently high value, the reverse current suddenly increases to a large value. This voltage at which breakdown of junction diode occurs (suddenly large current flow) is called zener breakdown voltage or inverse voltage. The breakdown voltage may^tarts from one volt to several hundred volts, depending upon dopant density and the depletion layer.

Diagram

Procedure
For forward-bias

1. Make circuit diagram as shown in diagram.
2. Make all connections neat, clean and tight.
3. Note least count and zero error of voltmeter (V) and milli-ammeter (mA).
4. Bring moving contact of potential divider (rheostat) near negative end and insert the key K. Voltmeter V and milli-ammeter mA will give zero reading.
5. Move the contact a little towards positive end to apply a forward-bias voltage (VF) of
   0. 1 V. Current remains zero.
6. Increase the forward-bias voltage up to 0.3 V for Ge diode. Current remains zero, (It is due to junction potential barrier of 0.3 V).
7. Increase VF to 0.4 V. Milli-ammeter records a small current.
8. Increase VF in steps of 0.2 V and note the corresponding current. Current increases first slowly and then rapidly, till VF becomes 0.7 V.
9. Make VF = 0.72 V. The current increases suddenly. This represents “forward break-down” stage.
10. If the VF increases beyond “forward breakdown” stage, the forward current does not change much. Now take out the key at once.
11. Record your observations as given ahead.
    For reverse-bias
12. Make circuit diagram as shown in diagram.
13. Make all connections neat, clean and tight.
14. Note least count and zero error of voltmeter (V) and micro-ammeter (μA).
15. Bring moving contact of potential divider (rheostat) near positive end and insert the key K Voltmeter V and micro-ammeter μA will give zero reading.
16. Move the contact towards negative end to apply a reverse-bias voltage (VR) of 0.5 V, a feebly reverse current starts flowing.
17. Increase VR in steps of 0.2 V. Current increases first slowly and then rapidly till VR becomes 20 V. Note the current.
18. Make VR = 25 V. The current increases suddenly. This represents “reverse break-down” stage. Note the current and take out the key at once.
19. Record your observations as given ahead.

Observations
For forward-bias
Range of voltmeter                        = 0 to  3V
Least count of voltmeter              = 0.1V
Zero error of voltmeter                =0V
Range of milli-ammeter              = 0 to 100mA
Least count of milli-ammeter    =  1 mA
Zero error of milli-ammeter      = ….0.mA

1.                         Table for forward-bias voltage and forward current

Sl. No.	Forward bias voltage, Vf (V)	Forward current, If (mA)
1
2
3
4
5
6
7
8
9
10

Note. The readings are as a sample.
For reverse-bias
Range of voltmeter                     =0 to 50V
Least count of voltmeter           = 1V.
Zero error of voltmeter              = 0V
Range of micro-ammeter          = 0 to 100μA
Least count of micro-ammeter = 2μA
Zero error of micro-ammeter    = 0…..
2.                        Table for reverse-bias voltage and reverse current

Sl. No.	Reverse bias voltage, Vr  (V)	Reverse current, Ir   (µA)
1
2
3
4
5
6
7
8
9
10

Note. The readings are given as a sample.

Calculations
For forward-bias
Plot a graph between forward-bias voltage VF (column 2) and forward current IF (column 3) taking VF along X-axis and IF along Y-axis.
This graph is called forward-bias characteristic curve a junction diode.


For reverse-bias
Plot a graph between reverse-bias voltage VR (column 2) and reverse current IR (column 3) taking VR along X-axis and IR along Y-axis.
This graph is called reverse-bias characteristic curve of a junction diode.

Precautions

1. All connections should be neat, clean and tight.
2. Key should be used in circuit and opened when the circuit is not being used.
3. Forward-bias voltage beyond breakdown should not be applied.
4. Reverse-bias voltage beyond breakdown should not be applied.

Sources of error
The junction diode supplied may be faulty.

Result
Junction resistance for forward-bias = 40 ohms
Junction resistance for reverse-bias = 2 x 106 ohms.

 

To determine angle of minimum deviation for a given prism by plotting a graph between angle of incidence and the angle of deviation

Aim
To determine angle of minimum deviation for a given prism by plotting a graph between angle of incidence and the angle of deviation.

Apparatus:

Drawing board, a white sheet of paper, prism, drawing pins, pencil, half-metre scale, office pins, graph paper and a protractor.

Theory
The refractive index in) of the material of the prism is given by

where, Dm angle of minimum deviation and A angle of the prism.

Diagram

Procedure

1. Fix a white sheet of paper on the drawing board with the help of drawing pins or tape.
2. Draw a straight line XX’ parallel to the length of the paper nearly in the middle of the paper.
3. Mark points Q1, Q2, Q3,… on the straight line XX’ at suitable distances of about 5 cm.
4. Draw normals N1Q1, N2Q2, N3Q3,… on points Q1, Q2, Q3,…  as shown in diagram.
5. Draw straight lines R1Q1, R2Q2, R3Q3,… making angles of  35°, 40°, … 60° (write value of the angles on the paper) respectively with the normals.
6. Mark one comer of the prism as A and take it as the edge of the prism for all the observations.
7. Put it prism with its refracting face AB in the line XX’ and point Q1 in the middle of AB.
8. Mark the boundary of the prism.
9. Fix two or more office pin P1 and P2 vertically on the line R1Q1. The distance between the pins should be 10 mm or more.
10. Look the images of point P1 and P2 through face AC.
11. Close your left eye and bring open right eye in line with the two images.
12. Fix two office pins P3 and P4 vertically, and 10 cm apart such that the open right eye sees pins P4 and P3 and images of P2 and P1 in one straight line.
13. Remove pins P3 and P4 and encircle their pricks on the paper.
14. Repeat steps 7 to 13 with points Q2, Q3,… for i = 40°,…, 60°.                                         To measure D in different cases
15. Draw straight lines through points P4 and P3 (pin pricks) to obtain emergent rays S1T1,S2T2,S3T3,……
16. Produce T1S1, T2S2, T3S3, … inward in the boundary of the prism to meet produced incident rays R1Q1, R2Q2, R3Q3,… at points F1, F2, F3,…
17. Measure angles K1F1S1,K2F2S2,K3F3S3,…….. These give angle of deviation D1,D2,D3,….
18. Write values of these angles on the paper.
    To measure A
19. Measure angle BAC in the boundary of the prism. This gives angle A.
20. Record your observations.

Observations
Angle of prism ‘A’ =……

Trial No.	Angle of Incidence (i)	Angle of Deviation (d)
1	35	48
2	40	42
3	45	37
4	50	40
5	55	43
6	60	45

 Calculations

Plot a graph between angle of incidence ∠i and angle of deviation ∠D by taking ∠i along X-axis and ∠D along Y-axis. From this graph, find the value of single of minimum deviation Dm corresponding to the lowest point of the graph.

Let we find the angle of minimum deviation from above graph  Dm 

Find the refractive index of prism by using below formula

 

Result

1. i-D graph indicates that as the angle of incidence (i) increases, the angle of deviation (D) first decreases, attains a minimum value (Dm ) and then starts increasing for further increase in angle of incidence.
2. Angle of minimum deviation, Dm = …37…..
3. Refractive index of the material of the prism, n = 1.49

Precautions

1. The angle of incidence should lie between 30°-60°.
2. The pins should be fixed vertical.
3. The distance between the two pins should not be less than 10 mm
4. Arrow heads should be marked to represent the incident and emergent rays.
5. The same angle of prism should be used for all the observations.

Sources of error

1. Pin pricks may be thick.
2. Measurement of angles may be wrong.

Result:

1. i-D graph indicates that as the angle of incidence (i) increases, the angle of deviation (D) first decreases, attains a minimum value (Dm ) and then starts increasing for further increase in angle of incidence.
2. Angle of minimum deviation, Dm =37 Degree
3. Refractive index of the material of the prism, n = 1.49