Experiment 5 : To determine the angle of minimum deviation for a given prism by plotting a graph between angle of incidence and angle of deviation.
Aim :
To determine the angle of minimum deviation for a given prism by plotting a graph between angle of incidence and angle of deviation.
Apparatus :
A glass prism, drawing boards, white sheet of paper, paper pins, drawing pins, half metre rod and a protector, a drawing board.
Theory :
A prism is a wedge-shaped body made from a refracting medium (glass) bounded by two plane faces inclined to each other at some angle. A ray PQ incident on face AB of prism ABC making angle of incident i and is refracted along QR making angle of refraction $r_1$. At point R, the ray is refracted along RS making angle of emergence e and angle of refraction $r_2$ an shown in Fig. A is angle of prism and $\delta$ is angle of deviation .
It can be proved that$
$A + \delta = i + e$
and
$A = r_1 + r_2$
At minimum deviation ($\delta = \delta_m$)
$i = e$ and $r_1 = r_2 = r$
So Eq. (i) becomes
$A + \delta_m = i + i$
or
$i = \frac{A + \delta_m}{2}$
From Eq. (ii), A = r + r
or
$r = \frac{A}{2}$
The refractive index (\mu) of the prism is given by :
$\mu = \frac{\sin \frac{A + \delta_m}{2}}{\sin \frac{A}{2}}$
Procedure :
To find angle of prism
- Fix a sheet of white paper on the drawing board with the help of drawing pins.
- Place the prism on the paper and mark its boundary ABC with a sharp pencil as shown in Fig
- Remove the prism. Mark a point Q at about centre of AB. Draw normal to the surface AB at the point Q. Draw a line PQ making an angle of incidence with the normal $QN_1$. Fix two pins $P_1$ and $P_2$ on this line. See that the distance between the pins is not less than 10 cm. Now place the prism on boundary marked as ABC
- Close one eye (say left) and looking through the face AC, bring your right eye in line with the images of $P_1$ and $P_2$. Fix pins $P_3$ and $P_4$ about 10 cm apart vertically on the white paper sheet with their tips in line with the tips of the images of pins $P_1$ and $P_2$.
- Remove the pins one by one and draw small circles around their positions as shown in Fig Remove the prism.
- Join the points $P_3$ and $P_4$ with a sharp pencil and obtain the emergent ray RS. Produce it backward to meet the incident ray at point T.
- Draw a normal to the surface AC at point R and measure the angle of emergence e, angle of deviation \delta and angle of prism A with protector.
- Change the angle of incidence to 40°, 45°, 50°, 55°, 60° and find the corresponding angle of deviation $\delta$ and emergence (e) each time as explained above.
Observations :
Angle of prism A = 60°.
Calculations :
Plot a graph with angle of incidence i along x-axis and corresponding angle of deviation $\delta$ along y-axis by taking suitable scale.
The graph will be as shown in Fig, and note the value of angle of minimum deviation $\delta_m$ from the graph value of $\delta$ = $37^\circ$
Given the formula for refractive index:
$n = \frac{\sin\left(\frac{A + D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
And the given values:
$A = 60^\circ$
$D_m = 37^\circ$
Substitute these values into the formula:
$n = \frac{\sin\left(\frac{60^\circ + 37^\circ}{2}\right)}{\sin\left(\frac{60^\circ}{2}\right)}$
First, calculate the terms inside the sine functions:
$\frac{60^\circ + 37^\circ}{2} = \frac{97^\circ}{2} = 48.5^\circ$
$\frac{60^\circ}{2} = 30^\circ$
Now, substitute these back into the equation:
$n = \frac{\sin(48.5^\circ)}{\sin(30^\circ)}$
Calculate the sine values:
$\sin(48.5^\circ) \approx 0.74896$
$\sin(30^\circ) = 0.5$
Finally, perform the division:
$n = \frac{0.74896}{0.5}$
$n \approx 1.49792$
Rounding to one decimal place as suggested in the problem statement ("we get n= 1.5"), we get:
$n \approx 1.5$
Result :
(i) The angle of deviation $\delta$ first decreases with increase in i, attains a minimum value $\delta_m$ and then increases for further increase in i.
(ii) Refractive index of prism material = 1.5
Precautions
(i) The boundary of prism must be drawn with a sharp pencil.
(ii) The point of incidence should be in the middle portion of the prism.
(iii) The angle of incidence should be between 30° and 60°.
(iv) The same angle of prism should be used for all observations and, therefore, it should be marked with ink before the start of experiment.
(v) Pins P and Q should be well illuminated and all pins should be fixed vertically.
(vi) Encircle the pin-pricks immediately after removing them.
Sources of error
1. There may be an error in measuring the values of the angles.
2. The distance between the pins $P_1$, $P_2$ and $P_3$, $P_4$ may be less than 10cm.
3. Pins fixed on the paper may not be vertical.
VIVA VOCE :
Q.1. Define Prism ?
Ans. It is a portion of transparent medium bounded by two plane faces inclined to each other at some angle.
Q. 2. What is refraction?
Ans. The phenomenon in which light ray changes its path when it passes from one medium to another is called refraction.
Q. 3. What are the laws of refraction?
Ans. (i) Incident ray, refracted ray and normal lie in the same plane.
(ii) $\frac{\sin i}{\sin r}$ = constant = $\mu$ (called Snell's law).
Q. 4. What is Snell's law ?
Ans. $\frac{\sin i}{\sin r}$ = constant.
Q. 5. What is meant by angle of deviation ?
Ans. It is the angle through which the incident ray turned from its original path after passing through the prism.
Q. 6. Define refracting edge of a prism.
Ans. The plane where the two refracting surfaces meet.
Q. 7. What is the relation between angle of incidence, angle of refraction, angle of deviation and angle of prism, when a ray of light passes through it?
Ans. $A + \delta = i + e$
and
$A = r_1 + r_2$
Q. 8. What is minimum deviation?
Ans. The smallest value of angle of deviation is called minimum deviation.
Q. 9. What is the relation between angle of incidence and angle of emergence when the light ray suffers minimum deviation?
Ans. i = e.
Q. 10. Is it possible to have minimum deviation for two values of angle of incidence?
Ans. No.
Q. 11. On which factors does $\delta_m$ depend ?
Ans. (i) Wavelength of colour of light.
(ii) The nature of transparent material of which the prism is made of.
Q. 12. What is the name given to a phenomenon of splitting of white light into component colours ?
Ans. Dispersion of light.
Q. 13. What do you mean by angle of prism?
Ans. Refracting angle of the prism is called angle of the prism. It is generally 90°.
Q. 14. What is an emergent ray?
Ans. This is that ray which comes out of the prism.
Q. 15. On what factors does the angle of deviation depend?
Ans. It depends upon
(i) Angle of incidence
(ii) The material of the prism
(iii) Refracting angle of the prism
(iv) Colour of the light used.
Q. 16. What happens in the case of minimum deviation position?
Ans. At minimum deviation, the angle of emergence is equal to the angle of incidence and the refracted ray is parallel to the base of the prism.
Q. 17. Should the heads of the pins or feet of the pins be in straight line?
Ans. We look towards the feet. They have to be in a straight line.
Q. 18. What does \delta-i graph indicate?
Ans. $\delta-i$ graph indicates that \delta decreases on increasing 'i', attains a minimum value $\delta_m$ and then increases for further increase in 'i'.
Q. 19. What are the units of refractive index?
Ans. Since it is the ratio of two similar quantities and have no unit.
Q. 20. What is dispersion of light?
Ans. Dispersion of light is the splitting of white light into its constituent colours on passing through a prism.