`color{blue} ✍️`Figure 9.23 shows the passage of light through a triangular prism ABC. The angles of incidence and refraction at the first face AB are `i` and `r_1,` while the angle of incidence (from glass to air) at the second face AC is `r_2` and the angle of refraction or emergence `e.`
`color{blue} ✍️`The angle between the emergent ray RS and the direction of the incident ray PQ is called the angle of deviation, `δ.`
In the quadrilateral AQNR, two of the angles (at the vertices Q and R) are right angles. Therefore, the sum of the other angles of the quadrilateral is 180º.
`color{purple}{∠A + ∠QNR = 180º}`
`color{blue} ✍️`From the triangle `QNR`,
`color{purple}{r_1 + r_2 + ∠QNR = 180º}`
`color{blue} ✍️`Comparing these two equations, we get
`color {blue}{r_1 + r_2 = A}`
..............(9.34)
`color{blue} ✍️`The total deviation `δ` is the sum of deviations at the two faces,
`color{purple}{δ = (i – r_1 ) + (e – r_2 )}`
that is,
`color {blue}{δ = i + e – A}`
............(9.35)
`color{blue} ✍️`Thus, the angle of deviation depends on the angle of incidence. A plot between the angle of deviation and angle of incidence is shown in Fig. 9.24. You can see that, in general, any given value of `δ,` except for `i = e,` corresponds to two values i and hence of `e.`
`color{blue} ✍️`This, in fact, is expected from the symmetry of i and e in Eq. (9.35), i.e., `δ` remains the same if i and e are interchanged. Physically, this is related to the fact that the path of ray in Fig. 9.23 can be traced back, resulting in the same angle of deviation.
`color{blue} ✍️`At the minimum deviation Dm, the refracted ray inside the prism becomes parallel to its base. We have
`color{purple}{δ = D_m, i = e}` which implies `r_1 = r_2.`
`color{blue} ✍️`Equation (9.34) gives
`color {blue}{2r = A or = A/2}`
.............(9.36)
In the same way, Eq. (9.35) gives
`color {blue}{D_m = 2i – A, or i = (A + D_m)//2}`
...............(9.37)
`color{blue} ✍️`The refractive index of the prism is
`color {blue}{n_(21) = (n_2)/(n_1) = (sin[(A+D_m)//2])/(sin [A//2])}`
............(9.38)
`color{blue} ✍️`The angles `A` and `D_m` can be measured experimentally. Equation (9.38) thus provides a method of determining refractive index of the material of the prism. For a small angle prism, i.e., a thin prism, `D_m` is also very small, and we get
`color{purple}{n_(21) = (sin[(A+D_m)//2])/(sin [A//2]) = ((A+D_m) //2)/(A//2)}`
`color{purple}{D_m = (n_(21) –1)A}`
`color{blue} ✍️`It implies that, thin prisms do not deviate light much.
`color{blue} ✍️`Figure 9.23 shows the passage of light through a triangular prism ABC. The angles of incidence and refraction at the first face AB are `i` and `r_1,` while the angle of incidence (from glass to air) at the second face AC is `r_2` and the angle of refraction or emergence `e.`
`color{blue} ✍️`The angle between the emergent ray RS and the direction of the incident ray PQ is called the angle of deviation, `δ.`
In the quadrilateral AQNR, two of the angles (at the vertices Q and R) are right angles. Therefore, the sum of the other angles of the quadrilateral is 180º.
`color{purple}{∠A + ∠QNR = 180º}`
`color{blue} ✍️`From the triangle `QNR`,
`color{purple}{r_1 + r_2 + ∠QNR = 180º}`
`color{blue} ✍️`Comparing these two equations, we get
`color {blue}{r_1 + r_2 = A}`
..............(9.34)
`color{blue} ✍️`The total deviation `δ` is the sum of deviations at the two faces,
`color{purple}{δ = (i – r_1 ) + (e – r_2 )}`
that is,
`color {blue}{δ = i + e – A}`
............(9.35)
`color{blue} ✍️`Thus, the angle of deviation depends on the angle of incidence. A plot between the angle of deviation and angle of incidence is shown in Fig. 9.24. You can see that, in general, any given value of `δ,` except for `i = e,` corresponds to two values i and hence of `e.`
`color{blue} ✍️`This, in fact, is expected from the symmetry of i and e in Eq. (9.35), i.e., `δ` remains the same if i and e are interchanged. Physically, this is related to the fact that the path of ray in Fig. 9.23 can be traced back, resulting in the same angle of deviation.
`color{blue} ✍️`At the minimum deviation Dm, the refracted ray inside the prism becomes parallel to its base. We have
`color{purple}{δ = D_m, i = e}` which implies `r_1 = r_2.`
`color{blue} ✍️`Equation (9.34) gives
`color {blue}{2r = A or = A/2}`
.............(9.36)
In the same way, Eq. (9.35) gives
`color {blue}{D_m = 2i – A, or i = (A + D_m)//2}`
...............(9.37)
`color{blue} ✍️`The refractive index of the prism is
`color {blue}{n_(21) = (n_2)/(n_1) = (sin[(A+D_m)//2])/(sin [A//2])}`
............(9.38)
`color{blue} ✍️`The angles `A` and `D_m` can be measured experimentally. Equation (9.38) thus provides a method of determining refractive index of the material of the prism. For a small angle prism, i.e., a thin prism, `D_m` is also very small, and we get
`color{purple}{n_(21) = (sin[(A+D_m)//2])/(sin [A//2]) = ((A+D_m) //2)/(A//2)}`
`color{purple}{D_m = (n_(21) –1)A}`
`color{blue} ✍️`It implies that, thin prisms do not deviate light much.