`color{blue} ✍️`Consider two lenses A and B of focal length `f_1` and `f_2` placed in contact with each other. Let the object be placed at a point O beyond the focus of the first lens A (Fig. 9.21).

`color{blue} ✍️`The first lens produces an image at `I_1.` Since image `I_1` is real, it serves as a virtual object for the second lens B, producing the final image at I. It must, however, be borne in mind that formation of image by the first lens is presumed only to facilitate determination of the position of the final image.

`color{blue} ✍️` In fact, the direction of rays emerging from the first lens gets modified in accordance with the angle at which they strike the second lens. Since the lenses are thin, we assume the optical centres of the lenses to be coincident. Let this central point be denoted by P. For the image formed by the first lens A,

`color {blue}{1/(v_1) - 1/u = 1/(f_1)}`

...............(9.27)

`color{blue} ✍️`For the image formed by the second lens `B`, we get

`color {blue}{1/v - 1/(v_1) = 1/(f_2)}`

..............(9.28)

`color{blue} ✍️`Adding Eqs. (9.27) and (9.28), we get

`color {blue}{1/v - 1/u = 1/(f_1) + 1/(f_2)}`

...............(9.29)

`color{blue} ✍️`If the two lens-system is regarded as equivalent to a single lens of focal length f, we have

`color{purple}{1/v - 1/u = 1/f}`

`color{blue} ✍️`so that we get

`color {blue}{1/f = 1/(f_1) + 1/(f_2)}`

..............(9.30)

`color{blue} ✍️`The derivation is valid for any number of thin lenses in contact. If several thin lenses of focal length `f_1, f_2, f_3,..`. are in contact, the effective focal length of their combination is given by

`color {blue}{1/f = 1/(f_1) + 1/(f_2) + 1/(f_3)}`

...........(9.31)

`color{blue} ✍️`In terms of power, Eq. (9.31) can be written as

`color {blue}{P = P_1 + P_2 + P_3 + …}`

............(9.32)

`color{blue} ✍️`where `P` is the net power of the lens combination. Note that the sum in Eq. (9.32) is an algebraic sum of individual powers, so some of the terms on the right side may be positive (for convex lenses) and some negative (for concave lenses).

`color{blue} ✍️`Combination of lenses helps to obtain diverging or converging lenses of desired magnification. It also enhances sharpness of the image. Since the image formed by the first lens becomes the object for the second, Eq. (9.25) implies that the total magnification m of the combination is a product of magnification `(m_1, m_2, m_3,...)` of individual lenses

`color {blue}{m = m_1 m_2 m_3 ...}`

............(9.33)

`color{blue} ✍️`Such a system of combination of lenses is commonly used in designing lenses for cameras, microscopes, telescopes and other optical instruments.

`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} ✍️`It has been known for a long time that when a narrow beam of sunlight, usually called white light, is incident on a glass prism, the emergent light is seen to be consisting of several colours.

`color{blue} ✍️`There is actually a continuous variation of colour, but broadly, the different component colours that appear in sequence are: violet, indigo, blue, green, yellow, orange and red (given by the acronym `"VIBGYOR"`).

`color{blue} ✍️`The red light bends the least, while the violet light bends the most (Fig. 9.25).

`color{blue} ✍️`The phenomenon of splitting of light into its component colours is known as dispersion.

`color{blue} ✍️`The pattern of colour components of light is called the spectrum of light. The word spectrum is now used in a much more general sense. we discussed the electromagnetic spectrum over the large range of wavelengths, from γ-rays to radio waves, of which the spectrum of light (visible spectrum) is only a small part.

`color{blue} ✍️`In a classic experiment known for its simplicity but great significance, Isaac Newton settled the issue once for all. He put another similar prism, but in an inverted position, and let the emergent beam from the first prism fall on the second prism (Fig. 9.26).



`color{blue} ✍️`The resulting emergent beam was found to be white light. The explanation was clear the first prism splits the white light into its component colours, while the inverted prism recombines them to give white light.

`color{blue} ✍️`Thus, white light itself consists of light of different colours, which are separated by the prism. It must be understood here that a ray of light, as defined mathematically, does not exist.

`color{blue} ✍️`An actual ray is really a beam of many rays of light. Each ray splits into component colours when it enters the glass prism. When those coloured rays come out on the other side, they again produce a white beam.

`color{blue} ✍️`We now know that colour is associated with wavelength of light. In the visible spectrum, red light is at the long wavelength end (~700 nm) while the violet light is at the short wavelength end (~ 400 nm).

`color{blue} ✍️`Dispersion takes place because the refractive index of medium for different wavelengths (colours) is different. For example, the bending of red component of white light is least while it is most for the violet. Equivalently, red light travels faster than violet light in a glass prism.

`color{blue} ✍️` Table 9.2 gives the refractive indices for different wavelength for crown glass and flint glass. Thick lenses could be assumed as made of many prisms, therefore, thick lenses show chromatic aberration due to dispersion of light.



`color{blue} ✍️`The variation of refractive index with wavelength may be more pronounced in some media than the other. In vacuum, of course, the speed of light is independent of wavelength.

`color{blue} ✍️`Thus, vacuum (or air approximately) is a non-dispersive medium in which all colours travel with the same speed. This also follows from the fact that sunlight reaches us in the form of white light and not as its components. On the other hand, glass is a dispersive medium.