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  Lens Formula and Spectra, Physics tutorial

The Lens Formula:

It has been seen that there is mathematical relationship linking focal length of the lens (f), object distance from lens (u) and image distance from the lens (v).

This relation is provided as

1/f = 1/v - 1/u..................................Eq. 1

The Equation given above is the same as that for curved mirrors (concave or convex). Therefore, if any two of the parameter f, u and v are known, equation can be utilized to find out third unknown parameter. As a result, this equation can be utilized to derive same pieces of information attained by ray tracing.

Lens Makers' Equation:

The best way to signify focal length of the lens is by using radius of curvature of two faces (or surfaces). Now deduce the expression for focal length f of the convex lens. Here, it is supposed that ray falls on flat surface of lens and these surfaces at which a ray enters and leaves like the surfaces of the prism. So, use prism formula to find out deviation d.

1658_Lens Makers Equation.jpg

When the ray of light enters the prism, it deviates. Consider d be the angle of deviation of light from a small angle prism. The small angle of the prism is A. Let m is the refractive index of the glass. Then the expression for the deviation of a ray passing through a prism can be written as

d = (μ - 1) A..................................Eq. 2 Θ

It is observed that the light rays are parallel to the principal axis. To focus these light rays on the focal point f, each ray is deflected by an angle Θ, then

d = h/f..................................Eq. 3

(Here the value of Θ is small < 15o)

It signifies that all light rays don't hit lens too far from centre. It is also known that the transparent material whose surface is spherical will deflect light rays according to Eq.3 i.e. will make helpful lens.

Combining Eq.2 and Eq.3, we get another expression

h/f = (μ - 1)A

Rewriting the above equation in another form, we get

1/f = (μ - 1)A/h................Eq.4

from (b) Figure given above, it can be observed that r1 and r2 are radius of curvature and

d = α + β.........................Eq.5

(The sum of two interior opposite angles is equal to the exterior angle) and d = A ..............Eq.6

Now substituting Eq.6 in Eq.5, and also substituting values of α = h/r1 and β = h/r2

We get

α = h/r1 and β = h/r2 or A/h = 1/r1 + 1/r2...........Eq.7

Substituting Eq.7 into Eq.4, we get

1/f = (μ - 1)(1/r1 + 1/r2) ...........Eq.8

Equation shows relation between focal length of lens in terms of its refractive index and radius of curvature. Now, it can be seen from the Eq. 8 that to get the short focal length f lens must have a small value of r1 and r2 and refractive index of material must be high. The Eq.8 is called as lens maker's equation. It can also be stated that values of radii of curvature of two spherical surfaces, which the lens of required focal length must have, can be found by using formula. Then two surfaces of glass can be given the calculated value of radii of curvature. Therefore, lens so produced will have required focal length.

Dispersion and Spectra:

It is found that if white light, like light from sun, passes through the prism, and an elongated colored patch of light is attained on the screen placed behind the prism.

This patch of light is known as a spectrum and it comprise of following colors arranged in the order of presentation with red light rays having the least of deviation and violet light rays comprising the most angle of deviation. Incident white light really comprises of the colors combined but as each component color has different speed in glass medium. Though, in ordinary air, speed of each of the components is almost same and the components can't be distinguished as they travel together with the common speed. Though, in the medium like glass the components travel with different speeds (that is red travel with the fastest speed and violet is the slowest) and thus they are related with different angles of deviation. Thus, component splits in their different colors as they travel through glass and separated the white light in its different colors.

Wavelength of red light in air is 750 x 10-9m (the longest wavelength) and violet light has wavelength in air 450 x 10-9m (the shortest wavelength).

The band of impure colors is attained on the screen S. Separation of colors (red, orange, yellow, green, blue, indigo and violet) by the prism is known as dispersion. In simple words, spreading of white light into full spectrum is known as dispersion. Formation of rainbows is the natural example of dispersion. Prism alone produces what is known as impure spectrum or continuous spectrum in which different colors overlap. Hence to generate a pure spectrum, the set up is given below:

153_Prism to obtain pure spectrum.jpg

Spectra:

Spectra is the study of wavelengths of radiation coming out from the hot body. There are two kinds of spectra, the emission spectra and absorption spectra.

Emission spectra:

When the atom is heated, it's electrons gain thermal energy until it gets to excited state. And in a very short time, electron can go back to the lower energy level, thus emitting energy in form of photons. For instance, iron has 400 different wavelengths in spectrum, but a very element has unique spectrum features of it's atoms. As a result a study of spectrum of the substance allows its composition to be readily determined. Spectroscopy is analysis of mixtures or compounds by the study of their spectra.

Kinds of Spectra:

i) Line Spectra:

This is the kind of spectra obtained from atoms or molecules and is displayed in form of lines. Line spectrum is attained when the gas is heated or large electric current is passed through it. It is significant to note that, only certain wavelengths of light are emitted and these are different far different elements and compounds. These lines are actually the image of narrow slit of spectrometer in which the light is incident. These lines happen in series.

The different series and their wavelength are:

Lyman series: 1/λ = R(1/12 - 1/μ)  μ = 2,3....

Balmer series: 1/λ = R(1/22 - 1/μ) μ = 3,4,5....

Paschen Series: 1/λ = R(1/32 - 1/μ) μ = 4,5.....

Bracket series: 1/λ = R(1/42 - 1/μ) μ = 5,6......

Where R is a Rydberg constant = 1.097 x 107 m-1

ii) Band Spectra:

This is the type of spectra obtained from molecules and it comprises of bands (fine lines) sharp atom end and at the other end.

iii) Continuous Spectra:

This is type of spectra attained from solids and liquids. As atoms and molecules are closely packed, there interaction exists between neighboring atoms, such that all radiations of different wavelength are emitted. For instance, light bulb filament produces the continuous spectrum.

iv) Absorption Spectra:

When the white light (for instance the continuous spectrum) is passed through sodium, the flame absorbs from the light, the wavelength equal to that which it can emit at that state (temperature). This will generate the dark line with in continuous spectrum when viewed with the spectrometer. This dark line is the natural feature of absorbing substance (sodium). Also atom and molecules absorb light at same wavelength at which they emit light that is if we look at sun's continuous spectrum, it will include the number of dark lines known as absorption lines. This type of spectra is known as absorption spectra.

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