JEE Physics Ray Optics and Optical Instruments Notes Set 05

Light as a Ray

Light travels in straight lines as long as the apertures and obstacles it meets are much larger than its wavelength. Under this condition we draw light as rays, which are straight lines with arrows showing the direction of energy flow. A narrow slit of width \(a\) spreads light by an angle with \(\sin\theta \approx \frac{\lambda}{a}\), so bending is noticeable only when \(a\) is comparable with \(\lambda\); this is why ray optics works for ordinary mirrors and lenses.

1. Speed of light in vacuum \(c = 3\times10^8\ \text{m/s}\). Light is an electromagnetic wave and carries energy and momentum.
2. \(v = f\lambda\) holds for light. The frequency never changes on reflection or refraction; the speed and wavelength change on refraction only.
3. A smooth surface gives regular reflection (a definite reflected beam). A rough surface gives diffused reflection, which lets us see an object from every direction.

Reflection and Plane Mirrors

1. Laws: the incident ray, reflected ray and normal lie in one plane, and the angle of incidence equals the angle of reflection \(i = r\), both measured from the normal.
2. Deviation of a reflected ray is \(\delta = 180^\circ - 2i\) (turning one way) or \(180^\circ + 2i\) (other way). For normal incidence the ray retraces its path.
3. Object and image: the point where incident rays meet (or appear to meet) is the object, and the point where reflected or refracted rays meet (or appear to meet) is the image. An image is real if the rays actually pass through it and virtual otherwise.

Properties of the image in a plane mirror:

1. Image distance equals object distance and the line joining them is normal to the mirror.
2. A real object gives a virtual image and a virtual object gives a real image. The image is the same size and laterally inverted.
3. The image of a point is seen only from within the field of view, the region bounded by the reflected rays from the mirror edges.

Two Plane Mirrors

Two mirrors inclined at angle \(\theta\) form images by repeated reflection. All images and the object lie on a circle whose centre is the line of intersection of the mirrors. Let \(\frac{360^\circ}{\theta} = N\).

1. If \(N\) is an even integer, the number of images is \(N - 1\) for every position of the object. If \(N\) is an odd integer, the count is \(N - 1\) when the object is on the bisector and \(N\) when it is not.
2. If \(N\) is not an integer, the number of images is the integer part of \(N\), or one more depending on the object position.
3. For parallel mirrors facing each other, an infinite number of images is formed, each farther and fainter.

Example: two mirrors inclined at \(60^\circ\) give \(\frac{360}{60} - 1 = 5\) images of an object placed between them.

Spherical Mirrors

A spherical mirror is a part of a sphere of radius \(R\). The pole \(P\) is the centre of the mirror, the centre of curvature \(C\) is the centre of the sphere, and the line \(PC\) is the principal axis. Rays close to the axis (paraxial rays) meet at the principal focus \(F\), with \(f = \frac{R}{2}\).

Sign convention (Cartesian): distances are measured from the pole, positive along the direction of the incident light and negative against it. Heights above the axis are positive. So for a mirror with the object on the left, \(u\) is negative, \(f\) and \(R\) are negative for a concave mirror and positive for a convex mirror.

1. Mirror formula: \(\frac{1}{v} + \frac{1}{u} = \frac{1}{f}\).
2. Lateral magnification: \(m = \frac{h_i}{h_o} = -\frac{v}{u}\). A negative \(m\) means a real, inverted image; a positive \(m\) means a virtual, erect image.
3. Longitudinal magnification of a small object along the axis: \(\frac{dv}{du} = -m^2\) (image and object move in the same direction along the axis).
4. Velocity of the image along the axis: \(v_{I} = -m^2 v_{O}\); across the axis it depends on \(m\) and on the transverse velocity of the object.
5. Standard rays for construction: a ray parallel to the axis goes through (or appears to come from) \(F\); a ray through \(F\) emerges parallel; a ray through \(C\) retraces its path; a ray striking the pole reflects with \(i = r\).

A convex mirror always gives a virtual, erect and diminished image between \(P\) and \(F\), whatever the object distance. Because of its wide field of view it is used as a rear-view mirror.

Worked example: a concave mirror of focal length 15 cm has an object 25 cm in front. Then \(\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = -\frac{1}{15} + \frac{1}{25} = -\frac{2}{75}\), so \(v = -37.5\) cm (real image in front of the mirror) and \(m = -\frac{v}{u} = -1.5\): the image is inverted and 1.5 times the size of the object.

Refraction at a Plane Surface

1. Refractive index \(n = \frac{c}{v}\). Snell's law at a boundary: \(n_1\sin i = n_2\sin r\). The incident ray, refracted ray and normal are coplanar.
2. On entering a denser medium the ray bends towards the normal. The deviation is \(\delta = |i - r|\).
3. Reversibility: a ray retraces its path if its direction is reversed. Hence the index of medium 2 with respect to medium 1 is the reciprocal of that of medium 1 with respect to medium 2.
4. Frequency is unchanged; wavelength in a medium is \(\lambda' = \frac{\lambda_0}{n}\).

Parallel Slab and Apparent Depth

1. A ray passing through a parallel-sided slab emerges parallel to its original direction but is displaced sideways by \(d = \frac{t\sin(i - r)}{\cos r}\).
2. Looking normally into a medium of index \(n\) at real depth \(d\), the apparent depth is \(\frac{d}{n}\). The object appears raised by \(d\left(1 - \frac{1}{n}\right)\).
3. For an object in a medium \(n_1\) seen from a medium \(n_2\), the apparent depth is \(d\,\frac{n_2}{n_1}\) (normal viewing).
4. A number of layers of thicknesses \(t_1, t_2, \dots\) and indices \(n_1, n_2, \dots\) viewed normally appear to be at \(\sum \frac{t_i}{n_i}\).
5. A glass slab of thickness \(t\) and index \(n\) placed in the path shifts the image of a point by \(t\left(1 - \frac{1}{n}\right)\) towards the observer.

Worked example: a coin lies at the bottom of water 12 cm deep (\(n = \frac{4}{3}\)). Viewed from above, it appears at \(\frac{12}{4/3} = 9\) cm below the surface.

Critical Angle and Total Internal Reflection

When light goes from a denser medium (index \(n_1\)) to a rarer medium (\(n_2 < n_1\)), the refracted ray bends away from the normal. At the critical angle \(C\) the refracted ray grazes the surface: \(\sin C = \frac{n_2}{n_1}\) (\(= \frac{1}{n}\) for a medium against air). For angles of incidence greater than \(C\) all the light is reflected back.

1. Conditions for total internal reflection: light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle.
2. Applications: shining of diamonds, mirage, optical fibres, totally reflecting prisms, bright appearance of an air bubble in water.
3. A point source at depth \(h\) in a liquid of index \(n\) lights a circle of radius \(R = \frac{h}{\sqrt{n^2 - 1}}\) on the surface, because \(\tan C = \frac{R}{h}\).

Worked example: for glass of index 1.5, \(\sin C = \frac{2}{3}\), so \(C \approx 41.8^\circ\).

Prism

A prism is a transparent medium bounded by two plane faces inclined at the angle of the prism (apex angle) \(A\). A ray entering at angle \(i\) refracts at angle \(r_1\), meets the second face at \(r_2\) and emerges at angle \(e\).

1. \(r_1 + r_2 = A\) and the total deviation is \(\delta = i + e - A\).
2. The graph of \(\delta\) against \(i\) is a U-shaped curve. For a given \(\delta\) (other than the minimum) there are two angles of incidence, and they are interchanged by reversing the ray.
3. At minimum deviation \(i = e\), \(r_1 = r_2 = \frac{A}{2}\) and the ray inside is parallel to the base. Then \(\delta_{min} = 2i - A\) and \(n = \frac{\sin\left(\frac{A + \delta_{min}}{2}\right)}{\sin\frac{A}{2}}\) (with \(n\) the index of the prism relative to its surroundings).
4. For a thin prism (\(A \le 10^\circ\)) with small \(i\): \(\delta = (n - 1)A\), or \((n_{rel} - 1)A\) when immersed in a medium.
5. A ray cannot emerge from the second face if \(r_2\) exceeds the critical angle. For a ray incident from the first face, the largest angle of prism that still lets a ray emerge is \(A = 2C\), and emergence is possible for any \(i\) only if \(A < C\).

Worked example: a prism with \(A = 60^\circ\) and \(n = \sqrt{2}\). At minimum deviation \(\sin\frac{A + \delta}{2} = \sqrt{2}\sin 30^\circ = \frac{1}{\sqrt{2}}\), so \(\frac{A + \delta}{2} = 45^\circ\) and \(\delta_{min} = 90^\circ - 60^\circ = 30^\circ\).

Dispersion

White light splits into colours on passing through a prism because the refractive index depends on wavelength. Cauchy's relation: \(n = a + \frac{b}{\lambda^2}\), with positive constants \(a\) and \(b\), so violet is bent most and red least. All colours travel at the same speed in vacuum but at different speeds in a material.

1. Angle of dispersion between violet and red for a thin prism: \(\theta = \delta_v - \delta_r = (n_v - n_r)A\).
2. Mean deviation (yellow): \(\delta_y = (n_y - 1)A\).
3. Dispersive power: \(\omega = \frac{n_v - n_r}{n_y - 1} = \frac{\delta_v - \delta_r}{\delta_y}\). It depends only on the material, not on the prism angle.
4. Dispersion without deviation (direct vision combination): two thin prisms of angles \(A\), \(A'\) of different materials with \((n_y - 1)A + (n_y' - 1)A' = 0\) (placed inverted against each other); there is then no net mean deviation but a net dispersion.
5. Deviation without dispersion (achromatic combination): \((n_v - n_r)A + (n_v' - n_r')A' = 0\), which leaves a net deviation \((n_y - 1)A + (n_y' - 1)A'\).

Worked example: flint glass has \(n_r = 1.613\) and \(n_v = 1.632\). A thin prism of \(A = 5^\circ\) disperses by \((1.632 - 1.613) \times 5^\circ = 0.095^\circ\).

Spectrum

1. Emission spectrum: light emitted by a source. A hot solid gives a continuous spectrum, an excited gas gives a line spectrum (atoms) or band spectrum (molecules).
2. Absorption spectrum: dark lines or bands on a continuous background, seen when white light passes through a cooler gas. The dark lines occur at wavelengths that the same gas would emit.
3. A spectrometer uses a collimator, a prism and a telescope to measure angles of deviation and to find refractive indices.

Refraction at a Spherical Surface

For paraxial rays at a spherical surface of radius \(R\) separating media of index \(n_1\) (object side) and \(n_2\) (image side), with Cartesian signs:

\[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \]

1. Lateral magnification: \(m = \frac{n_1 v}{n_2 u}\).
2. A plane surface is the special case \(R = \infty\), giving \(v = \frac{n_2}{n_1}u\).
3. For a glass sphere or hemisphere an object at the centre of curvature is not deviated at all.

Thin Lens

1. Lens formula: \(\frac{1}{v} - \frac{1}{u} = \frac{1}{f}\). Magnification: \(m = \frac{v}{u}\).
2. Lens maker's formula: \(\frac{1}{f} = (n_{rel} - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)\), where \(n_{rel} = \frac{n_{lens}}{n_{medium}}\). A convex lens in a denser medium behaves as a diverging lens when \(n_{rel} < 1\).
3. Power \(P = \frac{1}{f}\) with \(f\) in metres, measured in dioptre. Converging lenses have positive power and diverging lenses negative power.
4. Longitudinal magnification of a small axial object: \(m^2\) (the image and object move in the same direction).
5. Cutting a lens into two halves along the axis leaves \(f\) unchanged but halves the brightness. Cutting across the axis into two thin halves doubles \(f\) of each half, since each half has a single curved surface of the same radius.

Worked example: a biconvex lens with \(n = 1.5\), \(R_1 = 20\) cm and \(R_2 = -30\) cm has \(\frac{1}{f} = 0.5\left(\frac{1}{20} + \frac{1}{30}\right) = \frac{1}{24}\), so \(f = 24\) cm.

Displacement method: for a fixed object-screen distance \(D > 4f\), a convex lens forms a sharp image at two positions separated by \(d\). Then \(f = \frac{D^2 - d^2}{4D}\). The two image sizes satisfy \(h_o = \sqrt{h_1 h_2}\).

Combination of Lenses and Mirrors

1. Thin lenses in contact: \(\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} + \dots\), so powers add: \(P = P_1 + P_2 + \dots\).
2. Two thin lenses a distance \(d\) apart: \(\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}\).
3. Lens in front of a mirror (light passes through the lens, reflects, and passes through again): the system acts as a mirror with \(\frac{1}{f} = \frac{1}{F_m} - \frac{2}{F_{lens}}\), where \(F_m\) is the focal length of the mirror.

Worked example: a convex lens of 10 cm and a concave lens of 20 cm focal length are 5 cm apart. Then \(\frac{1}{F} = \frac{1}{10} - \frac{1}{20} + \frac{5}{200} = \frac{3}{40}\), so \(F = 13.3\) cm. A convex lens of \(f = 20\) cm in front of a plane mirror gives \(\frac{1}{f_{sys}} = 0 - \frac{2}{20}\), so the system behaves like a concave mirror of focal length 10 cm.

Atmospheric Refraction

The atmosphere is denser near the ground, so light from the sun bends towards the earth. We therefore see the sun a few minutes before it actually crosses the horizon and a few minutes after it has set, and the sun looks flattened at the horizon.