Optics, Waves, Electromagnetism, and AC Circuits Flashcards
OPTICS
Light is a form of energy that travels in a straight line.
Luminous and Non-Luminous Objects
Luminous Objects: Objects that produce their own light.
Examples: Sun, fireworms, firefly.
Non-Luminous Objects: Objects that do not produce light on their own but reflect it from luminous objects.
Examples: Moon, stars.
Classification of Objects Based on Light Transmission
Opaque Objects: Objects that do not allow light to pass through them.
Examples: Wood, wall, people.
Transparent Objects: Objects that allow most of the light to pass through them.
Examples: Glass, clear water, clear polythene.
Translucent Objects: Objects that allow some light to pass through them.
Examples: Paper, bathroom glasses, tinted glass.
Rays and Beams
Ray: The direction of the path taken by light.
Indicated by a straight line with an arrow on it.
Beam: A collection of light rays.
Alternatively: A stream of light energy.
Types of Beams
Parallel Beam: Rays are parallel to each other.
Obtained from light from a distant source (e.g., sun) and searchlights.
Convergent Beam: Rays from different directions meet at a common point.
Example: Light behind a convex lens after passing through it.
Divergent Beam: Rays start from a common point and separate into different directions.
Example: Light from a torch and car lights.
Reflection of Light
Reflection: The bouncing of light as it strikes a reflecting surface.
Types of Reflection
Regular Reflection: An incident parallel beam is reflected as a parallel beam when light falls on a smooth surface.
Examples: Plane mirror, still water, highly polished surfaces.
Irregular (Diffuse) Reflection: An incident parallel beam is reflected in different directions when light falls on a rough surface.
Examples: Iron sheets, unclear water.
Reflection from a Plane Mirror
Angle of Incidence (i): The angle between the normal and the incident ray.
Angle of Reflection (r): The angle between the normal and the reflected ray.
Laws of Reflection
Law 1: The incident ray, the reflected ray, and the normal at the point of incidence all lie on the same plane.
Law 2: The angle of incidence is equal to the angle of reflection (i = r).
Deviation of Light by a Plane Mirror
Consider a ray AO incident on a plane mirror M at a glancing angle g (angle between the incident ray and the mirror surface).
Derivation of Deviation (d):
The total deviation d = \angle BOY + \angle ZOY
\angle ZOY = g (vertically opposite angles)
The angle between the reflected ray and the mirror is \angle B = 90 - r
Since i = r (Law of Reflection), \angle B = 90 - i
Also, i = 90 - g
Substitute i into the expression for \angle B: \angle B = 90 - (90 - g) = g
Therefore, d = g + g = 2g
Deviation of a Reflected Ray by a Rotated Mirror
A constant ray OA is incident onto a plane mirror in position m1 and is reflected along OR1
The glancing angle is g and the initial deviation d = 2g
When the mirror is turned through an angle \alpha , the new glancing angle is (g + \alpha).
The new deviation is 2(g + \alpha).
The angle \beta through which the reflected ray is rotated is given by:
\beta = 2(g + \alpha) - 2g
\beta = 2g + 2\alpha - 2g
\beta = 2\alpha
Deviation by Successive Reflection at Two Inclined Mirrors
A ray is incident onto a plane mirror m_1 and is reflected.
The glancing angle is \alpha and the deviation caused is 2\alpha
When the ray is reflected at mirror m_2, the new glancing angle is \beta and the new deviation is 2\beta
The net deviation, d = 2(\alpha + \beta) clockwise.
From the geometry of the triangle formed by the mirrors and the incident ray, \beta + \theta + \alpha = 180^{\circ}
Thus, \beta + \alpha = 180^{\circ} - \theta
Substituting this into the deviation equation: d = 2(180^{\circ} - \theta) = (360^{\circ} - 2\theta) clockwise.
Alternatively, the deviation can be expressed as 360^{\circ} - (360^{\circ} - 2\theta) = 2\theta anticlockwise.
The Sextant
Sextant: An instrument used for measuring the angle of elevation of heavenly bodies such as stars and the sun.
Setup: B is a half-silvered fixed mirror, while M is a movable mirror.
Procedure:
Rotate mirror M until the image of the horizon H_1 is seen to coincide with horizon H.
At this point, mirrors M and B are parallel. Note this initial position of mirror M.
Rotate mirror M again until the image of the star is seen to coincide with the horizon H.
Measure the angle of rotation \theta .
The angle of deviation is 2\theta (due to reflection from mirror M).
Formation of Images by a Plane Mirror
An image is formed by the intersection of at least two rays.
Consider an object O placed in front of a plane mirror.
Rays of light from O are reflected from the mirror and appear to come from I, which is the virtual image of O.
Proof of Image Properties (using geometry):
\angle i = \angle r (Law of Reflection)
\angle i = \angle \alpha (alternate angles)
\angle r = \angle \beta (corresponding angles)
Therefore, \angle \alpha = \angle \beta
Since side MP is common to both triangles \triangle OMP and \triangle IMP , and the angles are congruent, the triangles are congruent.
Hence, OM = MI. The image is as far behind the mirror as the object is in front.
Properties of Images Formed by a Plane Mirror
The images are the same distance behind the mirror as the distance of the object in front of the mirror.
The images have the same size as the object.
The images are erect (upright).
The images are laterally inverted (rotated through 180^{\circ} in the mirror).
The images are virtual (cannot be formed on a screen).
Real and Virtual Images
Real Image: An image that can be formed on a screen and is formed by the actual intersection of light rays.
Examples: Images formed by concave mirrors and convex lenses.
Virtual Image: An image that cannot be formed on a screen and is formed by the apparent intersection of light rays.
Examples: Images formed by plane mirrors, concave lenses, and convex mirrors.
Location of an Image on Plane Mirrors (No Parallax Method)
An object pin O is placed in front of a plane mirror AB on a white sheet of paper.
Looking from side A of the mirror, two pins P1 and P2 are placed so that they appear to be in line with the image of pin O.
The experiment is repeated with pins P3 and P4 on side B.
The pins and the mirror are removed, and lines are drawn through the pin marks P1P2 and P3P4 to meet at I. I is the position of the image.
Minimum Vertical Length of a Plane Mirror (for a man to see his whole self)
Principle: The minimum length of a mirror for a person to see their full height is half their height.
Derivation: For a man of height H, with eye level E.
Rays from the top of the man (H) are reflected from the top of the mirror (A) and are incident on the man's eyes (E).
By similar triangles (or using law of reflection at point A), HA = AE.
Similarly, rays from the bottom of the man (F) are reflected from the bottom of the mirror (B) and are incident on the man's eyes (E).
By similar triangles (or using law of reflection at point B), EB = BF.
The height of the mirror needed is AB = AE + EB.
If the man's total height is H and his eye level is H_E from the ground:
Distance from top of head to eye is dH = H - HE
AE = d_H / 2
Distance from eye to foot is dF = HE
EB = d_F / 2
Total mirror length AB = (dH / 2) + (dF / 2) = (dH + dF) / 2 = H / 2.
Example Calculation: A man 2m tall whose eye level is 1.84m above the ground.
AE = (2 - 1.84) / 2 = 0.16 / 2 = 0.08m
EB = 1.84 / 2 = 0.92m
Minimum length of mirror = 0.08 + 0.92 = 1m
This is half the man's height (2m / 2 = 1m).
Formation of Multiple Images in Thick Plane Mirror
A thick plane mirror has two plane surfaces, say X and Y.
Reflection takes place at both surfaces (partial reflection and transmission).
Process:
Light from an object O is incident on surface X (N1).
Reflection at N1 leads to the formation of image I1
Transmitted light then strikes the silvered surface N (back surface).
It undergoes partial reflection and transmission at N_2
The light transmitted from N2 appears to originate from image I2
Successive internal reflections between surfaces lead to the formation of multiple images (I1, I2, I3, etc.).
Notes:
Thick mirrors form multiple images. The more distant images are faint.
Multiple images are due to successive reflections.
Faint images are due to energy absorbed at each reflection.
The disadvantages of using plane mirrors as reflectors in optical instruments (like submarine periscopes) are overcome by using reflecting prisms.
Comparison of Plane Mirrors and Reflecting Prisms
Plane Mirrors:
Produce multiple images.
Silvering wears out with time.
Exercise loss of brightness when reflection occurs at its surface.
Prisms:
Do not produce multiple images.
No silvering is required.
No loss of brightness (total internal reflection is more efficient than metallic reflection).
REFLECTION IN CURVED MIRRORS
Curved Mirrors (Spherical Mirrors)
Curved mirrors are mirrors whose surfaces are obtained from a hollow transparent sphere.
Types of Curved Mirrors
Concave Mirror (Converging Mirror): Part of a sphere whose center C is in front of its reflecting surface.
Convex Mirror (Diverging Mirror): Part of a sphere whose center C is behind its reflecting surface.
Terms Used in Curved Mirrors
P: Pole of the mirror (mid-point of the mirror surface).
F: Principal focus (focal point).
C: Center of curvature.
f: Focal length.
r: Radius of curvature.
APB: Aperture (length of the mirror surface).
PFC: Principal axis.
Definitions of Terms
Center of Curvature (C): It is the center of the sphere of which the mirror forms part.
Radius of Curvature (r): It is the radius of the sphere of which the mirror forms part.
Pole of the Mirror (P): It is the mid-point (center) of the mirror surface.
Principal Axis (CP): It is the line that passes through the center of curvature and the pole of the mirror.
Secondary Axis: A line through the center of a thin lens or through the center of curvature of a concave or convex mirror, other than the principal axis.
Paraxial Rays: These are rays close to the principal axis and make small angles with the mirror axis.
Marginal Rays: These are rays furthest from the principal axis of the mirror.
Principal Focus (F):
Concave Mirror: A point on the principal axis where paraxial rays incident on the mirror and parallel to the principal axis converge after reflection by the mirror. A concave mirror has a real (in front) principal focus.
Convex Mirror: A point on the principal axis where paraxial rays incident on the mirror and parallel to the principal axis appear to diverge from after reflection by the mirror. A convex mirror has a virtual (behind) principal focus.
Focal Length (f):
Concave Mirror: The distance from the pole of the mirror to the point where paraxial rays incident and parallel to the principal axis converge after reflection by the mirror.
Convex Mirror: The distance from the pole of the mirror to the point where paraxial rays incident and parallel to the principal axis appear to diverge from after reflection by the mirror.
Aperture of the Mirror: It is the length of the mirror surface.
Geometrical Rules for the Construction of Ray Diagrams (to locate image positions)
Rays are always drawn from the top of the object.
A ray parallel to the principal axis is reflected through the principal focus.
A ray through the principal focus is reflected parallel to the principal axis.
A ray through the center of curvature is reflected along its own path (it strikes the mirror normally).
Rays incident to the pole are reflected back, making the same angle with the principal axis.
At least two rays are used (e.g., rules 1 & 2, or 1 & 3). Their point of intersection is where the top of the image is located.
Note 1: The normal to the mirror surface at any point must pass through the center of curvature.
Note 2: The image position can be located by the intersection of two reflected rays initially coming from the object.
Images Formed by a Concave Mirror
The nature of the image formed by a concave mirror is either real or virtual, depending on the object distance from the mirror.
Object Position and Image Properties
Object between F and P:
Behind the mirror.
Virtual.
Erect.
Magnified.
Application: This property makes concave mirrors useful as shaving mirrors and for dentists (teeth examination).
Object at F:
Image at infinity.
Virtual.
Upright.
Object between F and C:
Beyond C.
Real.
Inverted.
Magnified.
Object at C:
At C.
Real.
Inverted.
Same size as the object.
Object beyond C:
Between C and F.
Real.
Inverted.
Diminished.
Object at infinity:
At F.
Real.
Inverted.
Diminished.
Uses of Concave Mirrors
As shaving mirrors.
By dentists for teeth examination.
As solar concentrators in solar panels.
In reflecting telescopes (for viewing distant objects).
In projectors (for showing slides on a screen).
Advantage: Forms magnified and erect images (when object is between F and P).
Images Formed by a Convex Mirror
The image of an object in a convex mirror is always erect, virtual, and diminished in size, regardless of the object's position.
Illustration: Rays appear to diverge from a virtual focus behind the mirror.
Wide Field of View: Convex mirrors have a wider field of view compared to plane mirrors.
Note: A convex mirror can form real images if it receives converging rays targeting any point between its focal point and its optical center (e.g., if a screen were placed at the position of the eye to capture such rays).
Uses of Convex Mirrors
As driving mirrors (due to erect images and wide field of view).
In supermarkets (to observe customer activities).
In security checkpoints (to inspect under vehicles).
Advantages of Convex Mirrors over Plane Mirrors
They have a wide field of view.
They form erect images.
Disadvantages of Convex Mirrors
They form diminished images, giving a wrong impression to drivers that objects behind are very far away.
Relation between Focal Length (f) and Radius of Curvature (r)
(a) Concave Mirror
A ray AX close and parallel to the principal axis is reflected through the principal focus F.
FP = \text{focal length }(f)
If C is the center of curvature, then CP is the radius of the mirror (r).
\angle AXC = \angle CXF = \alpha (due to law of reflection)
\angle XCP = \angle AXC = \alpha (alternate angles)
Therefore, triangle CXF is an isosceles triangle (FC = FX).
For AX close to CP (paraxial ray), FX \approx FP
Thus, CF = FP
And since CP = CF + FP = 2FP.
Also CP = r
Therefore, r = 2f
(b) Convex Mirror
A ray AX close and parallel to the principal axis is reflected, appearing to diverge from the principal focus F.
FP = \text{focal length }(f)
If C is the center of curvature, then CP is the radius of the mirror (r).
\angle AXB = \angle BXD = \alpha (due to law of reflection, where BXD is the angle of reflection)
\angle AXB = \angle XCP = \alpha (alternate angles, assuming reflected ray and principal axis are parallel)
Therefore, triangle CXF is an isosceles triangle (FC = FX).
For AX close to CP (paraxial ray), FX \approx FP
Thus, CF = FP
And since CP = CF + FP = 2FP.
Also CP = r
Therefore, r = 2f
Mirror Formula
(a) Concave Mirror
Consider a point object O on the principal axis of a concave mirror.
From triangle OZC:
\theta = \beta - \alpha (1)
From triangle ZCI:
\theta = \gamma - \beta (2)
Equating (1) and (2): \beta - \alpha = \gamma - \beta
2\beta = \gamma + \alpha (3)
For small angles in radians, \tan\alpha \approx \alpha, \tan\beta \approx \beta, \tan\gamma \approx \gamma.
Also, for a point on the mirror close to the pole, the height h (from the principal axis to the point of reflection) is small.
\alpha \approx h/OP = h/u
\beta \approx h/CP = h/r
\gamma \approx h/IP = h/v
Substituting into (3):
2(h/r) = (h/v) + (h/u)
2/r = 1/v + 1/u
Since r = 2f, then 2/(2f) = 1/v + 1/u
1/f = 1/v + 1/u
(b) Convex Mirror
Consider a point object O in front of a convex mirror, with image I.
From triangle OZC:
\theta = \beta + \alpha (This setup in the transcript seems to be from a ray from F, so the derivation is similar to concave but with appropriate sign changes for virtual image/focus/radius).
Let's re-derive based on the typical convex mirror image formation:
A ray from O to X (on mirror) is reflected, appearing to come from F.
A ray from O to C (center of curvature) strikes normally and reflects back.
A ray from O to P (pole) reflects such that angle of incidence = angle of reflection.
Based on diagram in transcript:
From triangle OXC: \beta = \alpha + \angle XCO (exterior angle) -> angle shown as \theta in triangle OXC, so \theta = \beta - \alpha
From triangle CXI: \gamma = \beta + \angle XIC (exterior angle) -> angle shown as \theta in triangle CXI, so \theta = \gamma - \beta
Equating them: \beta - \alpha = \gamma - \beta
2\beta = \gamma + \alpha
This is the same intermediate step as for a concave mirror, but the signs for r, v, f are handled by sign convention.
Using the height h from the principal axis to point X on mirror:
\alpha \approx h/u
\beta \approx h/(-r) (radius for convex is negative by convention if P is origin and C is behind mirror)
\gamma \approx h/(-v) (virtual image behind mirror is negative)
So, 2(h/(-r)) = h/(-v) + h/u
-2/r = -1/v + 1/u
1/f = 1/u + 1/v
Self-correction: The derivation in the transcript for convex mirror contains a similar error as often seen. The general formula 1/f = 1/u + 1/v holds for both, with proper sign conventions.
Where:
u = object distance
v = image distance
f = focal length
r = radius of curvature
Sign Convention (Cartesian Convention)
Distances of real objects (u) and real images (v) are positive.
Distances of virtual objects (u) and virtual images (v) are negative.
Focal length (f) for a concave mirror is positive, and for a convex mirror, it is negative.
(Note: Some conventions use positive for real, negative for virtual, and positive for converging, negative for diverging. The text's convention seems to be for real distances to be positive for the respective object/image, and focal length for concave mirrors is positive, convex negative).
Linear Magnification (m)
Definition: The ratio of the image height to the object height.
m = \text{height of image} / \text{height of object} (hi / ho)
Alternatively: The ratio of the distance of the image from the mirror (v) to the distance of the object from the mirror (u).
m = \text{image distance} (v) / \text{object distance} (u)
Proof of Magnification Formula
Consider a ray AP incident onto the pole P of a concave mirror from an object of height ho at distance u. It is reflected back, making the same angle \theta with the principal axis, forming an image of height hi at distance v.
From \triangle OAP (object at O, ray to P):
\tan\theta = h_o / u (i)
From \triangle IPB (image at I, reflected ray from P):
\tan\theta = h_i / v (ii)
Equating (i) and (ii): ho / u = hi / v
Thus, magnification, m = v / u = hi / ho
Notes on Magnification
No signs need to be inserted in the magnification formula (magnification value is positive).
Using the mirror formula, a connection relating magnification to the focal length of the mirror with either the object distance or the image distance can be established:
From 1/f = 1/v + 1/u:
Multiply by v: v/f = 1 + v/u
m = v/f - 1
Multiply by u: u/f = u/v + 1
1/m = u/f - 1
Examples of Mirror Calculations (with solutions provided in transcript)
Concave mirror: h_o = 1cm, u = 30cm, f = 20cm (concave, so f is positive).
1/v = 1/20 - 1/30 = 1/60 \implies v = 60cm (Real image, in front of mirror).
m = v/u = 60/30 = 2
hi = m \times ho = 2 \times 1 = 2cm
Convex mirror: h_o = 10cm, u = 30cm, f = -20cm (convex, so f is negative).
1/v = 1/(-20) - 1/30 = -5/60 \implies v = -12cm (Virtual image, behind mirror).
m = v/u = 12/30 = 0.4
hi = m \times ho = 0.4 \times 10 = 4cm
Convex mirror: u = 15cm, r = 20cm \implies f = -10cm.
1/v = 1/(-10) - 1/15 = -5/30 \implies v = -6cm (Virtual image, behind mirror).
Concave mirror: r = 40cm \implies f = 20cm, u = 25cm.
f = 20cm
1/v = 1/20 - 1/25 = 1/100 \implies v = 100cm (Real image).
Concave mirror: u = 10cm, f = 15cm.
1/v = 1/15 - 1/10 = -1/30 \implies v = -30cm (Virtual image, behind mirror).
m = v/u = 30/10 = 3
Convex mirror: v = -6cm, r = 20cm \implies f = -10cm.
1/u = 1/(-10) - 1/(-6) = 1/6 - 1/10 = 2/30 \implies u = 15cm
m = v/u = 6/15 = 0.4
Show that an object and its image coincide in position at the center of curvature of a concave mirror.
At the center of curvature, object distance u = r.
Since r = 2f, we have 1/f = 1/r + 1/v \implies 2/r = 1/r + 1/v \implies 1/v = 1/r \implies v = r.
Thus, the image is also formed at the center of curvature, coinciding with the object.
Magnification m = v/u = r/r = 1 (image is same size as object).
Complex Magnification Problems (using m = v/f - 1 and 1/m = u/f - 1)
Problem: Concave mirror. Real image 4x magnified (m1 = 4). Moved 10cm towards mirror, virtual image 4x magnified (m2 = -4).
u1 = (1/m1 + 1)f = (1/4 + 1)f = (5/4)f
u2 = (1/m2 + 1)f = (1/(-4) + 1)f = (3/4)f
u1 - u2 = (5/4 - 3/4)f = (2/4)f = (1/2)f = 10cm
f = 20cm
Problem: Concave mirror. Real image 3x (m1 = 3). Displaced by d, real image 4x (m2 = 4). Distance between images is 20cm (v2 - v1 = 20cm).
v1 = (m1 + 1)f = (3 + 1)f = 4f
v2 = (m2 + 1)f = (4 + 1)f = 5f
v2 - v1 = 5f - 4f = f = 20cm
u1 = (1/m1 + 1)f = (1/3 + 1)f = (4/3)f = (4/3)20 = 26.67cm
u2 = (1/m2 + 1)f = (1/4 + 1)f = (5/4)f = (5/4)20 = 25cm
d = u1 - u2 = 26.67 - 25 = 1.67cm
Problem: Concave mirror. Real image 3x (m1 = 3). Object/screen moved, image 5x (m2 = 5). Shift of screen (image) is 30cm (v2 - v1 = 30cm).
f = (v2 - v1) / (m2 - m1) = 30 / (5 - 3) = 30 / 2 = 15cm
Shift of object d = f(1/m1 - 1/m2) = 15(1/3 - 1/5) = 15(2/15) = 2cm
Problem: Concave mirror P (fP = 15cm), convex mirror Q (fQ = -10cm), distance 25cm apart. Object O 20cm from P.
Action of Concave Mirror P: u = 20cm, f = 15cm
1/v = 1/15 - 1/20 = 1/60 \implies v = 60cm (Image I_1 is real, 60cm from P, on the side of Q).
Action of Convex Mirror Q: I1 acts as an object for Q. Its distance from P is 60cm, so from Q is 60 - 25 = 35cm. Since I1 is on the side of Q, it is a real object for Q. Thus u = 35cm.
1/v = 1/(-10) - 1/35 = (-7 - 2)/70 = -9/70 \implies v = -70/9 = -7.78cm (Final image is virtual, 7.78cm behind Q).
Self-correction: The transcript calculation used u = -35cm and got v = -14cm. This either implies that (a) the sign convention for object distance to the second mirror is being applied such that if the image of the first mirror falls behind the second mirror (as if passing through it), it's treated as a virtual object (negative u). Or (b) the diagram is misleading or the calculation is applying a specific sign convention not clearly stated here.
Magnification: m = m1 \times m2 = (v1/u1) \times (v2/u2)
m = (60/20) \times (14/35) = 3 \times 0.4 = 1.2
Caustic Surface
When a wide parallel beam of light is incident on a concave mirror, the different reflected rays are converged to different points.
These reflected rays appear to touch a surface known as a caustic surface.
Definition: A surface on which every reflected ray from the mirror forms a tangent to.
The caustic surface has an apex at the principal focus F.
Note: Marginal rays (furthest from principal axis) are converged closer to the pole than paraxial rays.
Similarly, for a convex mirror, a wide parallel beam of light will appear to diverge from different points, forming a virtual caustic surface.
Comparison of Concave and Parabolic Mirrors (Reversing Light)
Concave Mirror: When a lamp is placed at the principal focus, only rays close to the principal axis are reflected parallel. Rays further from the axis are reflected in different directions, leading to a diminished intensity of the reflected beam at a distance.
Parabolic Mirror: When a lamp is placed at the principal focus, all rays from the lamp (close to and far from the axis) are reflected parallel to the principal axis. The intensity of the reflected beam remains undiminished.
Conclusion: Parabolic mirrors are used as searchlights and in headlights instead of concave mirrors due to their ability to produce a strong, undiminished parallel beam.
Uses of Parabolic Mirrors
Reflectors in searchlight torches.
Determination of the Focal Length of a Concave Mirror
Method 1: Using a Pin at C (Centre of Curvature)
Place a concave mirror on a horizontal bench with its reflecting surface upwards.
Clamp a pin horizontally on a retort stand so its pointed end lies along the principal axis.
Move the pin vertically until it coincides with its own image (no parallax).
Measure the distance r from the mirror to the pin (this is the radius of curvature).
Calculate the focal length: f = r / 2
Notes:
No Parallax: When there is no relative motion between the object pin and its image as the observer moves their head side-to-side.
When the pin coincides with its image, rays are incident normally to the mirror and are reflected along their own path. This occurs at the center of curvature.
Method 2: Using an Illuminated Object at C
An object is a hole cut in a white screen made of cross-wires, illuminated from behind.
Mount a concave mirror in a holder and move it back and forth in front of the screen.
Adjust until a sharp image of the cross-wire is formed on the screen, adjacent to the cross-wire.
At this point, both object and image are at the same distance from the mirror, and are effectively at the center of curvature.
Measure the distance between the mirror and the screen, which is the radius of curvature r.
Calculate f = r / 2
Method 3: Using No Parallax Method in Locating V (Image Position)
Place an object pin P_1 at a distance u in front of a mounted concave mirror. Its tip should be along the principal axis, forming an inverted image.
Measure u.
Place a search pin P2 between the mirror and pin P1. Adjust P2 until it coincides with the image of P1 by the no-parallax method.
Measure the distance v of pin P_2 from the mirror.
Repeat for several values of u. Tabulate results including uv and u+v.
Plot a graph of uv against u+v. The slope s of this graph is the focal length f (because uv = f(u+v), so uv / (u+v) = f).
Note: If a graph of 1/u against 1/v is plotted, each intercept C (1/u or 1/v axis) is equal to 1/f. Hence f = 1/C.
Measurement of Focal Length of a Convex Mirror
Method 1: Using a Converging Lens
Place object O in front of a convex lens to form a real image at I_1.
Measure the distance LI_1.
Place the convex mirror between the lens and image I_1, with its reflecting surface facing the lens.
Move the mirror along the axis until the image I_1 coincides with the object O (i.e., rays from the lens strike the mirror normally).
Measure distances PI (distance from pole of convex mirror to I_1) and LP (distance from lens to pole of convex mirror).
Here, I_1 is the center of curvature of the convex mirror, so PI = r. The focal length is obtained from f = PI / 2.
Note: When incident rays from an object (or first image) are reflected back along the incident path, they must strike the mirror normally. This means they are directed towards the center of curvature of the mirror.
Method 2: Using a Plane Mirror and No Parallax Method
Place an object pin O in front of convex mirror m_2 such that it forms a virtual, diminished image at I.
Measure the distance u of object O from convex mirror.
Place a plane mirror m_1 between object O and the convex mirror, covering half its aperture.
Adjust plane mirror m_1 until its own image of O coincides with I (the image from the convex mirror) by the no-parallax method.
Measure distances x (from O to m1) and y (from m1 to I).
The image I formed by the convex mirror is virtual, so its distance v = -(x-y).
The object distance for the convex mirror is u = x+y.
Calculate f from 1/f = 1/u + 1/v
Substituting: 1/f = 1/(x+y) + 1/(-(x-y)) \implies f = (y^2 - x^2) / (2y).
REFRACTION OF LIGHT
Refraction: The change of direction of light propagation as it travels from one medium to another.
Explanation of Refraction
The bending of light is a result of the change in speed as light travels from one medium to another.
This change in speed typically leads to a change in direction, unless the ray is incident normally to the surface.
The speed of light in air is higher than in glass or water.
Glass and water are considered denser than air. Glass is also denser than water.
Laws of Refraction
Consider a ray of light incident on an interface between two media.
O: Point of incidence.
OA: Incident ray.
OB: Refracted ray.
ON: Normal at O.
\angle i : Angle of incidence.
\angle r : Angle of refraction.
Law 1: The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.
Law 2 (Snell's Law): The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media.
This constant ratio is called the Refractive Index (n).
n = \sin i / \sin r
Refractive Index (n)
Definition 1: The ratio of the sine of the angle of incidence to the sine of the angle of refraction for a ray of light traveling from a vacuum to a given medium.
Definition 2: The ratio of the speed of light in a vacuum (c) to the speed of light in a medium (v).
n = c / v
Speed of light in a vacuum, c = 3.0 \times 10^8 \text{ m s}^{-1}.
Note:
The refractive index n for a vacuum is 1.
If light travels from air to another medium, the value of n is slightly greater than 1.
Examples: n = 1.33 for water, n = 1.5 for glass.
The Principle of Reversibility of Light
Statement: The paths of light rays are reversible.
Meaning: A ray of light can travel from medium 1 to medium 2, and from medium 2 to medium 1, along the exact same path.
General Relation between n and sin i (Extended Snell's Law)
Consider a ray of light moving from air through a series of media (1, 2) and then finally emerging into air.
At air - medium 1 interface: \sin ia / \sin i1 = n1 \implies \sin ia = n1 \sin i1 (i)
At medium 1 - medium 2 interface: (Assuming i1 is angle of incidence in medium 1, i2 in medium 2) n1 \sin i1 = n2 \sin i2
This generalizes to: n \sin i = \text{a constant}
N.B.:
For a ray traveling from a less dense medium to a denser medium (e.g., air to glass), it is refracted towards the normal because the speed of light decreases.
For a ray traveling from a denser to a less dense medium (e.g., glass to air), it is refracted away from the normal because the speed of light increases.
When the incident ray meets a refracting surface at 90^{\circ} (i.e., along the normal), it is not refracted at all (passes straight through).
The speed of light reduces when it travels into a denser medium.
Side-wise Displacement of Light Rays (Lateral Displacement)
When light travels from one medium to another and then back to the original medium through parallel boundaries, its direction is displaced sideways. This is called lateral displacement.
Consider a ray of light incident at an angle i on the upper surface of a glass block of thickness t. It is refracted through an angle r and emerges parallel to the incident ray, but displaced laterally by a distance d.
Derivation of d:
Let the path within the block be AC.
From \triangle ABC (where B is where ray exits block at bottom, C is point on emergent ray such that BC is perpendicular to original path):
AC = t / \cos r (assuming AB is perpendicular to BC, and t is normal thickness)
From the diagram, the angle between the incident ray's extended path and the emergent ray is (i - r).
In triangle A CD (where D is the lateral shift point on emergent ray such that AD is perpendicular to extended incident ray):
AD = d / \cos(i-r) (if angle of emergent ray with normal is i). *Self-correction: often, the triangle used involves the refracted point *inside* the block and the emergent point, with the lateral shift d being the perpendicular distance between the incident and emergent rays.*
A common derivation uses the perpendicular distance from the point of emergence to the line of the incident ray extended. Let the point of incidence be P, and point of emergence be Q.
Distance PQ = t / \cos r
The lateral displacement d is the perpendicular distance from Q to the extended incident ray.
This perpendicular distance is PQ \sin(i - r).
So, d = (t / \cos r) \sin(i - r)
Note: The horizontal displacement of the emergent ray, BC = t \tan r (this appears to be the distance projected onto the interface, not the lateral shift).
Explanation of Some Effects of Refraction
Appearance of the Sun when Setting/Rising: As the sun sets, air layers near the Earth are warmer and less dense. Higher layers are cooler and denser. Light from the sun is continuously refracted towards the normal.
Due to the gradient in refractive index of the atmosphere, rays of light that would normally propagate away from Earth are refracted onto it.
This gives the impression of the sun being above the horizon even after it has geometrically set.
A similar effect occurs when the sun rises.
A Stick Partially Immersed in Water Appears Bent: When light rays pass from an object under water (denser medium) to air (less dense medium), they are refracted away from the normal.
As a result, the part of the stick under water appears to be raised up.
The part above water is seen in its normal position, making the stick appear bent or broken.
A Pond Appears Shallower than It Really Is: Light rays from an object at the bottom of a pond travel from water to air. They bend away from the normal as they exit the water.
To an observer, the object appears to be along the line of sight for the emergent rays, making the object seem shallower and the pond appear less deep than it actually is.
The Apparent Size of a Fish Situated in Water: This is often explained by the large surface area of water not being perfectly flat, but having ripples. These ripples, acting as convex surfaces to the air, can function like convex lenses of long focal length.
If the fish is within the focal length of such an