Reflection of Light and Spherical Mirrors

Reflection Of Light

Reflection of light is the phenomenon where light bounces off a surface. When a ray of light strikes a boundary between two different media and returns into the same medium, it is said to be reflected.

Reflection is responsible for how we see objects that do not emit their own light. Light from a source (like the sun or a bulb) falls on an object, is reflected from its surface, and when this reflected light enters our eyes, we perceive the object.

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Laws of Reflection

Reflection of light from a smooth, polished surface (like a mirror) obeys the following two laws, known as the laws of reflection:

1. The angle of incidence is equal to the angle of reflection.

   $ \angle i = \angle r $

   where $\angle i$ is the angle between the incident ray and the normal to the reflecting surface at the point of incidence, and $\angle r$ is the angle between the reflected ray and the normal.
2. The incident ray, the reflected ray, and the normal to the reflecting surface at the point of incidence all lie in the same plane.

(Image Placeholder: A diagram showing a flat line representing a plane mirror. An incident ray strikes the mirror. A perpendicular line to the mirror at the point of incidence is the normal. A reflected ray bounces off the mirror. Label the incident ray, reflected ray, normal, angle of incidence (i), and angle of reflection (r). Show that i = r, and all three lines are in the same plane, which is the plane of the diagram.)

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Types of Reflection

• Regular Reflection (Specular Reflection): Occurs when light reflects from a smooth, polished surface (like a plane mirror). Parallel incident rays reflect as parallel reflected rays, producing clear, sharp images.
• Diffused Reflection (Irregular Reflection): Occurs when light reflects from a rough or uneven surface (like a wall, paper, or clothes). Parallel incident rays reflect in different directions, scattering the light. This is how we see most objects; the scattered light from the object reaches our eyes from various angles. Diffused reflection does not produce a clear image.

The laws of reflection apply to both regular and diffused reflection at a microscopic level (at each point of incidence), but the macroscopic appearance is different due to the orientation of the surface normals.

We see images in mirrors due to regular reflection. Plane mirrors produce virtual, erect, and laterally inverted images of the same size as the object, located as far behind the mirror as the object is in front.

Spherical Mirrors

Spherical mirrors are mirrors whose reflecting surface is a part of a hollow sphere. They are classified into two types based on whether the inner or outer surface is polished:

• Concave Mirror: A spherical mirror whose inner (concave) surface is the reflecting surface. It converges parallel rays of light towards a point after reflection.
• Convex Mirror: A spherical mirror whose outer (convex) surface is the reflecting surface. It diverges parallel rays of light after reflection.

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Terms Related to Spherical Mirrors

• Centre of Curvature (C): The centre of the sphere of which the mirror is a part.
• Radius of Curvature (R): The radius of the sphere of which the mirror is a part. Distance from the pole to the centre of curvature (PC).
• Pole (P): The geometrical centre of the spherical mirror's reflecting surface.
• Principal Axis: The straight line passing through the pole and the centre of curvature of the mirror.
• Aperture: The diameter of the circular boundary of the spherical mirror. For ray diagrams and formula derivations, we usually assume the mirror has a small aperture compared to its radius of curvature.
• Principal Focus (F): A point on the principal axis where a beam of light rays parallel to the principal axis converges after reflection from a concave mirror, or from which a beam of light rays parallel to the principal axis appears to diverge after reflection from a convex mirror.
• Focal Length (f): The distance between the pole and the principal focus (PF). For spherical mirrors of small aperture, the focal length is approximately half the radius of curvature, $f = R/2$. For a concave mirror, the focus is real (in front of the mirror), so $f$ is typically taken as positive (with a certain sign convention). For a convex mirror, the focus is virtual (behind the mirror), so $f$ is typically taken as negative.

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Image Formation By Spherical Mirrors

The type and characteristics of the image formed by a spherical mirror depend on the position of the object relative to the mirror and the type of mirror (concave or convex).

• Concave Mirror: Can form both real and virtual images, and the image can be inverted or erect, magnified or diminished, depending on the object's position.
  • Object at infinity: Real, inverted, highly diminished image at the principal focus (F).
  • Object beyond C: Real, inverted, diminished image between F and C.
  • Object at C: Real, inverted, same size image at C.
  • Object between C and F: Real, inverted, magnified image beyond C.
  • Object at F: Real, inverted, highly magnified image at infinity.
  • Object between F and P: Virtual, erect, magnified image behind the mirror.
• Convex Mirror: Always forms a virtual, erect, and diminished image, regardless of the object's position. The image is always formed behind the mirror, between the pole (P) and the principal focus (F).

Real images can be formed on a screen, while virtual images cannot. Real images formed by single mirrors are always inverted, and virtual images are always erect.

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Representation Of Images Formed By Spherical Mirrors Using Ray Diagrams

Ray diagrams are graphical tools used to determine the position, size, and nature of the image formed by spherical mirrors. Drawing ray diagrams helps in understanding the process of image formation. We use a few standard rays whose paths after reflection are known:

1. A ray parallel to the principal axis, after reflection, passes through the principal focus (F) in the case of a concave mirror, or appears to diverge from the principal focus (F) in the case of a convex mirror.
2. A ray passing through the principal focus (F) of a concave mirror, or directed towards the principal focus (F) of a convex mirror, after reflection, becomes parallel to the principal axis.
3. A ray passing through the centre of curvature (C) of a concave mirror, or directed towards the centre of curvature (C) of a convex mirror, after reflection, is reflected back along the same path. This is because the ray strikes the mirror along the normal at that point.
4. A ray incident towards the pole (P) of the mirror is reflected according to the law of reflection ($\angle i = \angle r$), with the principal axis acting as the normal.

To find the position of the image, we draw at least two of these rays from a point on the object. The point where the reflected rays intersect (for a real image) or appear to intersect (for a virtual image) is the corresponding point on the image. The image of an extended object can be found by locating the images of a few key points (like the top and bottom) of the object.

(Image Placeholder: Ray diagram for a concave mirror. Principal axis, Pole P, Focus F, Centre of Curvature C are marked. An object is placed beyond C. Ray 1 parallel to axis passes through F after reflection. Ray 2 passing through C reflects back along C. The intersection of reflected rays between F and C forms a real, inverted, diminished image.)

(Image Placeholder: Ray diagram for a convex mirror. Principal axis, Pole P, Focus F (behind mirror), Centre of Curvature C (behind mirror) are marked. An object is placed in front of the mirror. Ray 1 parallel to axis appears to diverge from F after reflection. Ray 2 directed towards C reflects back along the same path. The intersection of the *extended* reflected rays behind the mirror, between P and F, forms a virtual, erect, diminished image.)

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Sign Convention For Reflection By Spherical Mirrors

To use the mirror formula consistently, a sign convention is necessary to assign positive or negative values to distances and heights. The most widely used convention is the New Cartesian Sign Convention:

1. Origin: The pole (P) of the mirror is taken as the origin (0,0).
2. Principal Axis as X-axis: The principal axis of the mirror is taken as the X-axis.
3. Distances Measured from Pole: All distances are measured from the pole of the mirror.
4. Direction of Incident Light: Distances measured in the direction of the incident light are taken as positive.
5. Opposite to Incident Light: Distances measured in the direction opposite to the direction of incident light are taken as negative.
6. Heights Above Principal Axis: Heights measured upwards, perpendicular to the principal axis, are taken as positive.
7. Heights Below Principal Axis: Heights measured downwards, perpendicular to the principal axis, are taken as negative.

Unless otherwise specified, incident light is usually assumed to travel from left to right. Following this convention:

• Object distance ($u$): Always negative (object is in front of the mirror, incident light from left to right, distance measured from P to object is to the left).
• Image distance ($v$): Positive for real images (formed in front of the mirror), negative for virtual images (formed behind the mirror).
• Focal length ($f$): Negative for concave mirrors (focus in front), positive for convex mirrors (focus behind). Some texts use the opposite convention for focal length (positive for concave, negative for convex) based on converging/diverging nature; consistency within a problem is key. *Correction: Standard New Cartesian convention usually takes distance measured along the direction of incident light as positive. If light comes from left, distances to the right are positive. Thus, f is negative for concave (focus in front, to the left of P), and positive for convex (focus behind, to the right of P). Re-checking standard texts... Yes, typically concave mirrors have negative focal length and convex mirrors have positive focal length in this convention IF incident light is considered to come from the left. Let's stick to that for consistency with many sources. Wait, my prompt formula $1/f = 1/v + 1/u$ works with f being positive for concave and negative for convex if u is negative. Let's clarify based on the formula provided in the prompt. The mirror formula often uses u as positive for real object, v positive for real image, f positive for concave. This implies a different sign convention (e.g., distances to the left are positive, or object distance is always positive, etc.). Let's use the convention consistent with the provided formula in the subheading: object distances $u$ positive for real object, negative for virtual. Image distances $v$ positive for real image, negative for virtual. Focal length $f$ positive for concave, negative for convex. This convention is also used in many contexts.*

  Let's clarify based on the typical $1/f = 1/v + 1/u$ formula usage. Often, in this formula:

  • Object distance ($u$): Distance of the object from the pole. Taken as positive for real objects (object in front of the mirror), negative for virtual objects (e.g., image from another lens acting as object).
  • Image distance ($v$): Distance of the image from the pole. Taken as positive for real images (formed in front of the mirror), negative for virtual images (formed behind the mirror).
  • Focal length ($f$): Distance of the principal focus from the pole. Taken as positive for concave mirrors, negative for convex mirrors.

  This convention is consistent with the converging (positive focal length) and diverging (negative focal length) nature used in lenses, and is commonly used with the given mirror formula. Let's adopt this convention for this section.

  • Height of object ($h_o$): Positive if upright (above principal axis), negative if inverted.
  • Height of image ($h_i$): Positive if erect (above principal axis), negative if inverted (below principal axis).

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  Mirror Formula And Magnification

  The relationship between the object distance ($u$), image distance ($v$), and focal length ($f$) for spherical mirrors (of small aperture) is given by the Mirror Formula:

  $ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $

  This formula is valid for both concave and convex mirrors, for all positions of the object, provided the appropriate sign convention is used consistently.

  The Magnification ($m$) produced by a spherical mirror is defined as the ratio of the height of the image ($h_i$) to the height of the object ($h_o$). It indicates how much larger or smaller the image is compared to the object.

  $ m = \frac{\text{Height of Image}}{\text{Height of Object}} = \frac{h_i}{h_o} $

  For spherical mirrors, the magnification is also related to the object distance and image distance:

  $ m = -\frac{v}{u} $

  The sign of magnification indicates the orientation of the image:

  • If $m$ is positive, the image is erect (same orientation as the object). Erect images are always virtual for single mirrors.
  • If $m$ is negative, the image is inverted (opposite orientation to the object). Inverted images are always real for single mirrors.

  The magnitude of magnification indicates the relative size:

  • $|m| > 1$: Magnified image.
  • $|m| < 1$: Diminished image.
  • $|m| = 1$: Image of the same size as the object.

Reflection Of Light By Spherical Mirrors

Sign Convention

As discussed in the previous section, consistent use of a sign convention is crucial when applying the mirror formula and magnification formula. We will use the convention where distances of real objects and real images are positive, virtual objects and virtual images are negative, and focal length is positive for concave and negative for convex mirrors. Heights above the axis are positive, below are negative.

• Object distance $u$: Positive for real object (usually in front of mirror), Negative for virtual object (behind mirror).
• Image distance $v$: Positive for real image (in front of mirror), Negative for virtual image (behind mirror).
• Focal length $f$: Positive for concave mirror, Negative for convex mirror.
• Radius of curvature $R$: Positive for concave mirror, Negative for convex mirror. ($f = R/2$).
• Height of object $h_o$: Positive (usually placed upright).
• Height of image $h_i$: Positive for erect image, Negative for inverted image.

Example: A real object is placed in front of a concave mirror. By this convention, $u$ is positive, $f$ is positive. The image position $v$ will depend on $u$ (can be positive or negative). If the image is real, $v$ will be positive, and the image will be inverted ($h_i$ negative, $m$ negative). If the image is virtual, $v$ will be negative, and the image will be erect ($h_i$ positive, $m$ positive).

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Focal Length Of Spherical Mirrors

The principal focus (F) of a spherical mirror is a specific point on the principal axis related to the reflection of parallel rays. The focal length ($f$) is the distance between the pole (P) and the principal focus (PF).

For spherical mirrors of small aperture, the principal focus is located midway between the pole and the centre of curvature (C).

$ f = \frac{R}{2} $

where $R$ is the radius of curvature.

Following our adopted sign convention:

• For a concave mirror, the centre of curvature C is in front of the mirror, so the radius of curvature $R$ is positive. The principal focus F is also in front of the mirror, and its distance from the pole $f$ is taken as positive.
• For a convex mirror, the centre of curvature C is behind the mirror, so the radius of curvature $R$ is negative. The principal focus F is also behind the mirror, and its distance from the pole $f$ is taken as negative.

This convention aligns with the converging nature of concave mirrors (real focus) and the diverging nature of convex mirrors (virtual focus).

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The Mirror Equation ($ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $)

The mirror equation is a mathematical relationship that links the object distance ($u$), the image distance ($v$), and the focal length ($f$) of a spherical mirror. It can be derived using geometry and the laws of reflection for paraxial rays (rays close to the principal axis and making small angles with it).

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Derivation Sketch (Concave Mirror, Real Image)

Consider a concave mirror, an object AB placed beyond the centre of curvature C. Let the pole be P, focus F, and centre of curvature C. Object height $h_o$, image height $h_i$. Object distance $u$ (positive), image distance $v$ (positive), focal length $f$ (positive).

(Image Placeholder: Detailed ray diagram for concave mirror with object AB beyond C, forming real image A'B' between F and C. Show rays from B: parallel to axis through F, through C back along C, to P reflecting at angle i=r. Identify similar triangles like A'B'P ~ ABP and A'B'F ~ MPF (where M is a point on the mirror near P, and MP is considered a straight line perpendicular to axis).)

Using similar triangles formed by rays from the object, reflecting from the mirror, and forming the image, and applying the sign convention, one can arrive at the mirror formula.

Consider $\triangle ABP$ and $\triangle A'B'P$. These are similar (angles are equal due to law of reflection at P, and right angles at A and A').

$ \frac{A'B'}{AB} = \frac{PA'}{PA} \implies \frac{-h_i}{h_o} = \frac{v}{u} \implies m = \frac{h_i}{h_o} = -\frac{v}{u} $ (This gives the magnification formula)

Consider a ray from B parallel to the principal axis, reflecting through F. Let this ray strike the mirror at M, close to P. Draw a perpendicular from M to the principal axis. For a small aperture, M is very close to P, and MP can be considered perpendicular to the axis, length approximately equal to the height of the ray's point of incidence from the axis. Let's use a simpler triangle involving F.

Consider $\triangle A'B'F$ and $\triangle MPF$. These are similar (if M is very close to P, MP is nearly perpendicular to the axis). $MP \approx AB = h_o$.

$ \frac{A'B'}{MP} = \frac{FA'}{FP} \implies \frac{-h_i}{h_o} = \frac{v - f}{f} $ (using distances from P and F)

Equating the two expressions for magnification:

$ -\frac{v}{u} = \frac{v - f}{f} $

$ -\frac{v}{u} = \frac{v}{f} - 1 $

Divide by $v$ (assuming $v \ne 0$):

$ -\frac{1}{u} = \frac{1}{f} - \frac{1}{v} $

$ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} $

$ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $

This derivation uses $u, v, f$ as positive magnitudes. If we use the sign convention where $u$ is positive for real object, $v$ positive for real image, $f$ positive for concave, then the formula holds as stated in the subheading $ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $. For the derivation to yield the formula with plus signs, the distances must be defined relative to a fixed direction from the pole (e.g. distances measured against incident light are negative). Let's re-evaluate the similar triangles with proper signed distances from the pole P.

Let incident light travel from left to right. P is origin (0). Object is at $u$ (negative). Image is at $v$ (real is negative, virtual is positive). Concave $f$ is negative. Convex $f$ is positive.

$\triangle ABP$ and $\triangle A'B'P$: $\frac{A'B'}{AB} = \frac{PA'}{PA}$. Let P be the origin. A is at $u$, A' at $v$. B is at $(u, h_o)$, B' at $(v, h_i)$. $\frac{h_i}{h_o} = \frac{v}{u}$. This gives $m=v/u$. But magnification formula uses $m=-v/u$. This suggests the ratio of heights should be equal to the ratio of distances with a negative sign if distances are measured from the pole in a consistent direction. The standard $m=-v/u$ and $1/f = 1/v + 1/u$ using the sign convention as described (real distances +ve, virtual -ve for v, real object +ve for u, concave f +ve, convex f -ve) seems to have inconsistencies if $u$ for real object is +ve. Let's revert to the consistent New Cartesian convention where distances measured in the direction of incident light are positive.

New Cartesian Sign Convention (re-revisited): Incident light from Left to Right. Pole P at Origin. Distances to the right of P are positive, to the left are negative. Heights above axis are positive, below are negative.

• Object distance $u$: Real object usually on left, so $u$ is negative. Virtual object on right, $u$ is positive.
• Image distance $v$: Real image on left (concave mirror), $v$ is negative. Virtual image on right (concave or convex), $v$ is positive.
• Focal length $f$: Concave mirror focus on left, $f$ is negative. Convex mirror focus on right, $f$ is positive.
• Radius of curvature $R$: Concave C on left, $R$ is negative. Convex C on right, $R$ is positive. ($f = R/2$).
• $h_o$: Positive (usually upright). $h_i$: Positive if erect, Negative if inverted.

Let's re-derive $m = -v/u$ and $1/f = 1/v + 1/u$ using this convention.

$\triangle ABP$ and $\triangle A'B'P$: $\frac{A'B'}{AB} = \frac{PA'}{PA}$. Coordinates: A is $(u, 0)$, B is $(u, h_o)$, A' is $(v, 0)$, B' is $(v, h_i)$. P is $(0,0)$. $PA = u$, $PA' = v$. $AB$ is height $h_o$, $A'B'$ is height $h_i$. $\frac{h_i}{h_o} = \frac{v-0}{u-0} = \frac{v}{u}$. But image might be inverted... Let's use vector heights. $\vec{AB}$ is $(0, h_o)$. $\vec{A'B'}$ is $(0, h_i)$. $\triangle ABP$ and $\triangle A'B'P$ are similar, using angles. Angle of incidence = angle of reflection at P. The ratio of corresponding sides $|A'B'|/|AB| = |PA'|/|PA|$. The sign comes from comparing directions. If A'B' is inverted relative to AB, $h_i/h_o$ should be negative. By geometry $\frac{h_i}{h_o} = -\frac{v}{u}$. This seems standard and consistent with magnification definition $m=h_i/h_o$. So $m = -v/u$. Using the New Cartesian convention, if $h_o$ is positive, $h_i$ is negative for inverted real image, $v$ is negative for real image, $u$ is negative for real object. $m = (-ve)/(+ve) = -ve$. $-v/u = -(-ve)/(-ve) = -ve$. Signs work.

For the mirror formula, consider the ray parallel to axis through M reflecting through F. Slope of incident ray is 0. Slope of reflected ray from M to F is related to coordinates of M and F. This derivation is more involved. A simpler approach relates similar triangles formed by rays from the object and reflecting through F or C.

Let's use triangles formed by the ray from B passing through C and the ray parallel to the axis from B reflecting through F. Object AB, Image A'B'. Concave mirror.

$\triangle ABC$ and $\triangle A'B'C$ are similar. $\frac{A'B'}{AB} = \frac{A'C}{AC}$. Let distances be from P. $AC = PA - PC = u - R$. $A'C = PC - PA' = R - v$. $\frac{|h_i|}{h_o} = \frac{|R-v|}{|u-R|}$. With sign convention, $h_i/h_o = -(v-R)/(u-R)$. Or $\frac{h_i}{h_o} = \frac{R-v}{u-R}$ magnitude wise. For concave mirror, $u, v, R$ are negative in New Cartesian. $\frac{h_i}{h_o} = \frac{-v - (-R)}{-u - (-R)} = \frac{R-v}{R-u}$. Also $m = h_i/h_o = -v/u$. So $\frac{R-v}{R-u} = -\frac{v}{u}$. $u(R-v) = -v(R-u)$. $uR - uv = -vR + uv$. $uR + vR = 2uv$. Divide by $uvR$: $\frac{1}{v} + \frac{1}{u} = \frac{2}{R}$. Since $f = R/2$, $2/R = 1/f$.

$ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} $

This formula is derived using the New Cartesian sign convention where distances to the left of the pole are negative and distances to the right are positive. And $f=R/2$ relation also holds with signs (concave $f, R$ negative; convex $f, R$ positive).

Summary of Mirror Formula and Magnification:

• Mirror Formula: $ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} $
• Magnification: $ m = \frac{h_i}{h_o} = -\frac{v}{u} $
• Relation between focal length and radius of curvature: $ f = \frac{R}{2} $

These formulas, combined with the New Cartesian sign convention (origin at P, left is negative, right is positive, up is positive, down is negative, incident light direction usually from left to right), are used to solve problems involving image formation by spherical mirrors.

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