Derivation of Lens and mirror formula X Reflection and Refraction of light
MIRROR FORMULA
Consider a concave mirror MM' (of small aperture).
An object AB of height h is placed on the left of the mirror beyond its center of curvature so that the image formed is real, inverted and diminished in size(height = h').
Here :
Object distance = PB = -u
Image distance = PB'= -v
Focal length = PF = -f
Radius of curvature = PC = -R
As shown in Figure, Δ ABP ∼ Δ A'B'P.
∴ A'B'/AB = PB'/PB
∴ A'B'/AB = (-v)/(-u)
∴ A'B'/AB = v/u.............(I)
Similarly, as shown in the figure, Δ ABC ∼ Δ A'B'C.
∴ A'B'/AB = CB'/CB
∴ A'B'/AB = (PC-PB')/(PB-PC)........[as CB'=(PC-PB') and CB=(PB-PC)]
∴ A'B'/AB = [-R - (-v)]/[-u - (-R)]
∴ A'B'/AB = (-R + v)/(-u + R)........(II)
By (I) & (II),
v/u = (-R + v)/(-u + R)
∴ -uv + vR = -uR + uv
∴ vR + uR = 2uv
∴ (vR/uvR) + (uR/uvR) =2uv/uvR.......[Dividing each side by uvR]
∴ (1/u) + (1/v) = (2/R)..............(III)
Now, for an object placed at infinite distance (u = ∞), the image is formed at the focus F (v = f).Thus, eq.(III) reduces to
(1/∞) + (1/f) = (2/R)
∴ (2/R) = (1/f)....................(IV)
comparing eq.(III) and (IV),
(1/u) + (1/v) = (1/f)
LENS FORMULA.
The relation between object distance(u), image distance(v) and the focal length(f) of a lens is known as LENS FORMULA.
Consider a (thin spherical) convex lens MN (of small aperture).
An object AB of height h is placed in front of the lens MN beyond 2F on its left side.
The image A'B' formed by the lens is real, inverted and diminished in size (h').
Here,
Object Distance = OB = -u
Image Distance = OB' = +v
Focal Length = OF = +f
As shown in figure, Δ ABO ∼ Δ A'B'O
∴ (AB/A'B') = (OB/OB')
∴ (AB/A'B') = (-u/v) _______________(I)
Similarly, Δ OCF ∼ Δ A'B'F
∴ (OC/A'B') = (OF/FB')
∴ (AB/A'B') = (OF/FB')........(as AB = OC)
∴ (AB/A'B') = (OF)/(OB' - OF)
∴ (AB/A'B') = (f)/(v - f) __________(II)
From Eq. (I) and (II), we have
(-u/v) = (f)/(v - f)
∴ -u(v - f) = vf
∴ -uv + uf = vf
∴ (-uv/uvf) + (uf/uvf) = (vf/uvf)...(dividing by uvf)
∴ (-1/f) + (1/v) = (1/u)
∴ (1/f) = (1/v) - (1/u)
This is known as LENS FORMULA.
The relation between object distance(u), image distance(v) and the focal length (f) of the mirror is known as MIRROR FORMULA.
Consider a concave mirror MM' (of small aperture).An object AB of height h is placed on the left of the mirror beyond its center of curvature so that the image formed is real, inverted and diminished in size(height = h').
Here :
Object distance = PB = -u
Image distance = PB'= -v
Focal length = PF = -f
Radius of curvature = PC = -R
As shown in Figure, Δ ABP ∼ Δ A'B'P.
∴ A'B'/AB = PB'/PB
∴ A'B'/AB = (-v)/(-u)
∴ A'B'/AB = v/u.............(I)
Similarly, as shown in the figure, Δ ABC ∼ Δ A'B'C.
∴ A'B'/AB = CB'/CB
∴ A'B'/AB = (PC-PB')/(PB-PC)........[as CB'=(PC-PB') and CB=(PB-PC)]
∴ A'B'/AB = [-R - (-v)]/[-u - (-R)]
∴ A'B'/AB = (-R + v)/(-u + R)........(II)
By (I) & (II),
v/u = (-R + v)/(-u + R)
∴ -uv + vR = -uR + uv
∴ vR + uR = 2uv
∴ (vR/uvR) + (uR/uvR) =2uv/uvR.......[Dividing each side by uvR]
∴ (1/u) + (1/v) = (2/R)..............(III)
Now, for an object placed at infinite distance (u = ∞), the image is formed at the focus F (v = f).Thus, eq.(III) reduces to
(1/∞) + (1/f) = (2/R)
∴ (2/R) = (1/f)....................(IV)
comparing eq.(III) and (IV),
(1/u) + (1/v) = (1/f)
LENS FORMULA.
The relation between object distance(u), image distance(v) and the focal length(f) of a lens is known as LENS FORMULA.
Consider a (thin spherical) convex lens MN (of small aperture).
An object AB of height h is placed in front of the lens MN beyond 2F on its left side.
The image A'B' formed by the lens is real, inverted and diminished in size (h').
Here,
Object Distance = OB = -u
Image Distance = OB' = +v
Focal Length = OF = +f
As shown in figure, Δ ABO ∼ Δ A'B'O
∴ (AB/A'B') = (OB/OB')
∴ (AB/A'B') = (-u/v) _______________(I)
Similarly, Δ OCF ∼ Δ A'B'F
∴ (OC/A'B') = (OF/FB')
∴ (AB/A'B') = (OF/FB')........(as AB = OC)
∴ (AB/A'B') = (OF)/(OB' - OF)
∴ (AB/A'B') = (f)/(v - f) __________(II)
From Eq. (I) and (II), we have
(-u/v) = (f)/(v - f)
∴ -u(v - f) = vf
∴ -uv + uf = vf
∴ (-uv/uvf) + (uf/uvf) = (vf/uvf)...(dividing by uvf)
∴ (-1/f) + (1/v) = (1/u)
∴ (1/f) = (1/v) - (1/u)
This is known as LENS FORMULA.
For some value of angle of incidence, the angle of refraction becomes 90°.The angle of incidence for which the angle of refraction is 90°, is called CRITICAL ANGLE.
ReplyDeleteIf the angle of incidence is greater than the critical angle, the angle of refraction becomes greater than 90° and the refracted ray is confined to the same medium only.This is called TOTAL INTERNAL REFLECTION.