Name: Sign Convention
Uploaded: 2019-07-19
Duration: 7 min 37 s
Description: All distances are measured from the optical centre of the lens. Distances measured in the direction opposite to the incident light's direction are considered...

Question

Draw a ray-diagram to show image formation of an object placed between the pole and the focus of a concave mirror. Derive the relation among

u

,

v

and

f

for this particular case. State the assumptions and sign convention used.

Accepted Answer

The diagram above gives the ray diagram of the image $(I)$ formed by an object $(O)$ placed between the focus and pole of the mirror.
The object distance is $u$ , the image distance is $v$ , and the focal length is $f$ .
Considering the sign convention, all the quantities measured behind the mirror will be positive and the quantities measured in front of the mirror will be negative.
Therefore, the magnitude of the distance between the pole and the object becomes, $O P = - u$
The magnitude of the distance between the pole and the image becomes, $I P = v$
The magnitude of the distance between the pole and the focus becomes, $F P = - f$

From the figure, using the paraxial approximation, we get, $a \approx \tan (a) = \frac{M P}{F P} = \frac{M P}{- f} - - (1)$ , $b \approx \tan (b) = \frac{M P}{I P} = \frac{M P}{v} - - (2)$ and $c \approx \tan (c) = \frac{M P}{O P} = \frac{M P}{- u} - - (3)$
In $∆ M O I$ , we get, $2 θ = c + b - - (4)$
In $∆ F M O$ , we get, $a + θ - α = c - - (5)$
In $∆ C M F$ , we get, $a = 2 α - - (6)$
Substituting the values of $α$ and $θ$ from equations $(6)$ and $(4)$ in equation $(5)$ , we get $a = c - b - - (7)$
By substituting the values of $a, b$ and $c$ from the equations $(1)$ , $(2)$ and $(3)$ in the equation $(7)$ , we get,
$\frac{M P}{- f} = \frac{M P}{- u} - \frac{M P}{v} \Rightarrow \frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

Draw a ray-diagram to show image formation of an object placed between the pole and the focus of a concave mirror. Derive the relation among $u$ , $v$ and $f$ for this particular case. State the assumptions and sign convention used.

Important Questions on Reflection of Light : Spherical Mirrors

Learn and Prepare for any exam you want !

Draw a ray-diagram to show image formation of an object placed between the pole and the focus of a concave mirror. Derive the relation among u, v and f for this particular case. State the assumptions and sign convention used.

Important Questions on Reflection of Light : Spherical Mirrors

Learn and Prepare for any exam you want !

Draw a ray-diagram to show image formation of an object placed between the pole and the focus of a concave mirror. Derive the relation among $u$ , $v$ and $f$ for this particular case. State the assumptions and sign convention used.