Question

A convex mirror has a radius of 20cm .An object is placed 30cm in front of the mirror. Find the distance where the image is formed​

Answer 1

Answer:

7.5 cm behind the convex mirror

Explanation:

As per the provided information in the given question, we have ;

• Radius of curvature of convex mirror (R) = 20 cm
• Object distance (u) = –30 cm

[Object Distance is negative because the object is always placed on the left side of the mirror and sign convention implies that the distance towards the left of the mirror is taken negative.]

We are asked to calculate the image distance (v).

We can calculate the image distance by using the mirror formula.

$\\ \longrightarrow \quad \pmb{\boxed{\sf {\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f} } }}$

• v denotes image distance
• u denotes object distance
• f denotes focal length

★ Calculating focal length :

$\\ \longrightarrow \quad \pmb{\boxed{\sf { 2f = R } }}$

• R = 20 cm

$\\ \longrightarrow \quad \sf { 2f = 20 \; cm}$

$\\ \longrightarrow \quad \sf { f = \dfrac{20 \; cm}{2} }$

$\\ \longrightarrow \quad \bf \underline{ f = +10 \; cm}$

Therefore, focal length of the convex mirror is 10 cm.

Now, we have

• u = – 30 cm
• f = +10 cm

Substituting values on the mirror formula,

$\\ \longrightarrow \quad \pmb{\boxed{\sf {\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f} } }}$

$\\ \longrightarrow \quad \sf { \dfrac{1}{v} + \dfrac{1}{(-30)} = \dfrac{1}{10} }$

$\\ \longrightarrow \quad \sf { \dfrac{1}{v} - \dfrac{1}{30} = \dfrac{1}{10} }$

$\\ \longrightarrow \quad \sf { \dfrac{1}{v} = \dfrac{1}{10} + \dfrac{1}{30} }$

$\\ \longrightarrow \quad \sf { \dfrac{1}{v} = \dfrac{3+ 1}{30} }$

$\\ \longrightarrow \quad \sf { \dfrac{1}{v} = \dfrac{4}{30} }$

On reciprocating both sides,

$\\ \longrightarrow \quad \sf { v =\cancel{ \dfrac{30}{5}} }$

$\\ \longrightarrow \quad \bf \underline{ v= 7.5\; cm}$

Therefore, the image is formed at 7.5 cm behind the convex mirror.