Question

A convex lens forms a real and inverted image of a needle at a distance of $50cm$ from it. Where is the needle placed in front of the convex lens if the image is equal to the size of the object? Also, find the power of the lens.

Answer 1

Step 1:Given data

Image distance, $v=50cm$

Height of image = $h_i$

Height of Object=$h_o$

$h_i=h_o$

Magnification, $m=-1$ (image is of the same size, real and inverted)

Step 2: To find

Object distance $u$ and power of the lens $P$

Step 3: Finding object distance $u$ using the equation of magnification

Magnification of a lens,

$$\begin{gathered} m=\frac{v}{u} \\ -1=\frac{50}{u} \\ u=-50cm \end{gathered}$$

Negative sign because the object is on the left side of the lens (sign convention )

Step 4: Using the lens formula to find the focal length

Applying Lens formula

$$\begin{gathered} \frac{1}{f}=\frac{1}{v}-\frac{1}{u} \\ \frac{1}{f}=\frac{1}{50}-\frac{1}{-50} \\ \frac{1}{f}=\frac{2}{50} \\ \frac{1}{f}=\frac{1}{25} \\ f=25cm \end{gathered}$$

Step 5: Finding the power of the lens

Power of lens$P=\frac{1}{f(inmeters)}=\frac{1}{25\times {10}^{-2}}$$=4m^{-1}$

The power of a lens is measured in Diopter D

$$\begin{gathered} 1D=1m^{-1} \\ 4m^{-1}=4D \end{gathered}$$

Therefore, the object's distance from the lens is $50\mathrm{cm}$ and the power of the lens is $4\mathrm{D}$.