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

Solution:

A Convex Lens forms a real and inverted image of a needle at a distance of $50\mathit{cm }$ from it, so that the size of the image is equal to the size of the object. The power of the lens is $4\mathit{ D}.$ Given that, Image distance $\left(v\right) = 50\mathit{ cm}$ Magnification $\left(m\right) = -1$ So, we will apply the formula $m=\frac{v}{u}$ From here we will determine the “u” $u=\frac{50}{-1}$   $u=-50\mathit{ cm}$ So, the needle is placed $50\mathit{ cm}$ in front of the lens. Now, to find the power first we will find the value of $“f”$ . So, we will apply the lens formula: $\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$ Where, $v=50\mathit{ cm}$ , $u= -50\mathit{ cm}$   $\frac{1}{f}=\frac{1}{50}-\frac{1}{-50}$   $f=0.25\mathit{ m}$ Now, to find the power of the lens we will apply the Power Formula. $P=\frac{1}{f}$   $P=\frac{1}{0.25}$   $P=4\mathit{ D}$