JEE Physics 2023 Question with Solution

Given: Radius of curvature of the concave mirror is $R = 40 \, \text{cm}$. The two objects are at $15 \, \text{cm}$ and $25 \, \text{cm}$ in front of the mirror.

Find: The distance between the two images formed.

For a concave mirror,

$$f = \frac{R}{2} = \frac{40}{2} = -20 \, \text{cm}$$

The focal length is taken negative by the sign convention used in the solution.

For object A, the object distance is $u_1 = -15 \, \text{cm}$. Using the mirror formula,

$$\frac{1}{v_1} + \frac{1}{u_1} = \frac{1}{f}$$ $$\frac{1}{v_1} + \frac{1}{-15} = \frac{1}{-20}$$ $$\frac{1}{v_1} = \frac{1}{15} - \frac{1}{20} = \frac{20 - 15}{300} = \frac{1}{60}$$ $$v_1 = 60 \, \text{cm}$$

For object B, the object distance is $u_2 = -25 \, \text{cm}$. Again,

$$\frac{1}{v_2} + \frac{1}{u_2} = \frac{1}{f}$$ $$\frac{1}{v_2} + \frac{1}{-25} = \frac{1}{-20}$$ $$\frac{1}{v_2} = \frac{1}{25} - \frac{1}{20} = \frac{-1}{100}$$ $$v_2 = -100 \, \text{cm}$$

So one image is at $+60 \, \text{cm}$ and the other is at $-100 \, \text{cm}$. Hence the distance between the images is

$$d = |v_1| + |v_2| = 60 + 100 = 160 \, \text{cm}$$

Therefore, the distance between the images is $160 \, \text{cm}$.

The solution states both "The Correct Option is B" and "$160 \, \text{cm}$ (Option 2)." Since the working clearly gives $160 \, \text{cm}$, the defensible matching option value is C.

Given: A concave mirror with $R = 40 \, \text{cm}$, so $f = -20 \, \text{cm}$. Objects are placed in front of the mirror at $u_1 = -15 \, \text{cm}$ and $u_2 = -25 \, \text{cm}$.

Find: Separation between the two image positions.

The mirror formula is

$$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$

For the first object,

$$\frac{1}{-20} = \frac{1}{v_1} + \frac{1}{-15}$$ $$\frac{1}{v_1} = -\frac{1}{20} + \frac{1}{15} = \frac{1}{60}$$ $$v_1 = 60 \, \text{cm}$$

For the second object,

$$\frac{1}{-20} = \frac{1}{v_2} + \frac{1}{-25}$$ $$\frac{1}{v_2} = -\frac{1}{20} + \frac{1}{25} = -\frac{1}{100}$$ $$v_2 = -100 \, \text{cm}$$

Because the image points lie on opposite sides relative to the sign convention used in the solution, the distance between them is the sum of magnitudes:

$$160 \, \text{cm} = 60 \, \text{cm} + 100 \, \text{cm}$$

So the correct option by value is C.