A thin convex lens \( L \) of focal length 10 cm and a concave mirror \( M \) of focal length 15 cm are placed coaxially 40 cm apart as shown in the figure. A beam of light coming parallel to the principal axis is incident on the lens. The final image will be formed at a distance of:
Show Hint
- 10 cm, left of lens
- 10 cm, right of lens
- 20 cm, left of lens
- 20 cm, right of lens
The Correct Option is B
Solution and Explanation
Image Formation by a Convex Lens and Concave Mirror
Step 1: Image Formation by the Convex Lens
- A parallel beam of light incident on a convex lens converges at the focus of the lens.
- Given focal length of the convex lens: \[ f_L = 10 \text{ cm} \] - Since parallel rays converge at the focal point, the image formed by the convex lens is: \[ I_1 = 10 \text{ cm (to the right of the lens)} \] - This image acts as the object for the concave mirror.
Step 2: Object Distance for the Concave Mirror
- Distance between the lens and the mirror: \[ d = 40 \text{ cm} \] - Distance of the image formed by the lens from the mirror: \[ u_M = 40 - 10 = 30 \text{ cm} \] - Focal length of the concave mirror: \[ f_M = -15 \text{ cm} \quad \text{(negative for concave mirror)} \] Using the mirror formula: \[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \] \[ \frac{1}{v} + \frac{1}{30} = \frac{1}{-15} \] \[ \frac{1}{v} = \frac{1}{-15} - \frac{1}{30} \] \[ \frac{1}{v} = -\frac{2}{30} - \frac{1}{30} = -\frac{3}{30} = -\frac{1}{10} \] \[ v = -10 \text{ cm} \]
Step 3: Final Image Position
- The negative sign indicates that the final image is on the same side as the mirror.
- Distance from the mirror: 10 cm.
- Since the mirror is 40 cm from the lens, the final image is: \[ 40 - 10 = 30 \text{ cm from the lens, on the right}. \] - Since the image distance from the lens is greater than the focal length, the final image is real and inverted.
Step 4: Conclusion
- Final image is formed at 10 cm to the right of the lens.
- Thus, the correct answer is: \[ \boxed{(B) \text{ 10 cm, right of lens}} \]
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