Question

A person suffering from short-sightedness can see up to 100 meters. Calculate the nature and the focal length of the lens in order to correct this defect so that he can see objects at infinity hence correct vision. Also, draw the ray diagram.

Hint:

To correct short-sightedness, a concave lens with a negative focal length is used. The focal length can be calculated using the lens formula.

Answer 1

A person suffering from short-sightedness (myopia) can see objects clearly only up to a certain distance, which in this case is 100 meters. In order to correct this defect, a diverging lens (concave lens) is used.
For short-sightedness, the image of a distant object is formed in front of the retina, so a diverging lens is required to diverge the light rays before they reach the eye, so that the image is formed on the retina.
Step 1: Determine the focal length of the lens.
In the case of myopia, the near point of the person is 100 meters, and the person wants to be able to see distant objects at infinity.
The lens formula is: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u}, \]
where:
- \( f \) is the focal length of the lens, - \( v \) is the image distance (infinity for distant objects), - \( u \) is the object distance (100 meters).
For distant objects, \( v = \infty \), and the formula becomes: \[ \frac{1}{f} = \frac{1}{\infty} - \frac{1}{u} = 0 - \frac{1}{100}. \] Thus, \[ f = -100 \, \text{meters}. \] The negative sign indicates that the lens is a diverging lens.
Step 2: Nature of the lens.
The focal length is negative, indicating that the lens is a concave lens, which is used to correct short-sightedness.
Step 3: Ray Diagram.
In the ray diagram for a concave lens: - The object is placed at a distance greater than the focal length. - The diverging rays from the object are converged by the concave lens to form a virtual image at the near point (100 meters), allowing the person to see the object clearly.