Spherical Mirrors

For a prism, the relation between the refracting angle , the refractive index , and the angle of minimum deviation is given by: The angle of minimum deviation is related to the refracting angle by the equation: Thus, the angle of minimum deviation is .

To determine the total power of lenses in contact, we use the fundamental formula for lens power:

Where is the power and is the focal length of the lens.

When multiple lenses are in contact, their combined power is the sum of their individual powers:

1. Analyzing the given situation:
   • For the combination of two biconvex lenses with focal lengths and , the total power is given by:

• For the combination of the first biconvex and the biconcave lens with focal lengths and respectively, the total power is:

• For the combination of the second biconvex and the biconcave lens with focal lengths and respectively, the total power is:

1. Justifying the Correct Option:
   • The derived expressions for the total powers , , and match the given correct answer option:

Conclusion: Thus, based on the above analysis and calculations, the correct answer is and .

To find the ratio of focal lengths of a glass lens when placed in water and in air, we start by understanding the lens maker's formula. The formula for thin lenses is given by:

Where:

• is the focal length of the lens.
• is the refractive index of the material of the lens with respect to the surrounding medium.
• and are the radii of curvature of the lens surfaces.

1. **In Air:**
If the lens is in air, the refractive index of glass with respect to air ( ) is .

The formula becomes:

2. **In Water:**
If the lens is in water, first find the refractive index of glass with respect to water ( ). This can be calculated using relative refractive indices:

The lens maker's formula for a lens in water becomes:

3. **Finding the Ratio of Focal Lengths:**
Given that the same radii are used, we can compare the two equations:

From these equations, the ratio of focal lengths is calculated as:

Therefore, the ratio of the focal lengths of a glass lens in water to air is .

The correct answer is: .

For a prism, the relation between the refracting angle , the refractive index , and the angle of minimum deviation is given by: The angle of minimum deviation is related to the refracting angle by the equation: Thus, the angle of minimum deviation is .