Refractive Index

For , light rays bend away from the normal upon passing from a denser to a rarer medium. The first diagram (i) correctly demonstrates this refraction.

1. Total Capacitance of the Combination:

   The initial total capacitance (without dielectric) is the series combination of and :

   After inserting the dielectric of constant into the capacitor of capacitance , its new capacitance becomes:

   The new total capacitance is the series combination of and :

   Simplifying:

   Thus:

2. Final Voltage Difference Across the Combination:

   The charge on the initial combination is given by:

   Since the battery is removed, the total charge remains the same. The final voltage across the new combination is:

   Substituting the values:

3. Total Energy Stored in the Combination:

   The total energy stored in a capacitor is given by:

   Substituting and :

   Simplifying:

   Thus, the total energy stored is:

The maximum induced emf is related to mutual inductance by: For : The maximum value of is . Substituting values:

Step 1: Understanding the Concept:
This problem requires the use of the Lens Maker's Formula, which relates the focal length of a lens to its refractive index and the radii of curvature of its two surfaces. It is essential to use the correct sign convention for the radii of curvature.

Step 2: Key Formula or Approach:
The Lens Maker's Formula is given by: where: - is the focal length of the lens.
- is the refractive index of the lens material with respect to the surrounding medium.
- is the radius of curvature of the first surface (where light enters).
- is the radius of curvature of the second surface.

Step 3: Detailed Explanation:
Let's apply the sign convention for a double convex lens. We assume light travels from left to right.
1. For the first surface, the center of curvature is on the right side. Thus, its radius of curvature is positive: .
2. For the second surface, the center of curvature is on the left side. Thus, its radius of curvature is negative: .
The given values are: - Refractive index, .
- Radius of curvature for both surfaces has magnitude .
Now, substitute these values into the Lens Maker's Formula: Taking the reciprocal of both sides, we get:

Step 4: Final Answer:
The focal length of the double convex lens is equal to its radius of curvature, .

A bi-convex lens is a type of lens in which both surfaces are convex. The focal length of a lens is determined by the lensmaker's formula:
Where: - is the focal length of the lens,
- is the refractive index of the material of the lens,
- and are the radii of curvature of the two surfaces of the lens.
For a bi-convex lens, (convex surface) and (concave surface). The refractive index of the glass is .
Substituting these values into the lensmaker's formula:
Thus, the focal length of the lens is:
The lens is a converging lens because it is bi-convex.
Now, when the lens is filled with a liquid of refractive index , we use the modified lensmaker's formula:
Where is the refractive index of the liquid. Substituting the values into the formula:
Thus, the new focal length of the lens when filled with liquid is:
The nature of the lens remains converging, but its focal length decreases due to the increased refractive index.
The power of the lens is given by:
The power in air is:
The power in the liquid is: