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To determine whether the biconvex lens made of a transparent material with a refractive index of 1.25 will behave as a converging or a diverging lens when immersed in water with a refractive index of 1.33, we need to consider the relationship between the refractive indices of the lens material and the surrounding medium.

In general, when a lens is immersed in a medium with a higher refractive index than the lens material itself, the lens tends to behave as a diverging lens. Conversely, when a lens is immersed in a medium with a lower refractive index than the lens material, the lens tends to behave as a converging lens.

In this case, the refractive index of the lens material (1.25) is lower than that of the surrounding medium (water, with a refractive index of 1.33). Therefore, the lens will behave as a converging lens.

This behavior can be explained by Snell's law, which states that when light passes from one medium to another, the change in the refractive index causes the light to bend. In this scenario, light passing through the lens will bend towards the normal as it enters the water, causing the light rays to converge. Hence, the lens behaves as a converging lens.

When a convex lens is placed in contact with a plane mirror and an object at a distance of 20 cm on the axis of this combination has its image coinciding with itself, it means that the combination of the convex lens and the plane mirror acts as a "catadioptric" system, where the lens forms a real image and the mirror forms a virtual image. These images coincide, resulting in no net displacement of the object's image.

In this case, the effective focal length (fefffeff) of the combination is equal to the distance between the object and its real image formed by the lens. Since the object's image coincides with itself, the distance between them is 0.

Given that the object is placed at a distance of 20 cm from the combination, the distance between the object and its image formed by the lens is also 20 cm.

Therefore, the effective focal length (fefffeff) of the combination is 20 cm.

However, the focal length of the convex lens (flensflens) can be determined using the lens formula:

1flens=1v−1uflens1=v1−u1

Where:

• flensflens is the focal length of the lens,
• vv is the image distance (which is 20 cm since the image coincides with the object),
• uu is the object distance (given as 20 cm).

Substituting the given values:

1flens=120−120flens1=201−201

1flens=0flens1=0

flens=∞flens=∞

The focal length of the convex lens is infinity. This result is consistent with the fact that the combination of the convex lens and the plane mirror forms an image coinciding with the object, indicating that the system acts as a mirror.

The relationship between the angle of incidence (ii), the angle of the prism (AA), and the angle of minimum deviation (DD) from a triangular prism can be expressed using the prism formula.

The prism formula states:

A + D = i + eA + D = i + e

Where:

• AA is the angle of the prism (apex angle),
• DD is the angle of minimum deviation,
• ii is the angle of incidence,
• ee is the angle of emergence.

For a prism in minimum deviation condition, the angle of incidence (ii) is equal to the angle of emergence (ee).

So, the relation can be written as:

A + D = i + iA + D = i + i A + D = 2iA + D = 2i

Therefore, the relationship between the angle of incidence (ii), the angle of the prism (AA), and the angle of minimum deviation (DD) from a triangular prism is given by:

A + D = 2iA + D = 2i

The focal length of a lens does not change when red light incident on it is replaced by violet light. The reason for this is that the focal length of a lens depends on the lens material and shape, not on the color (or wavelength) of the incident light.

The focal length of a lens is determined by its refractive index and curvature, both of which remain constant regardless of the color of light passing through the lens. When red light is replaced by violet light, the change in wavelength does not alter the refractive index of the lens material or its shape, so the focal length remains unchanged.

In other words, the focal length of a lens is a property of the lens itself and is not affected by the color of light passing through it. Therefore, the focal length of the lens remains the same regardless of whether the incident light is red or violet.