Ray Optics And Optical Instruments

Any image is formed by a large number of rays passing from the object. Therefore, if any part of the lens is covered, the brightness of the image will be less as compared to the image formed without wrapping the lens. 

Therefore option D) is the correct answer - If the central portion of the convex lens is wrapped in black paper, then a full image will be formed but it will be less bright.

Convex lens with central portion wrapped in black paper

The rainbow is an example of the dispersion of sunlight by the water droplets in the atmosphere. A Rainbowis a beautiful phenomenon that occurs due to a combination of the refraction of sunlightby spherical droplets of water and of internal (not total) reflection.

• The dispersion of light is the splitting of light into its constituent colors.
• Total internal reflection is when the ray of light travels in a denser medium towards the rare medium and gets reflected back inside the initial medium.

As the light shows Total internal reflection and dispersion by splitting into the colors in the rainbow. Therefore, option B is the correct answer.

To find the focal length of the lens, we can use the lens formula:

We know that:

Solving the lens formula for :

Now, let's consider the situation where the lens is placed somewhere between the object and the screen. The distance between the object and the screen is given as . We can divide this distance into two parts:

• u is the object distance, the distance between the object and the lens.
• v is the image distance, the distance between the lens and the screen.

Therefore,

Rearranging the equation, we have:

Substituting this value of into the lens formula equation:

Let's solve this equation for:

Now, we need to find the expression for the focal length, so we take the reciprocal of both sides:

To simplify this expression, we can find the common denominator:

Simplifying further:

Factoring out from the denominator:

Now, we can divide both the numerator and denominator by :

Since the given distance between the object and the screen is , and the object distance is , we can rewrite the equation as:

Therefore, the correct option is (B):

1/f = (n - 1) (1/R₁ - 1/R₂)

Given:

• Power of the lens (P) = -4.5 D
• Material refractive index (n) = 1.6

Since the lens is equi-concave, the radii of curvature of both surfaces will have the same magnitude but opposite signs (R₁ = -R₂).

Using the lens maker's formula and substituting the given values, we can solve for the magnitude of the radii of curvature:

Now, we can find the value of by rearranging the equation:

Substituting the value of from the given power:

Converting to centimeters, we have:

The radii of curvature of the lens is approximately -540 cm or -5.4 m. Among the given options, the closest value to -540 cm is (B) -26.6 cm.

For light diverging from a finite point source:

The wave front is spherical, not parabolic. When light spreads out from a point source, it forms concentric spherical wave fronts around the source.

The wave front appears cylindrical only in the far-field (Fraunhofer) region, where the distance from the source is significantly greater than the source's dimensions.

The claim that intensity at the wave front remains unchanged with distance is incorrect. As light propagates outward, its intensity decreases due to the spreading of light over a larger area.

The intensity decreases in proportion to the square of the distance. This follows the inverse-square law, meaning the intensity III is inversely proportional to the square of the distance rrr from the source:

Thus, the correct statement for light diverging from a finite point source is:
"The intensity decreases in proportion to the distance squared."

The focal length of a convex lens is influenced by the refractive index of the material it is composed of and the wavelength of light passing through it. The refractive index of a material varies slightly with different wavelengths of light, which in turn affects the focal length.

In general, the refractive index of a material decreases as the wavelength of light increases. This phenomenon is known as dispersion. Among the options given, red light has the longest wavelength (longer than green, blue, and yellow light), so the refractive index of the lens material will be the least for red light.

According to the lens maker's formula, a lower refractive index corresponds to a longer focal length for a convex lens. Therefore, the focal length of a convex lens will be maximum for (C) red light.

Given the relationships:

Angle of incidence (i) = Angle of emergence (e)

Angle of incidence (i) = Angle of emergence (e) = * Angle of prism (A)

We can set up the following equations:

i = e (1)

i = e = * A (2)

The angle of deviation (D) is related to the angle of incidence and angle of emergence by the formula:

D = (i + e) - A (3)

Substituting equation (2) into equation (3), we get:

D = * A + * A - A

D = * A - A

D =

From the given information, we know that each of the angles of incidence and emergence is equal to of the angle of prism. Therefore, the angle of deviation is equal to half of the angle of prism.

Since the given prism is an equilateral prism, the angle of prism (A) is 60 degrees. Substituting this value into the equation for the angle of deviation, we get:

D = = 30 degrees

Hence, the correct option is (C) 30 degrees.

Given Information:
Refractive index of glass,
Refractive index of water,

Step-by-Step Explanation:

Step 1: Focal length of the lens in air ( )

For an equiconvex lens (radius of curvature same for both sides), the focal length in air is given by the lens maker's formula:

Substitute the value of :

Thus,

Step 2: Focal length of lens in water ( ):

When immersed in water, the lens formula becomes:

Substitute the given values:

Thus,

Hence, the new focal length in water is:

Step 3: Calculate the percentage change in focal length clearly:

The initial focal length was , and it becomes .

Change in focal length:

Percentage increase in focal length:

Thus, the focal length increases by 300% when immersed in water.

The critical angle ( ) for total internal reflection is given by the relation:

where is the refractive index of the denser medium relative to the rarer medium.

Important: Refractive index ( ) varies with wavelength (colour) due to dispersion. For visible light, the refractive index is highest for violet (shortest wavelength) and lowest for red (longest wavelength).

Step-by-Step Logic:

• Red light has the longest wavelength, thus the smallest refractive index ( ).
• As refractive index decreases, the critical angle increases, because:

Therefore, among visible colours, red will have the smallest refractive index, and thus the maximum critical angle.

Concept:

According to Huygens' principle, every point on a primary wavefront acts as a source of secondary wavelets. The new position of the primary wavefront after a given time interval is given by the envelope (common tangent) of all these secondary wavelets.

Key Point:

• In a homogeneous medium, the speed of the primary wavefront and secondary wavelets are equal. This ensures that the new wavefront formed (by the secondary wavelets) moves forward smoothly and consistently.
• However, mathematically, the envelope (primary wavefront) can never move slower than the wavelets. It is either moving at the same speed (in homogeneous media) or faster (in specific geometrical conditions).

Hence, the correct option:

In a homogeneous medium, the velocities of primary wavefronts are greater than or equal to those of secondary wavelets.