Ray Optics And Optical Instruments

Lens Formula and Error Calculation

Given: The object distance is cm and the image distance is cm.

We apply the lens formula to determine the focal length:

Solving for , we get:

Now, differentiate the lens equation to calculate the uncertainty in the focal length:

Now, substituting the values for and , we get:

Calculating the expression, we get:

Thus, the answer is .

The correct options are (A),(C) and(D).
For no reflection,







For red lights,

Given that = speed of person = 60 cm/s

Shadow Speed Problem

Also, is the speed of the tip of the person's shadow.

Applying a similar triangle rule in and :

This simplifies to:

Rearranging the equation:

Now, differentiating both sides with respect to :

Given that , we can substitute this into the equation:

Thus, the speed of the tip of the shadow is:

Object distance cm Image distance cm
Applying lens formula ….. (1)

f =

differentiate equation 1:

= 1

So, the answer is .

The correct options are: (B),(C) and (D).
For blue light,















For red light,

Reflection and Refraction Calculations

Reflection from M2:

The equation for the reflection from M2 is:

Simplifying the equation:

Solving for :

Refraction from L:

The equation for refraction from L is:

Substituting the values for and :

Solving for :

For L2 (If L2 is at the focus of M1):

The equation becomes:

Solving for :

If L2 is at the pole of M1:

The equation is:

Solving for :

If L2 is at the center of curvature of M1:

The equation becomes:

Solving for :

Refractive Index and Angle Analysis for the Prism

For the second prism, the relationship between the refractive index , the angle of refraction , and the angle is:

where is the angle of the prism.

Using the condition for minimum deviation:

Given that , we have:

For the first prism:

From the refractive index relationship:

where is the angle of incidence. Substituting for , we find:

Simplifying:

Equating and solving, we determine:

Final Answer:

• The value of is:

To find the value of , we need to analyze the optics problem involving a solid glass sphere with a spherical air cavity, as shown in the diagram. Step 1: Understand the setup

• The glass sphere has a refractive index and radius .
• The air cavity inside the sphere has a radius and is centered at point , which is also the point where a very thin glass layer separates the air cavity from the glass.
• The light source emits unpolarized, unidirectional, and monochromatic light from inside the air cavity toward the periphery of the glass sphere.
• The light is reflected at point (the interface between the air cavity and the glass sphere), and the angle of incidence at this inner surface is .
• The reflected light is fully polarized, which implies that the angle of incidence must be the Brewster angle (also called the polarizing angle).

Step 2: Identify the Brewster angle condition The Brewster angle occurs when the reflected light is fully polarized, which happens when the angle of incidence satisfies the condition: Here:

• (refractive index of the glass),
• (refractive index of the air inside the cavity).

So, the Brewster angle condition becomes:
Step 3: Calculate From the Brewster angle condition, we have: We know from trigonometry that , and using the identity , we can find . Since , we can think of a right triangle where:

• The opposite side (related to ) is ,
• The adjacent side (related to ) is ,
• The hypotenuse is .

Thus: Alternatively, we can use the identity: Given , we have: Let . Then: Since is the angle of incidence (between and ), is positive.
Step 4: Final answer The value of is:
Final Answer: