Refraction Through A Prism

- Atomic Model of Magnetism: The atomic model of magnetism is based on the behavior of atomic magnetic moments, which arise from the motion of electrons (both orbital motion and spin). The magnetic properties of a material depend on how these magnetic moments align and interact. A material can exhibit different types of magnetism depending on the alignment of these moments. - Types of Magnetism:
1. Paramagnetic Substances:
- In paramagnetic materials, the magnetic moments of individual atoms tend to align with an external magnetic field but do not stay aligned when the external field is removed.
- These materials have a weak, positive susceptibility to magnetism.
- The atomic magnetic moments are typically not in alignment with each other due to thermal motion.
- Example: Aluminum. 2. Diamagnetic Substances:
- Diamagnetic materials do not have permanent atomic magnetic moments. When subjected to a magnetic field, they produce an opposing magnetic field and are repelled by the external field.
- These materials have a negative, weak susceptibility to magnetism.
- Example: Bismuth. 3. Ferromagnetic Substances:
- Ferromagnetic materials have permanent magnetic moments even without an external magnetic field. These magnetic moments can align to create a strong net magnetic moment, even without an external field.
- They exhibit strong positive magnetic susceptibility and can retain their magnetization even after the external magnetic field is removed.
- Example: Iron.

Step 1: Understanding the Concept:
Total Internal Reflection (TIR) is a phenomenon where a light ray traveling from a denser medium (e.g., prism with refractive index ) to a rarer medium (e.g., air with refractive index ) is completely reflected back into the denser medium. This occurs only if the angle of incidence in the denser medium is greater than a specific angle called the critical angle, C. The question asks for the condition on the prism's apex angle A for TIR to be possible at the second face.

Step 2: Key Formula or Approach:
1. Critical Angle (C): The critical angle is defined by Snell's Law when the angle of refraction is 90 . For a prism in air, , which gives or .
2. Condition for TIR: For TIR to occur at the second face of the prism, the angle of incidence there, , must be greater than or equal to the critical angle C. That is, .
3. Prism Angle Formula: In a prism, the apex angle is related to the angle of refraction at the first face ( ) and the angle of incidence at the second face ( ) by the geometric relation .

Step 3: Detailed Explanation:
Let a ray of light enter the first face of the prism and refract at an angle . This ray then travels to the second face, striking it at an angle of incidence .
The condition for TIR to take place at this second face is: Using the prism angle formula, we can express as . Substituting this into the TIR condition: Rearranging the inequality to find the condition on A: This inequality shows that for TIR to be possible, the prism angle A must be at least . The value of depends on the initial angle of incidence on the first face. To determine if TIR is possible at all, we must check if this condition can be met for any valid .
The minimum possible value for is 0, which occurs when the light ray enters the prism at normal incidence ( ). If we substitute this minimum value of into our condition, we get the minimum requirement for A: This means that for TIR to be even a possibility, the prism angle A must be at least equal to the critical angle C. If , then will always be less than C (since ), and TIR can never occur.

Step 4: Final Answer:
Thus, total internal reflection is possible only if the angle of the prism is greater than or equal to the critical angle . Substituting the expression for the critical angle, , we prove that TIR is possible only when . The question's use of "more than" implies the strict condition for practical observation.

The Lens Maker's formula relates the focal length of a lens to the refractive index of the lens material and the radii of curvature and of the two curved surfaces of the lens.
For a thin lens, the Lens Maker's formula is given by: where:
- is the focal length of the lens,
- is the refractive index of the material of the lens,
- is the radius of curvature of the first surface (convex or concave),
- is the radius of curvature of the second surface (convex or concave).
Effect of Refractive Index ( ):
- As the refractive index increases, the focal length of the lens decreases. A higher refractive index material bends light more sharply, thus focusing light at a shorter distance.
Effect of Radii of Curvature ( and ):
- The radii of curvature of the lens surfaces also affect the focal length. For a convex surface, is positive, and for a concave surface, is negative. The curvature of the surfaces influences the converging or diverging behavior of the lens.

Step 1: Understanding the Concept:
When a ray of light passes through a prism, it deviates from its original path. The angle between the incident ray and the emergent ray is called the angle of deviation ( ). This angle depends on the angle of incidence ( ). The ( ) curve is a graphical representation of this relationship.

Step 2: Detailed Explanation and Diagram:
The graph of the angle of deviation ( ) as a function of the angle of incidence ( ) has the following characteristics:
\begin{itemize} \item Axes: The angle of incidence ( ) is plotted on the X-axis, and the angle of deviation ( ) is plotted on the Y-axis.
\item Shape of the Curve: As the angle of incidence is increased from a small value, the angle of deviation first decreases, reaches a minimum value, and then starts to increase again. This results in a U-shaped curve that is not symmetric.
\item Angle of Minimum Deviation ( ): The lowest point on the curve corresponds to the minimum possible value of the angle of deviation, denoted by . At this specific point, the angle of incidence is equal to the angle of emergence ( ). For every other value of , there are two corresponding values of the angle of incidence.
\end{itemize} Graphical Representation:
\begin{center} \begin{tikzpicture} \begin{axis}[ xlabel={Angle of Incidence ( )}, ylabel={Angle of Deviation ( )}, xmin=20, xmax=100, ymin=30, ymax=60, axis lines=left, ticks=none, clip=false, width=0.7\textwidth, height=0.5\textwidth, ] \addplot[smooth, thick, blue, domain=30:90] {37 + 0.01*(x-50)^2}; \node[label={below: }, circle, fill, inner sep=1.5pt] at (axis cs:50,37) {}; \draw[dashed, gray] (axis cs:20,37) -- (axis cs:50,37) node[left, black] { }; \draw[dashed, gray] (axis cs:50,30) -- (axis cs:50,37); \end{axis} \end{tikzpicture} \end{center}

Step 3: Final Answer:
The ( ) curve for a prism shows that the angle of deviation initially decreases with an increase in the angle of incidence, reaches a minimum value ( ), and then increases. The point where the deviation is minimum is called the angle of minimum deviation.

An astronomical telescope is designed to view distant objects in the sky, such as stars and planets, at a magnified scale. It consists of two lenses: the objective lens and the eyepiece.
- The objective lens ( ) has a large focal length and is placed at a distance from the object. It forms a real, inverted image of the object at its focal plane.
- The eyepiece ( ) is used to magnify the image formed by the objective lens. It is placed close to the focal plane of the objective lens and is used as a magnifying glass.
In the diagram: - is the objective lens.
- is the eyepiece.
- and are the focal points of the objective and eyepiece, respectively.
- The image formed by the objective lens is at the focal plane , and it is magnified by the eyepiece.
The magnifying power of the telescope is the ratio of the angular size of the image seen through the telescope ( ) to the angular size of the object when viewed with the unaided eye ( ). The formula for magnifying power of the telescope is:
Where: - is the focal length of the objective lens,
- is the focal length of the eyepiece.
The magnifying power is directly proportional to the focal length of the objective lens and inversely proportional to the focal length of the eyepiece.

Step 1: Prism formula. Refractive index of prism material is given by: where = prism angle and = angle of minimum deviation.

Step 2: Given condition. Here, .

Step 3: Simplify using identity. Wait carefully — correction! Actually: But from given options, the correct simplification matches (iii). % Corrected Answer Correct Answer: (iii)

To achieve the condition of minimum deviation in a prism, the following conditions must be met:
1. The incident ray and the refracted ray inside the prism must make equal angles with the prism surface. This means that the angle of incidence at the surface must be equal to the angle of refraction inside the prism.
2. The angle of deviation is minimized when the ray passes symmetrically through the prism. This implies that the path of the light through the prism should be symmetric in nature with respect to the prism's surface.
3. The angle of incidence at the silvered surface (the reflecting surface) must allow total internal reflection. The light should reflect off the silvered surface in such a way that the ray retraces its original path inside the prism. This occurs when the angle of incidence on the surface is greater than the critical angle for total internal reflection.
The relationship between the angles is given by: where: - is the angle of incidence on surface , - is the angle of refraction inside the prism, - is the refracting angle of the prism. Also, the angle of deviation is given by: For minimum deviation, the deviation is at its smallest value when the ray enters and exits the prism symmetrically. This means that the angle of incidence and the angle of refraction must be equal, i.e., . Thus, for minimum deviation: Now, considering the silvered surface at the other end of the prism, the ray must undergo total internal reflection. The critical angle for total internal reflection is given by: where is the refractive index of the prism. Therefore, the critical angle is: For total internal reflection to occur, the angle of incidence at the silvered surface must be greater than this critical angle. Thus, we set the angle of incidence such that the reflected ray retraces its path. The angle must satisfy: Thus, the angle of incidence on the surface is 45° for the reflected ray to retrace its path after total internal reflection at the silvered surface.