Two light beams fall on a transparent material block at point 1 and 2 with angle \( \theta_1 \) and \( \theta_2 \), respectively, as shown in the figure. After refraction, the beams intersect at point 3 which is exactly on the interface at the other end of the block. Given: the distance between 1 and 2, \( d = 4/3 \) cm and \( \theta_1 = \theta_2 = \cos^{-1} \frac{n_2}{2n_1} \), where \( n_2 \) is the refractive index of the block and \( n_1 \) is the refractive index of the outside medium, then the thickness of the block is cm. 
As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:
The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____. 
Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
Refraction is the bending of a wave when it enters a medium where its speed is different. The refraction of light when it passes from a fast medium to a slow medium bends the light ray toward the normal boundary between the two media. The amount of bending depends on the indices of refraction of the two media and is described quantitatively by Snell's Law.
If you ever observe a pencil dipped into water, you’ll notice that it seems to be tilted at an angle at the interface of air and water, or the bottom of a tub or a tank that contains water seems to be raised. This phenomenon is caused due to the process of refraction of light. Refraction of light is the bending of the light wave, passing from one medium to another, which is caused due to the difference in the density of the two mediums.
The main cause of refraction is the variation in the velocity of the light when it enters different mediums. The speed of light in the air is faster than that of water. So, the speed of the light increases when it travels from water to air, and similarly, the speed decreases when it travels from air to water.
In the below figure, it is shown why the printed alphabets appear to have risen when seen through a glass slab. This is because when the light travels from air to glass, the speed gets reduced and the light moves toward the normal, that is the light rays move towards the NN’ normal from its original path. Likewise, when the light ray travels from glass to air, its speed gets increased and it moves away from the normal.
There are two Laws of Refraction. They are: