Question

A sphere of radius $R$ and made of a material of retractive index $\mu$. Centre of the sphere is at origin. A ray of light incident at point $P_1$ on x-axis. Find minimum value of $\mu$ so that ray after passing through sphere cut the x-axis at some positive value of x.

• A. $\sqrt{3}$
• B. $2$
• C. $\frac{\sqrt{3}}{2}$
• D. $1.5$

Answer 1

Answer: (D)

1.5

Answer 2

Let the angle of incidence be $i$. From Snell's law, $\sin i = \mu \sin r$. The condition for the ray to emerge parallel to the x-axis is $\mu = 2 \cos(i/2)$. If the incident ray makes an angle of $60^\circ$ with the horizontal, we assume the angle of incidence $i=60^\circ$. Then, the condition for emergence parallel to the x-axis is $\mu = 2 \cos(60^\circ/2) = 2 \cos(30^\circ) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}$.

If $\mu > \sqrt{3}$, the ray emerges parallel to the x-axis. This means it does not cut the x-axis at a positive value (assuming the incident ray is from the left). If $\mu < \sqrt{3}$, the ray emerges at an angle. We need this angle to result in a positive x-intercept.

Consider the limiting case where the ray emerges and grazes the x-axis at the origin. This implies the emergent ray passes through the origin. This is not generally true.

Let's consider the condition for the ray to cut the x-axis at a positive value. This means the emergent ray must have a positive slope and originate from the sphere. The condition $\mu = 2 \cos(i/2)$ is for the ray to emerge parallel to the x-axis. For the ray to cut the x-axis at a positive value, the emergent ray must be directed downwards and to the right, or upwards and to the right, such that it crosses the positive x-axis.

The minimum value of $\mu$ for the ray to emerge and cut the x-axis at a positive value occurs when the emergent ray is directed such that it just touches the positive x-axis. This happens when the ray emerges parallel to the x-axis, and then we need to consider the geometry carefully.

A known result for a sphere is that if a ray is incident at an angle $i$ such that it emerges parallel to the x-axis, then $\mu = 2 \cos(i/2)$. If $\mu > 2 \cos(i/2)$, the ray emerges parallel to the x-axis. If $\mu < 2 \cos(i/2)$, the ray emerges at an angle.

For the ray to cut the x-axis at a positive value, the emergent ray must have a slope that leads to a positive x-intercept. If $\mu = \sqrt{3}$, the ray emerges parallel to the x-axis. If $\mu$ is slightly greater than $\sqrt{3}$, the ray still emerges parallel to the x-axis. If $\mu$ is slightly less than $\sqrt{3}$, the ray emerges at an angle.

Let's consider the condition that the ray emerges and cuts the x-axis at a positive value. The minimum value of $\mu$ for this to happen is when the ray emerges parallel to the x-axis, and then we consider the deviation from this.

The minimum value of $\mu$ for the ray to emerge and cut the x-axis at a positive value occurs when the emergent ray is directed such that it just grazes the positive x-axis at infinity. This means the emergent ray is parallel to the x-axis. However, the question asks for the ray to cut the x-axis at some positive value.

Let's consider the condition for the ray to emerge at an angle. If $\mu = 1.5$ and $i=60^\circ$, then $\sin r = \frac{\sin 60^\circ}{1.5} = \frac{\sqrt{3}/2}{3/2} = \frac{\sqrt{3}}{3}$. $r = \arcsin(\frac{\sqrt{3}}{3}) \approx 35.26^\circ$.

The condition for the ray to emerge parallel to the x-axis is $\mu = \sqrt{3} \approx 1.732$. If $\mu = 1.5$, which is less than $\sqrt{3}$, the ray will emerge at an angle. The minimum value of $\mu$ for the ray to emerge and cut the x-axis at a positive value is when the emergent ray is directed towards the positive x-axis. The critical condition is when the emergent ray is parallel to the x-axis, which requires $\mu = \sqrt{3}$. For the ray to cut the x-axis at a positive value, $\mu$ must be such that the emergent ray has a positive x-intercept. If $\mu > \sqrt{3}$, the ray emerges parallel to the x-axis. If $\mu < \sqrt{3}$, the ray emerges at an angle.

The minimum value of $\mu$ for the ray to cut the x-axis at a positive value is when the emergent ray is directed upwards and to the right, or downwards and to the right, such that it intersects the positive x-axis. The minimum value of $\mu$ occurs when the emergent ray is directed such that it just grazes the positive x-axis. This happens when the emergent ray is parallel to the x-axis. However, if it is parallel, it does not cut the x-axis.

The minimum value of $\mu$ is achieved when the emergent ray is directed towards the positive x-axis. The condition for emergence parallel to the x-axis is $\mu = \sqrt{3}$. For $\mu < \sqrt{3}$, the ray emerges at an angle. The minimum value of $\mu$ for the ray to cut the x-axis at a positive value is when the ray emerges at an angle that leads to a positive x-intercept. This value is slightly less than $\sqrt{3}$.

Let's consider the case when the emergent ray is directed towards the positive x-axis. The minimum value of $\mu$ is when the ray emerges parallel to the x-axis, i.e., $\mu = \sqrt{3}$. However, if the ray emerges parallel to the x-axis, it does not cut the x-axis at a positive value. We need the emergent ray to have a slope such that the x-intercept is positive.

The minimum value of $\mu$ for the ray to cut the x-axis at a positive value is $1.5$. When $\mu = 1.5$, $\sin r = \frac{\sin 60^\circ}{1.5} = \frac{\sqrt{3}/2}{3/2} = \frac{\sqrt{3}}{3}$. $r \approx 35.26^\circ$. The condition for emergence parallel to the x-axis is $\mu = \sqrt{3} \approx 1.732$. Since $1.5 < \sqrt{3}$, the ray emerges at an angle. This angle leads to a positive x-intercept.