Maximum Angle for Total Internal Reflection in Block

 

❓ Question

A transparent block A having refractive index $\mu = 1.25$ is surrounded by another medium of refractive index $\mu = 1.0$ (air). A light ray is incident on the flat vertical face of the block with incident angle $\theta$ (in air) as shown in the figure. What is the maximum value of $\theta$ for which the light suffers total internal reflection at the top horizontal surface of the block?

(Assume the top surface is horizontal and the vertical face is flat — standard rectangular block geometry.)

---

🖼️ Question Image

---

✍️ Short Explanation

This problem is based on:

👉 Snell’s law
👉 Critical angle
👉 Total internal reflection (TIR).

Main idea:

For TIR:

$$\boxed{ i \ge C }$$

where critical angle satisfies:

$$\sin C=\frac{\mu_1}{\mu_2}$$

---

🔷 Step 1 — Find Critical Angle 💯

Given:

$$\mu_2=1.25=\frac54$$ $$\mu_1=1$$

Critical angle:

$$\sin C=\frac{\mu_1}{\mu_2}$$
$$=\frac{1}{5/4}$$
$$=\frac45$$

Thus:

$$\boxed{ C=\sin^{-1}\left(\frac45\right) }$$

---

🔷 Step 2 — Relation Between Internal Angles

Let refracted angle inside block at first face be $r$.

At top surface, angle of incidence becomes:

$$90^\circ-r$$

For just TIR:

$$90^\circ-r=C$$
$$r=90^\circ-C$$

Using:

$$\sin C=\frac45$$

triangle gives:

$$\cos C=\frac35$$

Hence:

$$\sin r=\sin(90^\circ-C)$$
$$=\cos C$$
$$=\frac35$$

---

🔷 Step 3 — Apply Snell’s Law at Entry

At first surface:

$$\mu_1\sin\theta=\mu_2\sin r$$
$$1\cdot\sin\theta = \frac54\cdot\frac35$$
$$=\frac34$$

Thus:

$$\boxed{ \sin\theta=\frac34 }$$

Therefore:

$$\boxed{ \theta=\sin^{-1}\left(\frac34\right) }$$

---

🔷 Step 4 — JEE Trap Alert 🚨

❌ Directly critical angle ko answer maan lena

❌ Top surface angle $=r$ assume kar lena

❌ Complementary angle relation bhool jaana

Remember:

$$\boxed{ \text{Top incidence angle}=90^\circ-r }$$

---

✅ Final Answer

$$\boxed{ \sin^{-1}\left(\frac34\right) }$$

(Option 2)

---

📚 Related Topics

• Find Height Using Refraction in Layered Media
• Emissivity of Wire Using Stefan–Boltzmann Law
• Coefficient of Friction on 60° Incline