Refraction through Prism — Revision Notes

⚡ 30-Second Revision

Angle of Deviation — $\delta = (i_1 + i_2) - A$ \n- Prism Angle Relation: $A = r_1 + r_2$ \n- Minimum Deviation Conditions: $i_1 = i_2 = i$, $r_1 = r_2 = r = A/2$, ray parallel to base. \n- Refractive Index (Min. Dev.): $\mu = \frac{\sin((A + \delta_m)/2)}{\sin(A/2)}$ \n- Thin Prism Deviation: $\delta = (\mu - 1)A$ (for $A < 10^\circ$) \n- Snell's Law: $\mu_1 \sin \theta_1 = \mu_2 \sin \theta_2$ \n- Dispersion: Splitting of white light due to $\mu$ varying with wavelength (VIBGYOR).

2-Minute Revision

Refraction through a prism involves light bending twice, once upon entering and once upon exiting, always towards the base. The total bending is the angle of deviation, $\delta = (i_1 + i_2) - A$, where $i_1$ and $i_2$ are angles of incidence and emergence, and $A$ is the prism angle.

Internally, the angles of refraction $r_1$ and $r_2$ sum up to the prism angle: $A = r_1 + r_2$. A special case is minimum deviation ($\delta_m$), where the deviation is smallest, occurring when the light ray travels symmetrically ($i_1 = i_2$ and $r_1 = r_2 = A/2$).

This condition allows us to determine the prism's refractive index using $\mu = \frac{\sin((A + \delta_m)/2)}{\sin(A/2)}$. For thin prisms (small $A$), the deviation simplifies to $\delta = (\mu - 1)A$.

Prisms also cause dispersion, separating white light into colors because different wavelengths have slightly different refractive indices.

5-Minute Revision

Refraction through a prism is a key concept in optics. A prism is a transparent medium with two inclined refracting surfaces forming an angle $A$. When a light ray enters from air into the prism, it refracts towards the normal, and upon exiting, it refracts away from the normal.

The net effect is a deviation of the light ray towards the base of the prism. The total angle of deviation $\delta$ is given by $\delta = (i_1 + i_2) - A$, where $i_1$ is the angle of incidence and $i_2$ is the angle of emergence.

Inside the prism, the angles of refraction $r_1$ and $r_2$ are related to the prism angle by $A = r_1 + r_2$. \n\nThe most important scenario is minimum deviation ($\delta_m$). This occurs when the light ray passes symmetrically through the prism, meaning $i_1 = i_2$ and $r_1 = r_2$.

Under these conditions, the ray inside the prism is parallel to its base. From the geometric relations, we find $r_1 = r_2 = A/2$ and $i_1 = i_2 = (A + \delta_m)/2$. Applying Snell's Law ($\mu = \frac{\sin i}{\sin r}$) at the first surface yields the crucial formula for the refractive index of the prism material: $\mu = \frac{\sin((A + \delta_m)/2)}{\sin(A/2)}$.

\n\nThe graph of deviation ($\delta$) versus angle of incidence ($i$) is a U-shaped curve, confirming the existence of a unique minimum deviation. For thin prisms (where $A$ is small, typically less than $10^\circ$), the deviation formula simplifies to $\delta = (\mu - 1)A$.

\n\nPrisms are also known for dispersion, the phenomenon of splitting white light into its constituent colors (VIBGYOR). This happens because the refractive index of the prism material varies slightly with the wavelength of light ($\mu_{violet} > \mu_{red}$).

Consequently, violet light deviates the most, and red light deviates the least, leading to the formation of a spectrum. \n\nWorked Example: An equilateral prism ($A=60^\circ$) has a refractive index of $\sqrt{3}$.

Find the angle of minimum deviation. \nWe use $\mu = \frac{\sin((A + \delta_m)/2)}{\sin(A/2)}$. \n$\sqrt{3} = \frac{\sin((60^\circ + \delta_m)/2)}{\sin(60^\circ/2)}$ \n$\sqrt{3} = \frac{\sin((60^\circ + \delta_m)/2)}{\sin 30^\circ}$ \n$\sqrt{3} = \frac{\sin((60^\circ + \delta_m)/2)}{1/2}$ \n$\sin((60^\circ + \delta_m)/2) = \sqrt{3} \times 1/2 = \sqrt{3}/2$ \nSince $\sin 60^\circ = \sqrt{3}/2$, we have $(60^\circ + \delta_m)/2 = 60^\circ$.

\n$60^\circ + \delta_m = 120^\circ$. \n$\delta_m = 120^\circ - 60^\circ = 60^\circ$. \nThus, the angle of minimum deviation is $60^\circ$.

Prelims Revision Notes

1
Prism Geometry — A prism has a refracting angle $A$ between its two refracting surfaces. \n2. Ray Path: Light undergoes two refractions, bending towards the base. \n3. **Angle of Deviation ($\delta$)**: The total change in direction of the light ray. Formula: $\delta = (i_1 + i_2) - A$. \n4. Internal Angles Relation: The sum of internal angles of refraction equals the prism angle: $A = r_1 + r_2$. \n5. Snell's Law: Applied at each surface: $\mu_{air} \sin i_1 = \mu_{prism} \sin r_1$ and $\mu_{prism} \sin r_2 = \mu_{air} \sin i_2$. \n6. **Minimum Deviation ($\delta_m$)**: \n * Occurs when $i_1 = i_2$ and $r_1 = r_2$. \n * Ray inside prism is parallel to the base. \n * $r_1 = r_2 = A/2$. \n * $i_1 = i_2 = (A + \delta_m)/2$. \n7. **Refractive Index Formula (at $\delta_m$)**: $\mu = \frac{\sin((A + \delta_m)/2)}{\sin(A/2)}$. This is critical for numerical problems. \n8. Thin Prism Approximation: For small $A$ (typically $A < 10^\circ$), $\delta = (\mu - 1)A$. \n9. **$\delta$ vs. $i$ Graph: A U-shaped curve, showing a minimum deviation point. \n10. Dispersion**: White light splits into VIBGYOR because $\mu$ depends on wavelength. Violet deviates most, Red deviates least. \n11. Total Internal Reflection (TIR): Possible if the angle of incidence at an internal surface exceeds the critical angle. \n12. Key Values: Remember $\sin 30^\circ = 1/2$, $\sin 45^\circ = 1/\sqrt{2}$, $\sin 60^\circ = \sqrt{3}/2$.

Vyyuha Quick Recall

To remember the minimum deviation formula for refractive index: \nMy Sin Angle Divided By Sin Angle. \n$\mu = \frac{\sin((A + \delta_m)/2)}{\sin(A/2)}$ \n(M for Mu, Sin for Sine, A for Angle of Prism, D for Deviation, B for By, Sin for Sine, A for Angle of Prism. The '/2' is implicitly remembered as 'half' the sum/angle.)