Snell's Law — Revision Notes

⚡ 30-Second Revision

Snell's Law: — $n_1 \sin \theta_1 = n_2 \sin \theta_2$
Refractive Index: — $n = c/v$ (where $c$ is speed of light in vacuum, $v$ in medium)
Relative Refractive Index: — $n_{21} = n_2/n_1 = v_1/v_2$
Bending: — Rarer to denser ($n_1 < n_2$) $\implies$ bends towards normal ($\theta_2 < \theta_1$). Denser to rarer ($n_1 > n_2$) $\implies$ bends away from normal ($\theta_2 > \theta_1$).
Critical Angle (for TIR): — $\sin \theta_c = n_{\text{rarer}}/n_{\text{denser}}$ (occurs when light goes from denser to rarer medium).
Apparent Depth: — $n = \text{Real Depth} / \text{Apparent Depth}$ (for normal viewing).
Unchanged Property: — Frequency ($f$) remains constant during refraction.
Changed Properties: — Speed ($v$), Wavelength ($\lambda$), Direction (unless $\theta_1 = 0^circ$).

2-Minute Revision

Snell's Law is the quantitative description of refraction, the bending of light as it passes from one transparent medium to another due to a change in its speed. The law states $n_1 \sin \theta_1 = n_2 \sin \theta_2$, where $n_1$ and $n_2$ are the refractive indices of the first and second media, and $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, respectively, measured from the normal.

The refractive index ($n$) is defined as $c/v$, where $c$ is the speed of light in vacuum and $v$ is its speed in the medium. If light goes from a rarer to a denser medium ($n_1 < n_2$), it bends towards the normal ($\theta_2 < \theta_1$).

If it goes from denser to rarer ($n_1 > n_2$), it bends away from the normal ($\theta_2 > \theta_1$). A crucial related concept is the critical angle ($\theta_c$), where $\sin \theta_c = n_{\text{rarer}}/n_{\text{denser}}$.

If the angle of incidence exceeds $\theta_c$ when going from denser to rarer, Total Internal Reflection (TIR) occurs. Apparent depth is another application, where $n = \text{Real Depth} / \text{Apparent Depth}$.

Remember that only the frequency of light remains constant during refraction; speed, wavelength, and direction all change.

5-Minute Revision

Snell's Law is the cornerstone of understanding how light behaves at the interface of two different transparent media. It mathematically describes refraction, the phenomenon where light bends due to a change in its speed.

The law is expressed as $n_1 \sin \theta_1 = n_2 \sin \theta_2$, where $n_1$ and $n_2$ are the refractive indices of the first and second media, and $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, respectively, always measured with respect to the normal (perpendicular to the surface).

The refractive index ($n$) of a medium quantifies its optical density and is defined as the ratio of the speed of light in vacuum ($c$) to its speed in that medium ($v$), i.e., $n = c/v$. This means a higher refractive index implies a slower speed of light.

Key Rules for Bending:

1
Rarer to Denser ($n_1 < n_2$): — Light bends *towards* the normal ($\theta_2 < \theta_1$). Example: Air to water.
2
Denser to Rarer ($n_1 > n_2$): — Light bends *away* from the normal ($\theta_2 > \theta_1$). Example: Water to air.

Total Internal Reflection (TIR): This is a special case of refraction. When light travels from an optically denser medium to an optically rarer medium, if the angle of incidence ($\theta_1$) exceeds a certain value called the critical angle ($\theta_c$), no refraction occurs. Instead, all light is reflected back into the denser medium. The critical angle is found using Snell's Law: $n_1 \sin \theta_c = n_2 \sin 90^circ \implies \sin \theta_c = n_2/n_1 = n_{\text{rarer}}/n_{\text{denser}}$.

Apparent Depth: A common application of Snell's Law. When an object is submerged in a liquid and viewed from above, it appears shallower. The relationship is $n = \text{Real Depth} / \text{Apparent Depth}$, assuming normal viewing.

Properties of Light during Refraction:

Frequency ($f$): — Remains constant (determined by source).
Speed ($v$): — Changes ($v = c/n$).
Wavelength ($\lambda$): — Changes (since $v = f\lambda$ and $f$ is constant).
Direction: — Changes (bends), unless incident normally ($\theta_1 = 0^circ$).

Worked Example: A light ray enters a diamond ($n=2.42$) from air ($n=1$) at an angle of incidence of $30^circ$. Find the angle of refraction.

Given: $n_1 = 1$, $\theta_1 = 30^circ$, $n_2 = 2.42$.
Snell's Law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$
$1 \cdot \sin 30^circ = 2.42 \cdot \sin \theta_2$
$1 \cdot 0.5 = 2.42 \cdot \sin \theta_2$
$\sin \theta_2 = 0.5 / 2.42 \approx 0.2066$
$\theta_2 = \arcsin(0.2066) \approx 11.9^circ$. (Bends towards normal, as expected, since air to diamond is rarer to denser).

Prelims Revision Notes

Snell's Law is fundamental for NEET Ray Optics. Remember the formula $n_1 \sin \theta_1 = n_2 \sin \theta_2$. Angles $\theta_1$ and $\theta_2$ are always measured with respect to the normal. The refractive index $n$ is a dimensionless quantity, $n = c/v$, where $c$ is the speed of light in vacuum and $v$ is in the medium. Air's $n \approx 1$. Water's $n \approx 1.33$ (or $4/3$). Glass's $n \approx 1.5$.

Key Facts for Quick Recall:

Light bending towards normal: — Occurs when light enters an optically denser medium ($n_2 > n_1$, so $\theta_2 < \theta_1$).
Light bending away from normal: — Occurs when light enters an optically rarer medium ($n_2 < n_1$, so $\theta_2 > \theta_1$).
Normal incidence ($\theta_1 = 0^circ$): — No bending, $\theta_2 = 0^circ$, but speed and wavelength still change.
Properties of light during refraction: — Frequency ($f$) is constant. Speed ($v$) and Wavelength ($\lambda$) change. Direction changes (unless normal incidence).
Relative refractive index: — $n_{21} = n_2/n_1 = v_1/v_2 = \lambda_1/\lambda_2$.
Critical Angle ($\theta_c$): — For light going from denser ($n_1$) to rarer ($n_2$) medium, $\sin \theta_c = n_2/n_1$. If $\theta_1 > \theta_c$, Total Internal Reflection (TIR) occurs.
Apparent Depth: — $n = \text{Real Depth} / \text{Apparent Depth}$. This formula is for normal viewing. The apparent shift is $\text{Real Depth} (1 - 1/n)$.

Practice numerical problems involving these concepts. Be careful with trigonometric values and units. Always check for TIR possibility when light moves from denser to rarer medium.

Vyyuha Quick Recall

N-S-T-A: No Speed Through Air. (Meaning, frequency is constant, speed changes, direction changes, angles are from normal). Or, Snell's Law: Sin Light Angles With Normals. ($n_1 \sin \theta_1 = n_2 \sin \theta_2$)