Understanding why the refractive index equals sin(i) over sin(r) and how Snell's Law works

Light loves to surprise us at boundaries. A beam moving through air hits a glass pane, and suddenly it changes direction. It’s not magic—just a bit of physics at work. That change in direction is governed by a quiet little number called the refractive index, usually denoted by μ (mu). For students eyeing NEET Physics, refractive index often shows up in questions, diagrams, and, yes, quick-fire checks like the one we’re about to unpack.

Let’s zoom in on the heart of the idea: the formula μ = sin(i) / sin(r).

What is μ, really?

Think of light as a speedy traveler. In a vacuum, light zips along at the speed c. In any other medium—air, water, glass—its speed drops a bit. The refractive index μ of a medium is defined as the ratio of light’s speed in a vacuum to its speed in that medium:

μ = c / v, where v is the speed in the medium.

That statement is a handy baseline. But when light meets a boundary between two media, a different but closely related relation comes into play: Snell’s Law. It connects the angles the light ray makes with the boundary to the refractive indices of the two media. Mathematically, Snell’s Law is written as:

n1 sin(i) = n2 sin(r),

where:

• i is the angle of incidence (the angle the incoming ray makes with the normal to the boundary),

• r is the angle of refraction (the angle the transmitted ray makes with the normal),

• n1 is the refractive index of the first medium,

• n2 is the refractive index of the second medium.

From this, you can rearrange to see a clean relationship between the two angles:

sin(i) / sin(r) = n2 / n1.

This ratio, n2/n1, is what many call the relative refractive index. If you’re comparing the second medium to air (where n1 is about 1), sin(i) / sin(r) roughly equals the refractive index of the second medium itself. In some contexts, especially when the first medium is air or vacuum, people simply write μ = sin(i) / sin(r). That’s the exact form in the multiple-choice you’re looking at.

Why the right answer is B

Let’s map the options to Snell’s Law:

• A. μ = tan(i) / tan(r) — not a standard relation in Snell’s Law. The tangents don’t line up with how speeds and angles relate across a boundary.

• B. μ = sin(i) / sin(r) — this is the correct expression for the relative refractive index of the second medium with respect to the first, derived directly from Snell’s Law.

• C. μ = sin(r) / sin(i) — this is just the reciprocal of B. It would correspond to n1/n2, not the usual μ that many contexts use for the second medium’s index with respect to the first.

• D. μ = cos(i) / cos(r) — again, not part of Snell’s Law. The cosine appears in some geometry stories, but it isn’t the link here.

A practical way to remember it

Here’s a quick mental model: Snell’s Law says the “n-weighted” sine must stay the same across the boundary. If light slows down a lot upon entering the new medium (a larger μ), the angle of refraction becomes smaller—the ray bends toward the normal. If it speeds up (lower μ), the ray bends away from the normal. The sine ratio captures that balance.

A real-world intuition

Imagine you’re walking from a crowded room into a hallway with a different vibe. Your pace changes, and your path tilts relative to the boundary as you head through. Light behaves similarly. The more different the medium’s optical density is from the one you came from, the more pronounced the bending. Water in air, glass in air, or a diamond in air—each boundary nudges the ray in its own distinctive way.

Connecting speed, index, and direction

Let’s tie the pieces together with a simple chain:

• μ = c / v, the speed-based definition of refractive index for a medium.

• Snell’s Law in the two-media form: n1 sin(i) = n2 sin(r).

• Rearranging gives sin(i) / sin(r) = n2 / n1.

• If you take the first medium as air (n1 ≈ 1) or vacuum, then sin(i) / sin(r) ≈ n2, which is the refractive index of the second medium. In contexts where μ is defined as that ratio, μ = sin(i) / sin(r) becomes a handy shorthand.

A quick worked thought experiment

Suppose light goes from air (n1 ≈ 1) into water (n2 ≈ 1.33). If the ray hits the boundary at some incidence angle i, Snell’s Law tells us:

sin(i) = 1.33 sin(r).

Rearranged, sin(i) / sin(r) = 1.33.

That ratio is the refractive index of water with respect to air. If you know i and r from a diagram or measurement, you can verify this relationship in a single glance.

The other big idea that pairs with this is total internal reflection. When light travels from a denser medium to a rarer one and i gets large enough, sin(r) would exceed 1 if you followed the sine rule, which is impossible. The ray instead reflects entirely back into the dense medium. That’s not just a trick of textbooks; it’s the principle behind optical fibers and many sensing devices. It all folds neatly from the same Snell’s Law backbone and the sin(i)/sin(r) ratio.

A few practical tips that help

• Always label your mediums. n1 and n2 aren’t just numbers; they’re about the path light takes. If you switch the order (going from medium 2 into medium 1), you’ll flip the ratio.

• When the first medium is air, μ often boils down to the refractive index of the second medium. But if you’re ever unsure, fall back on Snell’s Law in its full form and solve for the ratio you need.

• A quick sanity check: if i is large and r becomes small, sin(i) / sin(r) climbs. That’s exactly what happens when light moves into a much denser medium; it slows and bends more toward the normal.

A gentle detour into the visuals

If you’ve ever drawn a diagram with light striking a boundary, you’ve probably sketched in the “normal” — a line perpendicular to the boundary. It’s a tiny line, but it’s the anchor for i and r. The bigger the mismatch in the media’s optical density, the sharper the bend. That visual cue is often more memorable than the formula alone. And because you’re likely studying optics for its conceptual beauty as well as its numerical checks, a good diagram is half the work.

A few extra notes that fit naturally in the journey

• The speed-angle relationship is symmetric in a sense: going from denser to rarer media makes the ray tilt away from the normal; going the other way makes it tilt toward the normal.

• The cosine terms you sometimes see in other geometric setups aren’t the governing factors here. For refraction at a boundary, sines do the heavy lifting because they tie directly to the wavefront’s geometry and the speed ratio.

• When you push the idea a bit further, you land at the critical angle for total internal reflection. That’s a fascinating corner of the topic: beyond it, refraction vanishes and the light stays inside the original medium. It’s a neat reminder that even a simple boundary hides rich physics.

Bringing it home with a compact takeaway

• The correct relation for the refractive index across a boundary is μ = sin(i) / sin(r). This comes straight from Snell’s Law, which is the compass for how light bends at interfaces.

• The other options—tan(i)/tan(r), sin(r)/sin(i), cos(i)/cos(r)—don’t capture the boundary’s true link between angles and speeds.

• Understanding μ in terms of speeds and angles gives you a flexible handle: it lets you predict how light will behave when it enters new media and how devices like lenses and fibers actually work.

A closing thought

Light isn’t just something that travels; it negotiates its path. That negotiation—the bend at the boundary, the slowing down, the change in direction—is all encoded in a simple ratio: the sines of the incidence and refraction angles. When you see μ = sin(i) / sin(r), you’re peeking at the glue that holds the picture together. It’s a small formula with big implications, and it sits at the heart of a lot of everyday optical phenomena.

If you’re curious to see it in action, grab a glass of water and a pencil, shine a beam at the water’s surface, and sketch the incoming and refracted rays. You’ll spot i and r, and you’ll feel how the boundary nudges light along its journey. That moment of visual confirmation—the bend, the shift, the math all lining up—says this: physics isn’t just theory. It’s a practical language for understanding the world, one boundary at a time.