Refraction and Thin Convex Lenses

Refraction and the Convex Lens

When light passes from air into glass, it slows down and changes direction at the surface. This bending is called refraction. How much a material slows light is measured by its index of refraction $n=c/v$, and the bending follows Snell's law: $n_1\sin\theta_1=n_2\sin\theta_2$. Going from air into glass, the ray bends toward the normal. A thin convex lens (also called a converging lens) is thicker in the middle than at the edges. Each ray bends toward the thick part, so rays that enter side by side are pulled together.

A beam of rays parallel to the principal axis (the line through the center of the lens, perpendicular to it) meets at one point after passing through the lens.

Why is it called "converging"? Because the rays come together. A lens with a shorter focal length bends light more strongly.

Principal Rays and Image Types

To find an image, you only need to draw two or three special rays from the top of the object. Where they meet is where the top of the image forms.

1. Parallel ray: travels parallel to the principal axis, then refracts through the far focal point.

2. Central ray: goes through the center of the lens and keeps going straight (a thin lens does not bend it).

3. Focal ray: passes through the near focal point, then refracts to travel parallel to the axis.

The image can be described with two pairs of words:

• Real image: light rays actually meet there. You can project it on a card or screen.

• Virtual image: rays only appear to come from there. It cannot be projected, and you see it by looking through the lens.

• Upright (same orientation as the object) or inverted (upside down).

The type of image depends on where the object is compared with $f$ and $2f$:

The Thin Lens Equation and Magnification

Drawings give the idea. Numbers give the exact answer. For a thin lens:

The signs tell you what kind of image you have:

• $f>0$ for a convex lens, and $d_o>0$ for a real object.

• $d_i>0$: real image (on the opposite side from the object).

• $d_i<0$: virtual image (on the same side as the object).

• $m<0$: inverted. $m>0$: upright. $\lvert m\rvert>1$ means enlarged; $\lvert m\rvert<1$ means reduced.

Why the minus sign in $m$? A real image ($d_i>0$) is always inverted, so a negative $m$ records that flip.

Putting It Together: Checking a Drawing and Moving the Object

A good habit: use the ray diagram first to predict the type of image, then use the equation to get numbers and check the drawing.

Now watch what happens as the object slides toward the lens. Far away, the image is small, real, and close to $F$. As the object approaches $2f$, the image grows and moves out. At $f$ the image runs off to infinity. Past that, the image jumps to the same side as the object and becomes virtual, upright, and enlarged.

The dashed lines show where the rays seem to come from. Your eye traces them back and sees a larger, upright image behind the object.

Back to the start: Jayden's candle was farther from the lens than $f$ (at $3f$), so the image was real, inverted, and could be caught on a card. Emma's coin was inside $f$, so the image was virtual, upright, and larger.

Summary