Hey, guys. In this video, we're going to talk about refraction and image formation at spherical surfaces. We saw before when we talked about Snell's law how to apply refraction to a single ray of light across a flat boundary. Now we want to look at what happens when multiple rays of light come from an object, refract through a spherical boundary, and form an image inside of a different medium. Okay? Let's get to it.

A single ray of light passing through a transparent surface undergoes refraction. Alright? This is something we know. We know how to apply Snell's law to figure this out, but many rays of light will also undergo refraction together. An object placed in front of a surface, a transparent surface that allows transmission of light, will form an image based on the focal length that that surface has. Those images can be real or virtual based on the shape of the surface. So that surface can have a positive focal length or it can have a negative focal length. Alright.

The image distance equation for a spherical surface is this, where n₁ is the index of refraction that the object is in and n₂ is the index of refraction across the boundary that the light rays are passing into. Okay? Now there are some sign conventions that are important for this equation. First, for a convex surface, the radius is always considered positive when you plug it into this equation. For concave surfaces, the radius is considered negative. Okay? And just like before, just like we had for mirrors, if you calculate a positive image distance that is a real image that is inverted. And if you calculate a negative image distance, that's a virtual image that is upright. So this is exactly the same as it was for mirrors. Okay? Let's do an example.

An object in air is placed 5 centimeters in front of a transparent concave surface. If the radius of curvature is 7 centimeters and the refractive index behind the surface is 1.44, where is the image located? Is the image real or virtual? Okay. Because this is a single surface refraction, we want to use our equation for that. n₁o+n₂i=n₂-n₁r.

Now because this is a concave surface, the radius is going to be negative. Remember, that's one of our rules and it's important to remember the sign convention. Okay? What we have is our initial index of refraction, which our initial medium is air is 1. Our object distance we're told is 5 centimeters in front of the surface, so that's 5 centimeters. The index of refraction behind the boundary is 1.44. Our image distance we don't know and finally, our radius of curvature is -7 centimeters. It's important to remember that negative sign because if you don't, the answer is going to be completely off. It's not just going to be off by a sign. Okay?

So let's rearrange this equation to get n₂i=n₂-n₁r-n₁o and let's plug in those numbers. This is going to be 1.44-1 over negative 7 minus 1 over positive 5 which if you plug into your calculator equals negative 0.263. Okay?

So we have n₂i=-0.263. If I multiply the s I up and divide by negative 0.263, I get s I is in 2 over negative 0.263. The second refractive index is 1.44 and this answer is negative 5.5 centimeters. Okay? Very simple to just apply the equation even though the arithmetic can get a little bit hairy. Now is this image real or virtual? Don't forget the sign convention for images. If it's a negative image distance, it is virtual. The question didn't ask for it but the image is also upright because virtual images are always upright. Alright, guys. That wraps up our discussion on refraction at a single surface, a single spherical surface. Thanks for watching, guys.