# Question Video: Determining the Position of a Focus for a Concave Lens Science

The diagram shows a thin concave lens. The lens is symmetrical. The point marked 𝑃 is one of the foci of the lens. Using the grid, determine which of the points (I, II, III, or IV) is the other focus of the lens.

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### Video Transcript

The diagram shows a thin concave lens. The lens is symmetrical. The point marked 𝑃 is one of the foci of the lens. Using the grid, determine which of the points — I, II, III, or IV — is the other focus of the lens.

In this question, we are shown a diagram with a concave lens on a grid, with one of the lens’s foci labeled 𝑃 and four other points. We are asked to figure out which of the four points is the other focus of this concave lens.

Now, remember that the shape of a concave lens can be created by overlapping a cylinder and two spheres, like shown. If the overlapped material from the cylinder is removed, we are left with the shape of a concave lens. If we consider a two-dimensional cross section of a concave lens, it looks like the shape made by removing two equal parts of a circle from opposite sides of a rectangle. The shape that is produced is thicker along the edges and thinnest in the middle. This shape causes concave lenses to act as diverging lenses. This means that light rays traveling through a concave lens will change directions to diverge or spread out from each other.

The centers of the circles that we saw to determine the shape of the cross section of a concave lens are the centers of curvature of the lens. If we connect the centers of curvature with a line, the line passes through the center of the lens. This line is called the optical axis. If a ray of light traveled along this line through the lens, it would not change its direction as it passed through the lens.

Let’s see what happens when a few rays of light traveling parallel to the optical axis enter the lens. These outer rays change direction. The rays will appear to diverge from a point on the side of the lens from which they enter the lens. If we trace the path of these rays after they exit the lens back through the lens, we find that the extended paths do all converge at the same point. This point is called the focal point of the lens. A concave lens is symmetrical, so it has two focal points, one on either side of the lens, each equal distances from the center of the lens.

So now we know how to find the focal point of a concave lens. Let’s go back to the diagram given in the question and use our point 𝑃 to draw some rays. We can pick two points along the vertical axis of the lens and draw lines from the focal point 𝑃 through these points. Note that these aren’t light rays we’re drawing, but at this point just dashed lines. Let us suppose that two light rays were originally parallel to the optical axis of the lens when they entered the lens at these two points we’ve picked. The lines on the opposite side to point 𝑃 are showing us the paths that these light rays would travel after entering the lens. We can measure the horizontal distance from point 𝑃 to the center of the lens, which happens to be three squares on this grid.

Because concave lenses are symmetrical lenses, the focal length for the two foci of the lens will be the same. So we can measure three squares from the center of the lens to the right. We can see that point II is three squares away from the center of the lens. So this must be our second focal point. Let’s draw lines from point II through the points we made on the vertical axis of the lens. Because this lens is symmetrical, these lines should be reversed copies of the lines from point 𝑃. And this is exactly what we see when we draw them. Therefore, point II is the second focal point of the lens. Point II is the correct answer.