A diverging lens with ƒ = -36.5 cm is placed 14.0 cm behind a converging lens with ƒ = 20.0cm. Where will an object at infinity be focused?

Verified step by step guidance

1. Step 1: Understand the problem. We are dealing with a system of two lenses: a converging lens (focal length ƒ₁ = 20.0 cm) and a diverging lens (focal length ƒ₂ = -36.5 cm). The diverging lens is placed 14.0 cm behind the converging lens. The object is at infinity, meaning the light rays entering the first lens are parallel.
2. Step 2: Calculate the image position formed by the first lens (converging lens). For a lens, the lens equation is given by: $\frac{1}{f}=\frac{1}{d}$_o + 1d_i. Since the object is at infinity, $d$_o = ∞, so $\frac{1}{d}$_o = 0. This simplifies the equation to: $\frac{1}{d}$_i = 1f. Solve for $d$_i to find the image position formed by the first lens.
3. Step 3: Determine the object distance for the second lens (diverging lens). The image formed by the first lens acts as the object for the second lens. The distance between the lenses is 14.0 cm, so the object distance for the second lens is: $d$_o^′ = 14.0 \, \(\text{cm}\) - \(\text{image position from the first lens}\). Be mindful of the sign convention: if the image from the first lens is on the same side as the diverging lens, the object distance for the second lens will be negative.
4. Step 4: Use the lens equation again for the second lens (diverging lens): $\frac{1}{f}=\frac{1}{d}$_o + 1d_i. Substitute the focal length of the diverging lens ($f=\; -36.5\; \backslash ,\; \backslash (\backslash text\{cm\}\backslash )$) and the object distance for the second lens ($d$_o^′) to solve for the final image position $d$_i^′.
5. Step 5: Interpret the result. The final image position $d$_i^′ will tell you where the object at infinity is focused relative to the diverging lens. If the value is positive, the image is on the opposite side of the diverging lens (real image). If negative, the image is on the same side as the diverging lens (virtual image).

Key Concepts

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens, expressed as 1/f = 1/v - 1/u. This formula is essential for determining where light rays converge or diverge after passing through a lens, allowing us to find the image position for given object distances.

Converging and Diverging Lenses

Converging lenses (convex) have a positive focal length and focus parallel rays of light to a point, while diverging lenses (concave) have a negative focal length and spread parallel rays outward. Understanding the behavior of these lenses is crucial for predicting how they will interact when placed in combination.

Ray Diagrams

Ray diagrams are graphical representations used to illustrate how light rays travel through lenses. By drawing the principal rays (the parallel ray, the focal ray, and the central ray), one can visually determine the location and nature of the image formed by the lens system, aiding in the analysis of complex lens arrangements.