Two identical concave mirrors are set facing each other 1.0 m apart. A small lightbulb is placed halfway between the mirrors. A small piece of paper placed just to the left of the bulb prevents light from the bulb from directly shining on the left mirror, but light reflected from the right mirror still reaches the left mirror. A good image of the bulb appears on the left side of the piece of paper. What is the focal length of the mirrors?

Verified step by step guidance

1. Step 1: Understand the problem setup. Two identical concave mirrors are placed facing each other, 1.0 m apart, with a lightbulb positioned halfway between them. The lightbulb emits light, but a piece of paper blocks direct light from reaching the left mirror. However, light reflected from the right mirror reaches the left mirror, forming an image of the bulb on the left side of the paper.
2. Step 2: Recall the mirror equation: $\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$, where $f$ is the focal length, $u$ is the object distance, and $v$ is the image distance. For concave mirrors, the focal length is positive.
3. Step 3: Analyze the light path. The lightbulb is placed halfway between the mirrors, so its distance from each mirror is 0.5 m. Light from the bulb reflects off the right mirror, forming an image that acts as the object for the left mirror. The distance between the mirrors (1.0 m) helps determine the object and image distances for the left mirror.
4. Step 4: For the left mirror, the object distance $u$ is the distance from the image formed by the right mirror to the left mirror. Since the bulb is 0.5 m from the right mirror, the image formed by the right mirror is also 0.5 m away from it, making the object distance for the left mirror 0.5 m.
5. Step 5: Use the mirror equation for the left mirror. Substitute $u$ = 0.5 m and $v$ = 0.5 m (since the image appears on the left side of the paper, at the same distance as the bulb). Solve for the focal length $f$ using the equation $\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$.

Key Concepts

Concave Mirrors

Concave mirrors are curved mirrors that bulge inward, causing parallel incoming light rays to converge at a focal point. The focal length is the distance from the mirror's surface to this focal point, which is positive for concave mirrors. Understanding how concave mirrors reflect light is crucial for analyzing image formation and determining the focal length in this scenario.

Image Formation

Image formation in mirrors involves the interaction of light rays with the mirror's surface. For concave mirrors, the position and nature of the image (real or virtual) depend on the object's distance from the mirror and the mirror's focal length. In this case, the lightbulb's position relative to the mirrors is key to understanding how the image appears on the left side of the paper.

Mirror Equation

The mirror equation relates the object distance (d_o), image distance (d_i), and focal length (f) of a mirror: 1/f = 1/d_o + 1/d_i. This equation is essential for calculating the focal length of the mirrors based on the distances involved in the setup. By applying this equation, one can derive the focal length from the given distances in the problem.