Problem 22 - Physics

Question

A concave mirror \((f=45 \mathrm{~cm})\) produces an image whose distance from the mirror is onethird the object distance. Determine (a) the object distance and (b) the (positive) image distance.

Step-by-Step Solution

Answer

The object distance is 180 cm, and the image distance is 60 cm.

Step 1: Understanding the Problem

We are given a concave mirror with a focal length \( f = 45 \text{ cm} \). The image distance \( v \) is given as one third of the object distance \( u \). This can be written as \( v = \frac{u}{3} \). We are asked to find the object distance \( u \) and the image distance \( v \).

Step 2: Using Mirror Formula

The mirror formula is \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \). Since \( v = \frac{u}{3} \), substitute \( v \) into the mirror formula to get \[ \frac{1}{45} = \frac{1}{u} + \frac{3}{u} = \frac{4}{u} \].

Step 3: Solving for Object Distance \( u \)

Re-arrange the equation \( \frac{1}{45} = \frac{4}{u} \) to solve for \( u \). Multiply both sides by \( u \) and 45 to get \( u = 4 \times 45 \). Thus, \( u = 180 \text{ cm} \).

Step 4: Calculating Image Distance \( v \)

Since \( v = \frac{u}{3} \), substitute \( u = 180 \text{ cm} \) into the equation to find \( v \). Thus, \( v = \frac{180}{3} = 60 \text{ cm} \).

Key Concepts

Focal Length

Focal length is a key term when working with mirrors, particularly concave mirrors. It refers to the distance between the mirror's surface and its focal point, where parallel rays of light converge after reflecting. For a concave mirror, this is a positive value, and understanding it helps in calculations related to image formation.
In practical terms:

• A shorter focal length indicates that the mirror is more curved, focusing light more sharply.
• A longer focal length shows a gentler curve, indicating that light is less aggressively focused.

In the exercise, the concave mirror's focal length is given as 45 cm, which means it's quite curved, allowing for precise image formation calculations.

Mirror Formula

The mirror formula is an essential mathematical expression used in optics to relate the object distance, image distance, and focal length of spherical mirrors. It is given by \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]where

• \( f \) is the focal length,
• \( u \) is the object distance,
• and \( v \) is the image distance.

This equation helps determine unknown distances when any two are known. In our example, the use of this formula is crucial because it allows us to find the object and image distances from the given focal length.
When substituting values, ensure the correct formula manipulation to maintain balance on both sides.

Image Distance

Image distance refers to how far the image forms from the mirror itself. In a concave mirror, this can be positive or negative based on the image's nature, whether real or virtual. In our exercise, it is positive, indicating a real image formed on the same side as the light source.
Important insights:

• It is defined as the variable \( v \) in calculations.
• An image distance that is one-third of the object distance signifies a ratio relationship, which helps simplify calculations using the mirror formula.

With the given relationship \( v = \frac{u}{3} \), and knowing \( u \), we directly calculate \( v \) by substituting and rearranging the relationships as needed.

Object Distance

Object distance is the distance from the object to the mirror's surface and is denoted by \( u \). This is a crucial measurement as it affects how the image is formed by the mirror.
Key points to remember:

• It is always considered positive for real objects placed on the mirror's principal axis.
• Understanding how to determine this distance involves using the mirror formula effectively.

In this problem, the challenge was to calculate \( u \) using the mirror formula, where manipulating the given relationships (such as \( v = \frac{u}{3} \)) plays a vital role. Solving the equation \( \frac{1}{45} = \frac{4}{u} \) gives us the object distance of 180 cm, revealing how the mirror’s curvature impacts object placement.