Problem 16 - Physics

Question

A mirror produces an image that is located \(34.0 \mathrm{~cm}\) behind the mirror when the object is located \(7.50 \mathrm{~cm}\) in front of the mirror. What is the focal length of the mirror, and is the mirror concave or convex?

Step-by-Step Solution

Answer

The focal length is approximately 9.62 cm, and the mirror is convex.

Step 1: Understanding the Problem

We need to find the focal length of a mirror given the positions of the object and image. The object is located at a distance of \(d_o = 7.50 \text{ cm}\) in front of the mirror, and the image position is \(d_i = -34.0 \text{ cm}\) because it is formed behind the mirror.

Step 2: Using the Mirror Equation

The mirror equation is \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\). We will use this equation to find the focal length \(f\). Here, plug \(d_o = 7.50 \text{ cm}\) and \(d_i = -34.0 \text{ cm}\) into the equation.

Step 3: Calculating the Inverses

Calculate \(\frac{1}{d_o} = \frac{1}{7.50}\) and \(\frac{1}{d_i} = \frac{1}{-34.0}\). This gives \(\frac{1}{d_o} = 0.1333\) and \(\frac{1}{d_i} = -0.0294\).

Step 4: Finding the Focal Length

Substitute the values into the mirror equation: \(\frac{1}{f} = 0.1333 - 0.0294\), which simplifies to \(\frac{1}{f} = 0.104\). Therefore, \(f = \frac{1}{0.104}\approx 9.62 \text{ cm}\).

Step 5: Determining the Type of Mirror

Since the image is behind the mirror (a negative image distance) and the calculated focal length is positive, the mirror is a convex mirror.

Key Concepts

Focal Length

The term **focal length** refers to the distance between the center of a mirror and its focal point. This point is where parallel beams of light either converge or appear to diverge after reflecting off the mirror. Focal length is a crucial concept in understanding how mirrors form images.

In the context of mirrors, the focal length varies based on the mirror's shape. It's denoted by the symbol \(f\) in the mirror equation, which is:

• \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\)

Here, \(d_o\) is the distance of the object from the mirror, while \(d_i\) is the distance of the image. Understanding these distances helps us determine the focal length. Whether a mirror is concave or convex influences the calculation.

• Concave mirrors can have positive or negative focal lengths depending on the image position.
• Convex mirrors, like the one we calculated in the original exercise, typically yield a negative image distance \(d_i\) but results in a positive focal length \(f\).

Knowing how to correctly apply these values in the mirror equation allows one to accurately determine the mirror's focal length and type.

Concave Mirror

A **concave mirror** is a mirror that curves inwardly, similar to the inside of a bowl. This type of mirror reflects light inwards towards a single focal point.

Key characteristics of a concave mirror include:

• The ability to produce real, inverted images when an object is placed outside its focal point.
• When the object is within the focal length, it can produce virtual, upright images that appear larger.

Unlike convex mirrors, concave mirrors have a focal point positioned on the same side as the object. Thus, these mirrors often have a positive focal length.
Concave mirrors are frequently used in situations where light needs to be concentrated, such as in flashlights, solar ovens, and even car headlights.

Convex Mirror

A **convex mirror** is outwardly curved, resembling the exterior of a sphere. It makes reflected light rays spread out, creating a virtual image perceived as coming from a point behind the mirror.

Convex mirrors possess distinct properties:

• They produce images that are always virtual, smaller than the object, and upright.
• The focal point and center of curvature appear on the opposite side of the mirror from the object.
• The focal length is considered positive even though the image distance is negative.

This type of mirror finds use in various applications where a wider field of view is necessary, such as in vehicle side mirrors or security mirrors in stores. Understanding their characteristics is crucial for predicting where an object’s image will form, as seen in the solved exercise, where the positive focal length indicated a convex mirror.