Problem 38 - Physics

Question

A candle is placed \(15.0 \mathrm{~cm}\) in front of a convex mirror. When the convex mirror is replaced with a plane mirror, the image moves \(7.0 \mathrm{~cm}\) farther away from the mirror. Find the focal length of the convex mirror.

Step-by-Step Solution

Answer

The focal length of the convex mirror is approximately \(-17.14 \; \text{cm}\).

Step 1: Identify Given Values

We know the object distance for both mirrors is the same, \(d_o = 15.0 \; \text{cm}\). The image distance in a plane mirror is equal to the object distance, so \(d_{i \text{(plane)}} = -15.0 \; \text{cm}\). According to the problem, when replacing the convex mirror with the plane mirror, the image moves \(7.0 \; \text{cm}\) farther away, indicating \(d_{i \text{(convex)}} = -8.0 \; \text{cm}\).

Step 2: Use Mirror Equation

The mirror equation is given by:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]Substitute \(d_o = 15.0 \; \text{cm}\) and \(d_i = -8.0 \; \text{cm}\) for the convex mirror.

Step 3: Substitute and Solve

Substitute the known values into the mirror equation:\[ \frac{1}{f} = \frac{1}{15} + \frac{1}{-8} \]Calculate:\[ \frac{1}{f} = \frac{8 - 15}{120} = \frac{-7}{120} \]Thus, the focal length \(f\) is:\[ f = -\frac{120}{7} \approx -17.14 \; \text{cm} \]

Step 4: Determine Focal Length

The focal length of the convex mirror is \(-17.14 \; \text{cm}\). This negative sign confirms the nature of a convex mirror, where the focal length is on the same side as the object.

Key Concepts

Focal Length

The focal length of a mirror is a crucial concept in understanding how mirrors form images. In a convex mirror, the focal length is defined as the distance from the mirror's vertex to the focal point, where parallel rays of light appear to diverge from after reflecting off the mirror. It is important to note that for a convex mirror, the focal length is always negative.

This signifies that the focal point is located behind the mirror, a virtual location where reflected light rays meet when extended backward. Knowing how to calculate the focal length using the mirror equation is fundamental in determining how and where images will be formed.

Mirror Equation

The mirror equation is a valuable tool for describing the relationship between object distance, image distance, and focal length in mirrors. It is given by the formula:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]This equation applies to both concave and convex mirrors, with caution needed regarding the sign conventions:

• For convex mirrors, the focal length \(f\) and the image distance \(d_i\) are considered negative since the focal point and image form behind the mirror (a virtual position).
• The object distance \(d_o\) is typically positive as the object is placed in front of the mirror.

By substituting known values into this equation, one can find the missing variable, such as the focal length in this exercise.

Image Distance

The image distance \(d_i\) in the context of mirrors is the distance between the image and the mirror surface. In our exercise, two different mirrors are involved—convex and plane.

For a plane mirror, the image distance \(d_i\) is exactly the same magnitude as the object distance \(d_o\), but it is negative since images formed are virtual and appear behind the mirror.

With the convex mirror, the image distance is calculated to be \(-8.0 \, \text{cm}\). The negative sign indicates a virtual image that is upright and appears smaller than the object itself. This distinction helps in understanding how images vary between different types of mirrors.

Object Distance

Object distance \(d_o\) is the straightforward distance from the object to the mirror's surface, which in our context remains consistent at \(15.0 \, \text{cm}\) for both convex and plane mirrors.

This parameter helps form the basis of calculations for determining how images are created, relying on the precise application of the mirror equation.

• A positive object distance indicates that the object is located in front of the mirror.
• Knowing the object distance ensures a correct substitution in the mirror equation to find other unknowns like image distance or focal length.

Understanding object distance in congruence with focal length and image distance is vital to analyze how different mirrors behave in forming images.