Problem 18 - Physics

Question

A concave mirror has a focal length of \(12 \mathrm{~cm}\). This mirror forms an image located \(36 \mathrm{~cm}\) in front of the mirror. What is the magnification of the mirror?

Step-by-Step Solution

Answer

The magnification of the mirror is 4.

Step 1: Understand the Mirror Equation

The mirror equation is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. For concave mirrors, if the image is formed in front of the mirror, \(d_i\) is negative. Here, \(f = 12 \mathrm{~cm}\) and \(d_i = -36 \mathrm{~cm}\).

Step 2: Solve for Object Distance

Rearrange the mirror equation to solve for \(d_o\):\[\frac{1}{d_o} = \frac{1}{f} - \frac{1}{d_i}\]Plug in the known values:\[\frac{1}{d_o} = \frac{1}{12} - \frac{1}{-36}\]Calculate \( \frac{1}{d_o} = \frac{1}{12} + \frac{1}{36} = \frac{3}{36} + \frac{1}{36} = \frac{4}{36} = \frac{1}{9} \).Thus, \(d_o = 9 \mathrm{~cm}\).

Step 3: Calculate Magnification

The magnification \(m\) of the mirror is given by the formula \(m = -\frac{d_i}{d_o}\). Substitute the values:\[m = -\frac{-36}{9}\]Calculate the magnification:\[m = \frac{36}{9} = 4\]

Step 4: Interpret the Magnification

The magnification of 4 means that the image is four times the size of the object and is upright (since the magnification is positive, resulting from the double negative in the calculation).

Key Concepts

The Mirror Equation

The mirror equation is a fundamental concept when dealing with mirrors, particularly concave ones. It is expressed as \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Here, \( f \) represents the focal length, \( d_o \) stands for the object distance, and \( d_i \) signifies the image distance. This equation helps us understand how objects and images relate to each other in mirror setups. When working with concave mirrors, especially if the image forms in front of the mirror, \( d_i \) will be negative. In our exercise, the focal length \( f \) is \( 12 \mathrm{~cm} \), and the image distance \( d_i \) is \( -36 \mathrm{~cm} \). Knowing these values enables us to rearrange the equation to find unknown quantities like object distance.

Focal Length

The focal length of a concave mirror is crucial because it determines the mirror's ability to converge light. Represented by \( f \), the focal length is half the radius of curvature of the mirror. This distance is where parallel rays of light meet after reflecting off the mirror's surface. In our example, the focal length is given as \( 12 \mathrm{~cm} \). This means that any object placed at this distance will have its rays converge at a point directly in front of the mirror, forming a focused image. The focal length is always positive for concave mirrors, indicating the convergence of light.

Magnification

Magnification tells us how much larger or smaller an image is compared to the actual object. It is calculated using the formula \( m = -\frac{d_i}{d_o} \). This ratio compares the size of the image to the object itself. In the given exercise, the magnification comes out as 4. This means the image is four times larger than the object. The negative sign in the formula indicates image orientation: a positive magnification means the image is upright, while a negative one would imply an inverted image. Here, the double negative results in a positive magnification of 4, confirming an upright image.

Image Distance

Image distance \( d_i \) is the distance from the mirror to the image formed. It gives insight into the position of the image in relation to the mirror. In the context of concave mirrors, if \( d_i \) is negative, the image is real, meaning it forms on the same side as the incoming light and can be projected onto a screen. In this exercise, the image is located \( 36 \mathrm{~cm} \) in front of the mirror, leading to a \( d_i \) of \( -36 \mathrm{~cm} \). Understanding image distance is essential for determining how mirrors will affect the size and orientation of the resulting image.