Question

The magnification of a convex mirror is \(0.60 \times\) for an object \(2.0 \mathrm{~m}\) from the mirror. What is the focal length of this mirror?

Short answer

Answer: The focal length of the convex mirror is approximately 5 m.

Step-by-step solution

Write down the magnification formula

The magnification formula for mirrors is: \(m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\), where m is the magnification, \(h_i\) and \(h_o\) are the heights of the image and object respectively, and \(d_i\) and \(d_o\) are the distances of the image and object from the mirror respectively. Since we are given the magnification and the object distance, we can use this formula to calculate the image distance.

Calculate the image distance

Given the magnification \(m=0.60 \times\) and the object distance \(d_o = 2.0\ \mathrm{m}\), we can find the image distance \(d_i\) by rearranging the magnification formula: \(d_i = -\frac{d_o}{m}\) Plugging in the given values: \(d_i = -\frac{2.0\ \mathrm{m}}{0.60}\) \(d_i = -\frac{10}{3}\ \mathrm{m}\) The image distance is \(-3.33\ \mathrm{m}\). The negative sign indicates that the image is virtual, which is expected for a convex mirror.

Write down the mirror formula

The mirror formula relates the object distance, image distance, and focal length of a mirror: \(\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}\) We have found the image distance in Step 2. Now, we can use the mirror formula to calculate the focal length of the convex mirror.

Calculate the focal length

Substituting the given object distance \(d_o = 2.0\ \mathrm{m}\) and the calculated image distance \(d_i = -3.33\ \mathrm{m}\) into the mirror formula: \(\frac{1}{f}=\frac{1}{2.0\ \mathrm{m}}+\frac{1}{-3.33\ \mathrm{m}}\) Now, find the common denominator and simplify: \(\frac{1}{f} = \frac{-3.33+2.0}{2.0\cdot(-3.33)}\) \(\frac{1}{f} = \frac{-1.33}{-6.66}\) \(\frac{1}{f} = \frac{1.33}{6.66}\) Finally, find the focal length by taking the reciprocal of both sides: \(f = \frac{1}{\frac{1.33}{6.66}}\) \(f = \frac{6.66}{1.33}\) \(f \approx 5\ \mathrm{m}\) The focal length of the convex mirror is approximately \(5\ \mathrm{m}\).

Key concepts

Understanding the Magnification Formula

The magnification formula is a fundamental concept in optics that determines the ratio of the image size to the object size for mirrors and lenses. It is represented as:

\[\begin{equation}m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\right)\right),\end{equation}\]where \(m\) stands for magnification, \(h_i\) and \(h_o\) are the heights of the image and the object respectively, and \(d_i\) and \(d_o\) are the distances of the image and object from the mirror. For a convex mirror, the image formed is virtual and upright, hence \(m\) will be positive if we select the direction towards the mirror as negative.When solving problems involving magnification, it's important to keep track of the sign conventions. For convex mirrors, the image is always virtual and diminished, which means the image height \(h_i\) is less than the object height \(h_o\), and the image distance \(d_i\) is negative. This reflects in the magnification being less than 1 and also a positive value in this context, indicating a reduced size compared to the object.To find the focal length using the magnification given in an exercise, one must first ascertain the image distance using the magnification formula. With the image distance known, the mirror formula can then be used to determine the focal length. This step-by-step approach ensures a clear understanding of the relationship between object and image characteristics in reflective optics.

Deciphering the Mirror Formula

The mirror formula is an equation that relates the focal length (\(f\)), the object distance (\(d_o\)), and the image distance (\(d_i\)) of a spherical mirror:

\[\begin{equation}\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\right)\end{equation}\]This formula is incredibly useful in predicting where the image of an object will form in relation to a mirror and what the properties of this image will be. In the given problem, using the mirror formula requires a prior calculation of the image distance derived from the magnification formula.By inputting the known object distance and the calculated virtual image distance, the mirror formula enables us to compute the unknown quantity - the focal length of the convex mirror. Note that, for virtual images formed by convex mirrors, \(d_i\) will be negative according to the sign conventions used in optics. To solve the exercise, it's key to remember that convex mirrors always form the virtual and upright images behind the mirror (negative image distance), and the focal length for a convex mirror is also negative, indicating that the focal point is imaginary and exists behind the mirror.

Grasping Virtual Image Distance in a Convex Mirror

Virtual image distance, noted as \(d_i\), in the context of a convex mirror, holds a special meaning since convex mirrors always produce virtual images.A virtual image is formed when the outgoing rays from a point on the object diverge after reflection. For a viewer, these rays appear to be coming from a point behind the mirror, hence the term 'virtual'. The distance measured from the mirror along the direction of the reflected rays to this virtual point is the virtual image distance. In convex mirrors, the value of \(d_i\) is always negative because it forms on the same side as the object, which we treat as the negative side.In an exercise, once the magnification is known, we can determine the virtual image distance using the magnification formula. This calculated distance is essential for solving the rest of the problem, such as finding the focal length using the mirror formula. Remember that the negative sign of the virtual image distance reflects its direction relative to the mirror, not the physical nature of the image. Understanding this concept is vital since it ensures correct application of sign conventions when calculating the focal characteristics of spherical mirrors in optics.