Determining the Image Location and Size Using a Concave Mirror

In this article, we will walk through the process of determining the location and size of the image produced by a concave mirror. We will use the mirror formula and magnification formula to find the required parameters, ensuring a comprehensive understanding of the phenomenon. This is an essential concept for any student of physics or optics.

Given Data

Let's consider the given data for a concave mirror:

Height of the object, (h_o) 2.0 cm Object distance, (d_o) -30.0 cm (negative sign as the object is in front of the mirror) Radius of curvature, (R) -20.0 cm (negative for concave mirrors)

Step 1: Finding the Focal Length

The focal length, (f), of a mirror is related to the radius of curvature (R) by the formula:

[ f frac{R}{2} ]

Substituting the given value of the radius of curvature:

[ f frac{-20.0, text{cm}}{2} -10.0, text{cm} ]

Step 2: Using the Mirror Formula to Determine Image Distance

The mirror formula is given by:

[ frac{1}{f} frac{1}{d_o} frac{1}{d_i} ]

Rearranging to solve for (d_i):

[ frac{1}{d_i} frac{1}{f} - frac{1}{d_o} ]

Substituting the known values:

[ frac{1}{d_i} frac{1}{-10.0, text{cm}} - frac{1}{-30.0, text{cm}} -0.100 - 0.0333 -0.0667 ]

Calculating (d_i):

[ d_i frac{1}{-0.0667} approx -15.0, text{cm} ]

This indicates that the image is located 15.0 cm in front of the mirror, forming a real image.

Step 3: Calculating the Magnification

The magnification, (m), is provided by:

[ m -frac{d_i}{d_o} frac{h_i}{h_o} ]

Substituting the known values:

[ m -frac{-15.0, text{cm}}{-30.0, text{cm}} frac{1}{2} 0.5 ]

Step 4: Finding the Height of the Image

Using the magnification to find the height of the image:

[ h_i m cdot h_o 0.5 times 2.0, text{cm} 1.0, text{cm} ]

Summary of Results

The image is located 15.0 cm in front of the mirror and has a height of 1.0 cm. Since the magnification is positive, the image is inverted.

Conclusion

Understanding the formation of images using a concave mirror involves using the mirror and magnification formulas. This process helps in comprehending the behavior of light and the formation of real images, which is crucial for various applications in optics and everyday life.

Additional Notes

The negative signs in the calculations indicate the direction of the image distance and magnification, according to the sign convention used in optics. These negative values confirm the image is real, inverted, and located in front of the mirror.