Question

You hold a spherical salad bowl 60 cm in front of your face with the bottom of the bowl facing you. The bowl is made of polished metal with a 35 cm radius of curvature. (a) Where is the image of your 5.0 cm tall nose located? (b) What are the image's size, orientation, and nature (real or virtual)?

Short answer

The image is 24.7 cm from the bowl, 2.1 cm tall, inverted, and real.

Step-by-step solution

Understanding the Problem

The problem involves calculating the image distance and characteristics for an object placed in front of a concave mirror. The mirror has a radius of curvature of 35 cm, meaning the focal length \( f \) is half of that, which is \( 17.5 \) cm. The object distance \( d_o \) is 60 cm from the mirror.

Determine the Focal Length

The focal length \( f \) is calculated using the formula \( f = \frac{R}{2} \), where \( R \) is the radius of curvature. Substituting the given value, \( f = \frac{35}{2} = 17.5 \) cm.

Use the Mirror Equation to Find the Image Distance

The mirror equation relates the object distance \( d_o \), image distance \( d_i \), and focal length \( f \):\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]Substitute \( f = 17.5 \) cm and \( d_o = 60 \) cm:\[\frac{1}{17.5} = \frac{1}{60} + \frac{1}{d_i}\]Solve for \( \frac{1}{d_i} \):\[\frac{1}{d_i} = \frac{1}{17.5} - \frac{1}{60} = \frac{60 - 17.5}{17.5 \times 60}\]\[\frac{1}{d_i} = \frac{42.5}{1050}\]\[d_i \approx 24.7 \text{ cm (rounded to one decimal place)}\].

Calculate the Magnification and Determine Image Characteristics

The magnification \( m \) is given by the formula:\[m = -\frac{d_i}{d_o}\]Substituting the values:\[m = -\frac{24.7}{60} \approx -0.412\]The image height \( h_i \) is calculated by multiplying the magnification by the object height \( h_o \):\[h_i = m \times h_o = -0.412 \times 5 \approx -2.1 \text{ cm}\].The negative sign indicates the image is inverted relative to the object.

Determine the Nature of the Image

Since the image distance \( d_i \) is positive, the image is real and inverted. Its size is smaller than the object, demonstrated by the magnification \( m \approx -0.412 \).

Key concepts

Radius of Curvature

The radius of curvature is a fundamental concept when dealing with concave mirrors. It is denoted by the symbol \( R \) and refers to the radius of the sphere from which the mirror is a part. Understanding this helps in determining how the mirror converges light.
In the case of the salad bowl problem, the bowl acts as a concave mirror with a radius of curvature of 35 cm. This is an important parameter as it helps define the focal point of the mirror.
The radius of curvature is directly related to the focal length, which is critical for further image calculations.

• Key takeaway: The larger the radius of curvature, the less curved the mirror's surface.
• This concept is crucial for understanding how sharply the mirror focuses light and how it affects image formation.

Focal Length Calculation

Focal length is a crucial measurement used to determine how a concave mirror focuses light. It is derived from the radius of curvature and is given by the formula:\[ f = \frac{R}{2} \]This formula tells us that the focal length is half the radius of curvature.
For the polished metal bowl in our problem, with a radius of curvature of 35 cm, the focal length becomes 17.5 cm. This focal length is used in calculations involving the mirror equation.
By understanding this, you can better visualize where the focus of light would be, which aids in predicting how the image characteristics will appear.

• A shorter focal length indicates a mirror focusing light tightly, producing a more pronounced image.
• Focal length helps predict convergence or divergence of light rays, influencing the image properties.

Mirror Equation

The mirror equation is a pivotal tool used to find image properties in relation to object properties and the nature of the mirror. It is expressed as:\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]Here, \( f \) denotes the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. This equation enables us to calculate unknowns when two of the three variables are known.
In our case study, the focal length \( f \) is 17.5 cm and the object distance \( d_o \) is 60 cm. Solving the equation allows us to find the image distance \( d_i \) as approximately 24.7 cm.
Understanding this equation means you can correctly determine how light interacts with the mirror.

• This fundamental tool assists in predicting the position and nature of the image.
• Using the equation carefully ensures accuracy in optical calculations.

Image Distance

Image distance \( d_i \) represents how far the image is formed by the concave mirror from its vertex. This distance is crucial for understanding image characteristics such as size, orientation, and type (real or virtual).
It is determined using the mirror equation. In the exercise, we calculated \( d_i \) to be approximately 24.7 cm.
Here's why it's important:

• A positive image distance signifies a real image formed on the same side as the object. It means light rays are actually converging.
• A negative image distance would indicate a virtual image, where rays appear to diverge from a common point behind the mirror.

This understanding provides insight into the practical applications and behavior of concave mirrors in optics.