Problem 12 - University Physics with Modern Physics

Question

For a concave spherical mirror that has focal length \(f\) = \(+\)18.0 cm, what is the distance of an object from the mirror's vertex if the image is real and has the same height as the object?

Step-by-Step Solution

Answer

The object is 36.0 cm from the mirror's vertex.

Step 1: Understand the Mirror Equation

The mirror equation relates the object distance \(d_o\), the image distance \(d_i\), and the focal length \(f\) of the mirror. It is given by: \[\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}.\] In this problem, the focal length \(f\) is given as \(+18.0\) cm.

Step 2: Analyze the Given Conditions

The problem states that the image is real and has the same height as the object. For a real image with the same height, the magnification \(m\), which is \(\frac{h_i}{h_o} = \frac{-d_i}{d_o}\), must be \(-1\). Hence, \(d_i = -d_o\).

Step 3: Substitute and Solve

Substitute \(d_i = -d_o\) into the mirror equation: \[\frac{1}{d_o} + \frac{1}{-d_o} = \frac{1}{18.0}.\] Simplify and solve: \[\frac{1}{d_o} - \frac{1}{d_o} = \frac{1}{18.0}\rightarrow 0 = \frac{1}{18.0}.\] However, this approach indicates a mistake in substitution. Instead, solve keeping \(d_i = d_o\): \[2\left(\frac{1}{d_o}\right) = \frac{1}{18.0}.\] Thus, \[\frac{2}{d_o} = \frac{1}{18.0}.\] Solving for \(d_o\), we find \[d_o = 36.0 \text{ cm}.\]

Step 4: Confirm the Solution

Verify the solution by checking if the conditions are satisfied. Substituting \(d_o = 36.0\) cm back gives \(d_i = 36.0\) cm, and using the magnification \(m = \frac{h_i}{h_o} = \frac{-d_i}{d_o}\), confirming \(-1\). Therefore, the solution is consistent.

Key Concepts

Mirror Equation

The mirror equation is a fundamental formula used to relate the key elements of a concave mirror's setup. This equation links the object distance ( d_o ), the image distance ( d_i ), and the focal length ( f ) of the mirror. It is expressed as:

• \( \frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f} \)

This equation is crucial because it helps determine how an image will form based on where an object is placed relative to the mirror. In the context of concave mirrors, a positive focal length value indicates that the mirror converges light rays, allowing for real image formation. By substituting the known values into the mirror equation, we can solve for unknown distances, providing a mathematical basis to predict image placement.

Focal Length

The focal length ( f ) is a critical characteristic of any spherical mirror, particularly concave ones. It is the distance from the mirror's surface to its focal point, where parallel rays of light converge. The focal length's sign is vital:

• A positive focal length, as in this exercise with f = +18.0 cm, indicates a concave mirror that focuses light to a point.
• A negative focal length would indicate a convex mirror, which diverges light rays.

Understanding the focal length is essential for predicting how mirrors affect light pathways and image projection. In this problem, the positive focal length establishes that the mirror can produce a real image, emphasizing its role in shaping light behavior.

Real Image

A real image is one that can be projected onto a surface, existing where light rays physically converge. This occurs when the image distance ( d_i ) is positive, marking the image as on the opposite side of the object relative to the mirror. In this exercise, the image height equaling the object height implies a magnification ( m ) of -1 , satisfying the condition for a real image of the same size but inverted:

• \( m = \frac{-d_i}{d_o} = -1 \)

This equation shows that the distances are equal in magnitude but opposite in sign, giving physical significance to the image's realism and orientation. Understanding real images aids in applications ranging from optical instruments to everyday reflections.

Magnification

Magnification describes how much larger or smaller an image appears compared to the object itself. It is calculated with the formula:

• \( m = \frac{h_i}{h_o} = \frac{-d_i}{d_o} \)

Where \( h_i \) is the image height and \( h_o \) is the object height. In this scenario, the magnification is -1, indicating that the image is the same size but inverted relative to the object. This negative sign is characteristic of real images formed by concave mirrors. Magnification is essential in determining not only the size but also the nature of the image (real or virtual). By applying these principles, the relationships between object placement, image characteristics, and mirror properties become clearer, enhancing our understanding of optical systems.