Problem 17 - Applied Physics

Question

Find the focal length of a mirror that forms an image \(2.30 \mathrm{~m}\) behind a mirror of an object \(6.50 \mathrm{~m}\) in front of the mirror.

Step-by-Step Solution

Answer

The focal length of the mirror is approximately \(-3.56\) meters.

Step 1: Understand the Mirror Formula

The mirror formula relates the object distance \( (u) \), image distance \( (v) \), and the focal length \( (f) \) of a mirror. It is given by: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Step 2: Identify the Given Values

In this problem, the image distance \( v = -2.30 \) meters (negative because the image is behind the mirror), and the object distance \( u = 6.50 \) meters (positive because the object is in front).

Step 3: Substitute Values Into the Mirror Formula

Substitute the given values into the mirror formula: \[ \frac{1}{f} = \frac{1}{-2.30} + \frac{1}{6.50} \] This will allow us to solve for \( f \).

Step 4: Calculate Each Term

Calculate \( \frac{1}{-2.30} \) which equals approximately \(-0.4348\), and \( \frac{1}{6.50} \) which equals approximately \(0.1538\).

Step 5: Solve for Focal Length \( f \)

Add the calculated values: \(-0.4348 + 0.1538 = -0.2810\). Thus, \( \frac{1}{f} = -0.2810 \). Take the reciprocal to find \( f \): \( f = -\frac{1}{0.2810} \approx -3.56 \) meters.

Key Concepts

Focal Length Calculation

When dealing with curved mirrors, the focal length is a crucial measurement. It determines how light rays converge or diverge after reflecting off the mirror's surface. In the context of a mirror, the focal length (\( f \)) is tied to the mirror formula \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]where \( u \) is the object distance and \( v \) is the image distance.

It's important to remember that these distances are signed values, depending on the type of mirror and the position of the object and image. To find the focal length, we need to rearrange the formula and solve for \( f \).

• Step 1: Gather given information, like object distance \( u \) and image distance \( v \).
• Step 2: Substitute these values into the mirror formula.
• Step 3: Perform the arithmetic to isolate \( f \).

The reciprocal of the resultant value will yield the focal length, indicating whether the focal point is behind or in front of the mirror.

Reflection Physics

Reflection physics explores how light behaves when it encounters a surface. Mirrors, flat or curved, operate on the principle of reflection, where light returns instead of passing through.

There are laws to understand how light reflects:

• Law of Reflection: The angle of incidence (angle between incoming light and the normal to the surface) is always equal to the angle of reflection (angle between reflected light and the normal).
• Types of Reflection: Specular reflection (from smooth surfaces like a mirror) and diffuse reflection (from rough surfaces).

Mirrors utilize reflection to create images by directing light rays back to our eyes or cameras. Knowing how reflection operates at a basic level helps explain how mirrors form images and manage light paths.

Image Formation

In mirrors, the formation of images hinges on the reflection of light rays.

• Real Image: Formed when reflected rays converge; it can be projected onto a screen and appears inverted.
• Virtual Image: Created when rays appear to diverge from a point; it can't be projected since it's formed by extensions of the rays, usually seen in mirrors like the one in the exercise.

Image characteristics, like size and orientation, depend on the type of mirror (convex or concave) and object placement. The distances of these images from the mirror (\( v \)) are used alongside object distances (\( u \)) in the mirror formula, helping us calculate focal length and other essential properties related to image formation.