Problem 15 - Physics

Question

A plane mirror and a concave mirror \((f=8.0 \mathrm{~cm})\) are facing each other and are separated by a distance of \(20.0 \mathrm{~cm}\). An object is placed \(10.0 \mathrm{~cm}\) in front of the plane mirror. Consider the light from the object that reflects first from the plane mirror and then from the concave mirror. Using a ray diagram drawn to scale, find the location of the image that this light produces in the concave mirror. Specify this distance relative to the concave mirror.

Step-by-Step Solution

Answer

The final image is located 6.32 cm in front of the concave mirror.

Step 1: Understand the Setup

In this problem, an object is placed in front of a plane mirror, and there is a concave mirror facing the plane mirror. The key distance given between the two mirrors is 20 cm, and the focal length of the concave mirror is 8 cm. The light reflects from the plane mirror first, so consider the reflection off the plane mirror for the initial setup.

Step 2: Reflection from the Plane Mirror

The object is placed 10 cm from the plane mirror. In this step, the plane mirror creates a virtual image of the object which appears to be 10 cm behind the plane mirror because the distance to the image from the mirror is the same as the distance from the object to the mirror.

Step 3: Calculate Distance from Virtual Image to Concave Mirror

Since the plane mirror and the concave mirror are 20 cm apart, and the virtual image is 10 cm behind the plane mirror, the distance from this virtual image to the concave mirror (where it is considered the object for the concave mirror) is 20 cm + 10 cm = 30 cm.

Step 4: Use Mirror Formula for the Concave Mirror

For the concave mirror, use the mirror formula: \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), where \( f = 8.0 \) cm and \( u = -30 \) cm (negative because the direction of the incoming object as seen by the concave mirror is opposite the direction of light). Plug in the values: \[ \frac{1}{8} = \frac{1}{v} - \frac{1}{30} \] which simplifies to: \[ \frac{1}{v} = \frac{1}{8} + \frac{1}{30} \] \[ \frac{1}{v} = \frac{15 + 4}{120} = \frac{19}{120} \] Solve for \( v \) by inverting the fraction: \( v \approx 6.32 \text{ cm} \) (positive, indicating a real image).

Step 5: Calculate the Final Image Distance Relative to the Concave Mirror

Since the distance \( v \) indicates the position of the image formed by the concave mirror, and it is measured from the concave mirror, the final image is located 6.32 cm in front of the concave mirror, on the side where the object is located.

Key Concepts

Concave Mirrors

Concave mirrors are like magical bowls capturing and reflecting light in a unique way. They have a curved, inward surface resembling a part of a sphere. The cool thing about concave mirrors is how they focus light to a specific point. This point is called the "focal point." It's the place where rays of light, parallel to the mirror's principal axis, converge after reflection.

When dealing with concave mirrors, it's all about understanding the focus and center of curvature:

• The **Focus** is a central feature, and it's located halfway between the mirror's surface and its center of curvature. This distance is known as the "focal length."
• The **Center of Curvature** is like the mirror's center if you imagine completing the sphere from which the mirror segment was taken.
• The **Principal Axis** is the straight line passing through the center of curvature and focus and extending to the mirror's surface.

Concave mirrors can form real images, which are inverted and can be projected onto a screen. At times, they also form virtual images, which are upright and not projectable. It all depends on where the object is placed along the principal axis.

Plane Mirrors

Imagine a plane mirror as your classic flat mirror often seen at home. Plane mirrors work by reflecting light directly back in the direction it came. They don't alter the shape or orientation of the image in terms of flipping or convergence.

Here's how plane mirrors affect light reflection:

• They create images that are upright and same-sized relative to the object.
• The image forms behind the mirror at an equal distance as the object is in front. So, if an object is 10 cm in front of a plane mirror, the image appears 10 cm behind it.
• Plane mirrors create virtual images because the light seems to be coming from behind the mirror, despite no physical light rays traveling from that area.

This principle of reflection makes plane mirrors predictable and user-friendly tools in optics. They offer simple, clear reflections with little distortion, typically used for daily activities like grooming, design, and visual checks.

Mirror Formula

Let's talk about the mirror formula, a handy equation that connects an object's position, the image's position, and the mirror's focal length. The formula helps us solve problems related to mirrors, especially curved mirrors like concave ones.

The mirror formula is written as:\[\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\]
Where:

• \(f\) is the focal length of the mirror, the distance between the mirror and its focus.
• \(v\) is the image distance from the mirror.
• \(u\) is the object distance from the mirror.

Using this formula requires attention to the sign conventions:

• Object distances (\(u\)) are negative if placed in front of the mirror.
• Image distances (\(v\)) are positive for real images (formed on the same side as the object) and negative for virtual images (formed on the opposite side).
• The focal length (\(f\)) of a concave mirror is taken as negative.

This equation is a powerful tool for determining either the position of images or the characteristics of the mirror. It’s essential knowledge for anyone working with or studying optics.