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Chapter 24: Problem 20

A rectangular block of glass is placed on a mark made on the surface of the table and it is viewed from the vertical position of eye. If refractive index of glass be \(\mu\) and its thickness \(d\), then the mark will appear to be raised up by (a) \(\frac{(\mu+1) d}{\mu}\) (b) \(\frac{(\mu-1) d}{\mu}\) (c) \(\frac{(\mu+1)}{\mu d}\) (d) \(\frac{(\mu-1) \mu}{d}\)

Short Answer

Expert verified
The mark will appear to be raised by \(\frac{(\mu - 1) d}{\mu}\), matching option (b).

Step by step solution

01

Understand the Concept

The apparent depth of an object when viewed through a medium is different from its actual depth due to refraction. The effective thickness of the medium can be calculated using the refractive index.
02

Formula for Apparent Depth

The formula to find the apparent depth from the actual depth is: \[ \text{Apparent Depth} = \frac{\text{Actual Depth}}{\mu} \] where \(\mu\) is the refractive index of the medium.
03

Calculate Apparent Depth

Given the thickness of the glass \(d\), use the formula: \[ \text{Apparent Depth} = \frac{d}{\mu} \].
04

Determine the Raise in Appearance

The apparent raise is the difference between the actual thickness and the apparent depth: \[ \text{Raise} = d - \frac{d}{\mu} \].
05

Simplify the Expression

Simplify the expression \( d - \frac{d}{\mu} \): \[ \text{Raise} = d \left(1 - \frac{1}{\mu} \right) = \frac{(\mu - 1) d}{\mu} \].
06

Match with Given Options

Compare the expression obtained, \(\frac{(\mu - 1) d}{\mu}\), to the options provided. This matches option (b).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Refractive Index
In the world of optics, the refractive index is a key player. It tells us how much light bends when it enters a new medium. Think about light moving from air into glass. Air has a refractive index of about 1.0, and glass is around 1.5. This means light slows down and bends when it hits the glass.
The formula for refractive index is:
  • \[ \mu = \frac{c}{v} \]
  • where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium.
This bending of light is why objects under water or glass can look different. A straw in a glass of water seems bent because of this refraction. A higher refractive index means light bends more, which is why clear things like glass and water can make things look odd.
Apparent Depth
Have you ever looked straight into a pond and noticed that the bottom looks nearer than it really is? This is what's known as apparent depth. When light rays travel through water, air, or glass, their path changes. The apparent depth formula is:
  • \[ \text{Apparent Depth} = \frac{\text{Actual Depth}}{\mu} \]
  • where \( \mu \) is the refractive index.
Using this, if you know the actual depth of an object and the refractive index of the medium, you can calculate how much closer or further away it will look.
This concept is crucial for understanding how images are formed when looking through transparent objects like a glass block, turning it into a fascinating optical illusion!
Optics
Optics is a branch of physics focusing on light and vision. It answers questions about how we see and why things appear the way they do. When light passes through different materials, it bends and changes speed. Reflection and refraction are mainstays of optics.
Mirrors reflect light, and lenses refract it, focusing or dispersing energy. All these phenomena depend on the principles of refraction, including how we perceive objects through different media. When you see an object through a glass block, optics explain why objects may seem warped or shifted. Optics is also behind many tools and technologies, from glasses to microscopes.
Glass Block
A glass block, often seen in walls or decorative pieces, influences the path of light in intriguing ways. Imagine placing a coin beneath a glass block and trying to touch it. This block raises the image of the coin due to refraction, making the coin appear closer to the surface than it is.
The light from the coin bends as it goes from glass to air, spurring an optical illusion.
  • This phenomenon causes an apparent depth that differs from the true depth.
  • The difference between apparent and actual depth is calculated with the formula: \[ \text{Raise} = d - \frac{d}{\mu} = \frac{(\mu - 1) d}{\mu} \]
Knowing this explains not just a simple trick of light but also underpins principles vital to designing optical instruments or even constructing visually appealing structures with glass.

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Most popular questions from this chapter

In an optics experiment, with the position of the object fixed, a student varies the position of a convex lens and for each position, the screen is adjusted to get a clear image of the object. A graph between the object distance \(u\) and the image distance \(v\), from the lens, is plotted using the same scale for the two axes. A straight line passing through the origin and making an angle of \(45^{\circ}\) with the \(x\) -axis meets the experimental curve at \(P\). The coordinates of \(P\) will be (a) \((2 f, 2 f)\) (b) \(\left(\frac{f}{2}, \frac{f}{2}\right)\) (c) \((f, f)\) (d) \((4 f, 4 f)\)

An equiconvex lens is cut into two halves along (i) \(X O X^{\prime}\) and (ii) \(Y O Y^{\prime}\) as shown in the figure. Let \(f, f^{\prime}, f^{\prime \prime}\) be the focal lengths of complete lens, of each half in case (i) and of each half in case (ii), respectively. Choose the correct statement from the following (a) \(f^{\prime}=2 f\) and \(f^{\prime \prime}=f\) (b) \(f^{\prime}=f\) and \(f^{\prime \prime}=f\) (c) \(f^{\prime}=2 f\) and \(f^{\prime \prime}=2 f\) (d) \(f^{\prime}=f\) and \(f^{\prime \prime}=2 f\)

In astronomical telescope, the final image is formed at (a) the least distance of distinct vision (b) the focus of the objective lens (c) the focus of the eye lens (d) infinity

The plane face of a plano-convex lens is silvered. If \(\mu\) be the refractive index and \(R\), the radius of curvature of curved surface, then the system will behave like a concave mirror of radius of curvature (a) \(\mu \mathrm{R}\) (b) \(R /(\mu-1)\) (c) \(R^{2} \mu\) (d) \([(\mu+1) /(\mu-1)] R\)

A convex lens, a glass slab, a glass prism and a spherical solid ball have been prepared from the same optically transparent material. Dispersive power will be possessed only by (a) the glass slab and the prism (b) the lens and the solid ball (c) the prism only (d) all of the above
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