A bubble in a glass slab (μ = 1.5) when viewed from one side appears at 5 cm and from the other side at 2 cm. The thickness of the glass slab is:
10.5 cm
Finding Glass Slab Thickness Using Apparent Depth
This problem asks us to find the actual thickness of a glass slab given the apparent depths of a bubble inside it, viewed from two different sides. This involves understanding the concept of real depth and apparent depth.
Understanding Apparent Depth and Real Depth
When an object is placed inside a denser medium (like glass) and viewed from a rarer medium (like air), it appears to be closer than its actual position. This apparent distance is called the apparent depth, while the actual distance is called the real depth.
The relationship between real depth, apparent depth, and the refractive index ($\mu$) of the medium is given by the formula:
$$ \text{Apparent Depth} = \frac{\text{Real Depth}}{\text{Refractive Index}} $$
Rearranging this formula, we can find the real depth:
$$ \text{Real Depth} = \text{Apparent Depth} \times \text{Refractive Index} $$
Calculating the Thickness of the Glass Slab
Let the total thickness of the glass slab be $T$. The bubble is located at some point inside the slab. When viewed from one side, the apparent depth is 5 cm. Let the real distance of the bubble from this side be $d_1$.
- Apparent depth from side 1, $a_1 = 5$ cm
- Refractive index of glass, $\mu = 1.5$
- Real depth from side 1, $d_1 = a_1 \times \mu$
Calculating $d_1$:
$$ d_1 = 5 \text{ cm} \times 1.5 = 7.5 \text{ cm} $$
When viewed from the other side, the apparent depth is 2 cm. Let the real distance of the bubble from this side be $d_2$.
- Apparent depth from side 2, $a_2 = 2$ cm
- Refractive index of glass, $\mu = 1.5$
- Real depth from side 2, $d_2 = a_2 \times \mu$
Calculating $d_2$:
$$ d_2 = 2 \text{ cm} \times 1.5 = 3.0 \text{ cm} $$
The total thickness of the glass slab is the sum of the real depths from both sides to the bubble's position. Imagine the bubble is located at a depth $d_1$ from one face and $d_2$ from the opposite face. The total thickness is simply $d_1 + d_2$.
$$ T = d_1 + d_2 $$
Substituting the calculated values:
$$ T = 7.5 \text{ cm} + 3.0 \text{ cm} = 10.5 \text{ cm} $$
Summary of Calculations
| Parameter | From Side 1 | From Side 2 |
|---|---|---|
| Apparent Depth | 5 cm | 2 cm |
| Refractive Index ($\mu$) | 1.5 | 1.5 |
| Real Depth (Apparent Depth $\times$ $\mu$) | $5 \times 1.5 = 7.5$ cm | $2 \times 1.5 = 3.0$ cm |
Total Thickness = Real Depth from Side 1 + Real Depth from Side 2
Total Thickness = 7.5 cm + 3.0 cm = 10.5 cm.
Therefore, the thickness of the glass slab is 10.5 cm.
Revision Table: Apparent Depth Concepts
| Concept | Description | Formula |
|---|---|---|
| Real Depth | The actual distance of an object from the surface of a medium. | $d_{real}$ |
| Apparent Depth | The distance an object appears to be from the surface when viewed from another medium. | $d_{apparent}$ |
| Refractive Index ($\mu$) | The ratio of the speed of light in vacuum to the speed of light in the medium. Also relates real and apparent depth. | $\mu = \frac{d_{real}}{d_{apparent}}$ (when viewed from rarer to denser medium) |
| Condition | Viewing an object in a denser medium from a rarer medium (e.g., fish in water viewed from air). | $d_{apparent} < d_{real}$ |
Additional Information on Refractive Index and Depth
The refractive index ($\mu$) is a key property of a medium that determines how light bends when passing from one medium to another. A higher refractive index means light travels slower in that medium and bends more significantly towards the normal when entering from a rarer medium.
When light rays from the bubble in the glass slab travel towards the observer in air, they bend away from the normal at the glass-air interface. This bending causes the rays to appear to originate from a point closer to the surface than the actual position of the bubble. This is why the apparent depth is less than the real depth when viewing an object in a denser medium from a rarer medium.
The formula $\text{Real Depth} = \text{Apparent Depth} \times \mu$ specifically applies when viewing an object inside a medium of refractive index $\mu$ from air (which has a refractive index approximately equal to 1). In this case, $\mu$ is the refractive index of the medium relative to air ($\mu_{glass/air}$).
Understanding this relationship is crucial for solving problems involving apparent depth, such as calculating the shift in the apparent position of the bottom of a tank filled with water or the position of marks on a glass slab observed from above.
Important Questions from Electromagnetic Induction
The half-life period of a radioactive element 'X' is same as the mean life of another radioactive element Y. Initially both of them have the same no. of atoms, then:
A. X and Y have the same decay rate initially.
B. X and Y decay at the same rate always.
C. Y will decay at a faster rate than X.
D. X will decay at a faster rate than Y.
Choose the correct answer from the options given below:
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A Neutron is moving with a velocity of V in a non-uniform magnetic field as shown in the figure.
Velocity v̅ of neutron would be:
The graph between resistivity and temperature given below can be for the material:
Which phenomenon proves the particle nature of photons?
