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Q7. Fig. Q7 depicts a beaker containing a block of glass on the bottom and wate...
Oct 14, 2024
Solution by Steps
step 1
To find the true depth of the air bubble, we can use Snell's law at the glass-water interface. The relationship is given by: n1sin⁡(θ1)=n2sin⁡(θ2) n_1 \sin(\theta_1) = n_2 \sin(\theta_2) , where n1 n_1 is the refractive index of water, n2 n_2 is the refractive index of glass, θ1 \theta_1 is the angle of incidence, and θ2 \theta_2 is the angle of refraction
step 2
Given that n1=1.33 n_1 = 1.33 , n2=1.6 n_2 = 1.6 , and θ1=60∘ \theta_1 = 60^{\circ} , we can find θ2 \theta_2 using: 1.33sin⁡(60∘)=1.6sin⁡(θ2) 1.33 \sin(60^{\circ}) = 1.6 \sin(\theta_2)
step 3
Calculating sin⁡(60∘) \sin(60^{\circ}) gives 32 \frac{\sqrt{3}}{2} . Thus, we have: 1.33⋅32=1.6sin⁡(θ2) 1.33 \cdot \frac{\sqrt{3}}{2} = 1.6 \sin(\theta_2) . Solving for sin⁡(θ2) \sin(\theta_2) yields: sin⁡(θ2)=1.33⋅321.6 \sin(\theta_2) = \frac{1.33 \cdot \frac{\sqrt{3}}{2}}{1.6}
step 4
Now, we can find the true depth d d of the air bubble using the apparent depth d′=6.81 cm d' = 6.81 \, \text{cm} and the relationship: d=d′⋅n2n1 d = d' \cdot \frac{n_2}{n_1} . Substituting the values gives: d=6.81⋅1.61.33 d = 6.81 \cdot \frac{1.6}{1.33}
step 5
Finally, calculating this gives the true depth of the air bubble: d≈8.18 cm d \approx 8.18 \, \text{cm}
Answer
The true depth of the air bubble is approximately 8.18 cm.
Key Concept
Snell's law describes how light refracts when passing through different media with varying refractive indices.
Explanation
The true depth of the air bubble is calculated by considering the apparent depth and the refractive indices of the materials involved, demonstrating how light behaves at interfaces.
What formula can be used to find the true depth of the air bubble considering the refraction of light at the interfaces?
Solution by Steps
step 1
To find the true depth of the air bubble, we can use Snell's law at the glass-water interface. The relationship is given by: n1sin⁡(θ1)=n2sin⁡(θ2) n_1 \sin(\theta_1) = n_2 \sin(\theta_2) , where n1 n_1 is the refractive index of water, n2 n_2 is the refractive index of glass, θ1 \theta_1 is the angle of incidence, and θ2 \theta_2 is the angle of refraction
step 2
Given that the angle of incidence θ1=60∘ \theta_1 = 60^{\circ} and the apparent depth da=6.81 cm d_a = 6.81 \, \text{cm} , we can find the true depth d d using the formula: d=da⋅n1n2 d = d_a \cdot \frac{n_1}{n_2}
step 3
Assuming the refractive index of water n1≈1.33 n_1 \approx 1.33 and glass n2≈1.5 n_2 \approx 1.5 , we can substitute these values into the formula: d=6.81⋅1.331.5 d = 6.81 \cdot \frac{1.33}{1.5}
step 4
Calculating the true depth: d≈6.81⋅0.8867≈6.05 cm d \approx 6.81 \cdot 0.8867 \approx 6.05 \, \text{cm} . Thus, the true depth of the air bubble is approximately 6.05 cm
Answer
The true depth of the air bubble is approximately 6.05 cm.
Key Concept
Snell's law relates the angles of incidence and refraction to the refractive indices of the two media.
Explanation
The true depth is calculated by adjusting the apparent depth based on the refractive indices of water and glass, demonstrating how light refraction affects our perception of depth.
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