Find the ratio of permeability of the two media when the wave is incident on the boundary at 45 degree and reflected by the boundary at 60 degree.a)1:1b)√3:1c)1:√3d)1:√2Correct answer is option 'C'. Can you explain this answer?

Question

Find the ratio of permeability of the two media when the wave is incident on the boundary at 45 degree and reflected by the boundary at 60 degree.a)1:1b)&radic;3:1c)1:&radic;3d)1:&radic;2Correct answer is option 'C'. Can you explain this answer?

Answer

Understanding Wave Incidence and Reflection
When a wave travels from one medium to another and reflects at the boundary, the angles of incidence and reflection adhere to Snell's Law. Given that the wave is incident at 45 degrees and reflected at 60 degrees, we can derive the ratio of the permeabilities of the two media.

Applying Snell's Law
According to Snell’s Law:
$$ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) $$
Where:
- $ n_1 $ and $ n_2 $ are the refractive indices of the two media.
- $ \theta_1 $ is the angle of incidence (45 degrees).
- $ \theta_2 $ is the angle of reflection (60 degrees).
Since the refractive index is related to permeability ($ \mu $) and permittivity ($ \epsilon $) as follows:
$$ n = \sqrt{\frac{\mu}{\epsilon}} $$

Calculating the Ratio of Permeabilities
From Snell's Law, we have:
$$ n_1 \sin(45^\circ) = n_2 \sin(60^\circ) $$
Substituting the sine values:
- $ \sin(45^\circ) = \frac{\sqrt{2}}{2} $
- $ \sin(60^\circ) = \frac{\sqrt{3}}{2} $
This gives us:
$$ n_1 \frac{\sqrt{2}}{2} = n_2 \frac{\sqrt{3}}{2} $$
Rearranging leads to:
$$ \frac{n_1}{n_2} = \frac{\sqrt{3}}{\sqrt{2}} $$
Using the relation of refractive index to permeability:
$$ \frac{\mu_1}{\mu_2} = \left(\frac{n_1}{n_2}\right)^2 = \frac{3}{2} $$
Taking the square root, we find:
$$ \frac{\mu_1}{\mu_2} = \sqrt{\frac{3}{2}} $$
This implies that the ratio of permeabilities can be expressed as:
$$ 1:\sqrt{3} $$

Conclusion
Thus, the required ratio of the permeabilities when the wave is incident at 45 degrees and reflected at 60 degrees is $ 1:\sqrt{3} $, confirming option 'C' as the correct answer.

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