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conantee
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« on: December 31, 2009, 03:15:47 AM »

3. For blue light, a transparent material has a relative permittivity (dielectric constant) of 2.1 and a relative permeability of 1.0. If the speed of light in a vacuum is c, the phase velocity of blue light in an unbounded medium of this material is
(A) \sqrt{3.1} c
(B) \sqrt{2.1} c
(C) \frac{c}{\sqrt{1.1}}
(D) \frac{c}{\sqrt{2.1}}
(E) \frac{c}{\sqrt{3.1}}
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« Reply #1 on: December 31, 2009, 03:34:53 AM »

If you don't remember anything about phase velocity of light in material, you should at least remember that in E&M and optics, the relative permittivity and relative permeability are totally different quantities. So, it is unlikely to be add up or subtract from one another. Also, they are usually used as a fundamental constant. No more add or subtract. Hence, 1.1 and 3.1 are not good numbers. You can eliminate (A), (C) and (E) then make a guess.

However, if you know a bit more, phase velocity in optics (again, in optics, not matter wave - it's different story) relates to velocity used in dispersion. Remember how prisms disperses white light into color. The phase velocity is given by
v = c/n where n is the refractive index.

Now, you don't have to know the relation between refractive index and those constants. You just know that n is typically greater than 1. So, the only choice we have is (D).

The correct answer is (D). 55 out of 100 people got the right answer.
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« Reply #2 on: December 31, 2009, 03:44:10 AM »

Physics behind it:
By definition, the refractive index of a medium is
n = c_{vacuum}/c_{medium} where c is the speed of light.

Now, the E&M theory shows that  c =\frac{1}{\sqrt{\mu \epsilon}}, where \mu is permability and \epsilon is permittivity.

Therefore,  n = \sqrt{\frac{\mu_{medium} \epsilon_{medium}}{\mu_{vacuum} \epsilon_{vacuum}}} or
                  n = \sqrt{ \mu_{rel} \epsilon_{rel}
as relative quantities.

The phase velociy is then,  v = c/n = c/\sqrt{ \mu_{rel} \epsilon_{rel}} = c/\sqrt{1.0\cdot 2.1} = c/\sqrt{2.1}
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