Calculate Wavenumber of Radiation Through Glass

[Roodles01]

Homework Statement

Glass has a permeability of ε=2.0 and relative permeability μ=1.
Electromagnetic radiation of angular frequency ω=2.8x10⁹ s^-1. travels through the glass. Calculate
a) refractive index
b) speed of radiation
c) frequency of radiation
d) wavenumber of radiation

a, b and c are straightforward but d I have questions for

Homework Equations

n=√ε
n=c/v
f=ω/2∏
v=ω/k
ω=2∏f
k=n2∏f/v

The Attempt at a Solution

a) refractive index, n=√ε = √2

b) speed of radiation traveling through glass,
n = c/v
so v = c/n
v = 3.0x10⁸ ms^-1 / √2
v = 2.12x10⁸ ms^-1

c) frequency of radiation, f = ω/2∏
f = 2.8x10⁹ s^-1 / 2∏
f = 4.46x10⁶ Hz

d)wavenumber of radiation
now v = ω/k - where k is the wavenumber
so make k the subject
k = ω/v
k = 2.8x10⁹ s^-1 / 2.12x10⁸ ms^-1
k = 13.2

BUT

ω = 2∏f
and
k=n2∏f/v

so k = (2x2∏x 4.46x10⁶) / 2.12x10⁸ ms^-1
k = 0.132

So which is right, please?

 

[Simon Bridge]

So which is right, please?

The first one.

Note: $y(x,t)=A\sin(kx-\omega t + \delta)$ ... which means that $k=2\pi/\lambda$ ... i.e. inverse wavelength in the medium.

If v is the speed in the medium, then $v=f\lambda \implies c=nf\lambda$ is the speed in a vacuum.

So $k=2n\pi f/c$ ... recognize it?

 

[collinsmark]

Homework Statement

Glass has a permeability of ε=2.0

Don't you mean relative permittivity?

c) frequency of radiation, f = ω/2∏
f = 2.8x10⁹ s^-1 / 2∏
f = 4.46x10⁶ Hz

Try that one again. Keep an eye on the exponent. You're a couple orders of magnitude off there.

 

[Roodles01]

Yes, I see.
And yes, typo's.

Ta.

 

[HalfLostAndSearching]

Both of your attempts at calculating the wavenumber are correct. The first calculation (k = ω/v) gives you the wavenumber in units of m-1, while the second calculation (k = n2∏f/v) gives you the wavenumber in units of cm-1. This is because the speed of radiation (v) is given in meters per second, and the frequency (f) is given in hertz. To convert from m-1 to cm-1, you can simply multiply by 100. So in this case, the wavenumber is 13.2 m-1 or 132 cm-1. Both are correct, it just depends on the unit you prefer to use.