There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 nmnm to 400 nmnm. It is not so harmful to the skin and is necessary for the production of vitamin D. UVB, with a wavelength between 280 nmnm and 320 nmnm, is much more dangerous, because it causes skin cancer. Required:
a. Find the frequency ranges of UVA and UVB.
b. What are the ranges of the wave numbers for UVA and UVB?

a) Remember the relationship:

v = f*λ

where:

v = velocity of the wave, remember that for an electromagnetic wave, this always is equal to the speed of light, then:

v = c = 3*10^8 m/s

f is the frequency of the wave

λ is the wavelength.

So if we want to compute the frequency, we have the equation:

f = c/λ

then:

We know that UVA has the range from 320 nm to 400nm, converting these to meters we hav:

1nm = 1m*10^(-9)

then:

320mm = (320*10^(-9)) m = 0.00000032 m

400nm = (400*10^(-9)) m =  0.00000040 m

Replacing these in our equation we can find that the range of frequencies is:

f = (3*10^8 m/s)/( 0.00000032 m) = 9.375*10^(14) Hz

f =  (3*10^8 m/s)/( 0.00000040 m) = 7.5*10^(14) Hz

Then the range of frequencies is:

( 7.5*10^(14) Hz to 9.375*10^(14) Hz)

Similar for the case of UVB:

the wavelengths are:

280 nm to  320 nm

Rewriting these in meters we get:

280nm = 280*10^(-9) m = 0.00000028m

320nm = 320*10^(-9) m = 0.00000032 m

Then the frequencies are:

f = (3*10^8 m/s)/( 0.00000032 m) = 9.375*10^(14) Hz

f = (3*10^8 m/s)/( 0.00000028 m) = 1.07*10^(15) Hz

The range of frequencies for UVB is:

(9.375*10^(14) Hz to 1.07*10^(15) Hz)

B) We want to know the range of wavenumbers for both types of waves.

The wave number is calculated as:

1/λ

Which represents the number of waves in a given unit of length.

Then the range for UVA will be:

1/320nm = 0.0031 nm^-1

1/400nm = 0.0025 nm^-1

Then the range is:

( 0.0025 nm^-1, 0.0031 nm^-1)

And for the case of UVB we will have:

1/320 nm = 0.0031 nm^-1

1/280 nm = 0.0036 nm^-1

Then the range is:

(0.0031 nm^-1, 0.0036 nm^-1)