# How can I calculate the wavelength of electromagnetic radiation?

Aug 9, 2014

The equation that relates wavelength, frequency, and speed of light is

$c = \lambda \cdot \nu$

$c = 3.00 \times {10}^{8}$ $\text{m/s}$ (the speed of light in a vacuum)
$\lambda$ = wavelength in meters
$\nu$ = frequency in Hertz (Hz) or $\frac{1}{\text{s}}$ or $\text{s"^(-1)}$.

So basically the wavelength times the frequency of an electromagnetic wave equals the speed of light.

FYI, $\lambda$ is the Greek letter lambda , and $\nu$ is the Greek letter nu (it is not the same as a v).

To find wavelength ($\lambda$), the equation is manipulated so that $\lambda = \frac{c}{\nu}$.

EXAMPLE PROBLEM 1

What is the wavelength of a electromagnetic wave that has a frequency of $4.95 \times {10}^{14}$ $\text{Hz}$?

Given or Known:

frequency or $\nu = 4.95 \times {10}^{14}$ $\text{Hz}$ or $4.50 \times {10}^{14}$ $\text{s"^(-1)}$

$c = 3.00 \times {10}^{8}$ $\text{m/s}$

Unknown:

Wavelength or $\lambda$

Equation:

$c = \lambda \cdot \nu$

Solution:

$\lambda = \frac{c}{\nu} = \left(3.00 \times {10}^{8} \text{m"/color(red)cancel(color(black)("s")))/(4.95xx10^14color(red)cancel(color(black)("s"))^(-1)}\right) = 6.06 \times {10}^{- 7}$ $\text{m}$

But what if you don't know the frequency? Can you still find the wavelength? Yes. An equation that relates energy and frequency is:

$E = h \nu$

$E$ = energy in Joules $\left(\text{J}\right)$
$h$ = Planck's constant = $6.626 \times {10}^{- 34} \text{J"*"s}$
$\nu$ = frequency = $\text{Hz}$ or $\text{s"^(-1)}$

To find frequency, the equation is manipulated so that

$\nu = \frac{E}{h}$

Once you have frequency, you can use the first equation $c = \lambda \cdot \nu$ to find the wavelength.

EXAMPLE PROBLEM 2

What is the wavelength of an electromagnetic wave having $3.28 \times {10}^{- 19}$ $\text{J}$ of energy?

Given or Known:

$E = 3.28 \times {10}^{- 19}$ $\text{J}$
$h = 6.626 \times {10}^{- 34}$ $\text{J"*"s}$

Unknown:

frequency, $\nu$

Equation:

$E = h \nu$

Solution: Part 1

nu = E/h = (3.28xx10^(-19)color(red)cancel(color(black)("J")))/(6.626xx10^(-34)color(red)cancel(color(black)("J"))*"s") = 4.95 xx10^14 $\text{Hz}$ or $4.95 \times {10}^{14}$ ${\text{s}}^{- 1}$

Solution: Part 2

$\lambda = \frac{c}{\nu} = \left(3.00 \times {10}^{8} \text{m"/color(red)cancel(color(black)("s")))/(4.95xx10^14color(red)cancel(color(black)("s"))^(-1)}\right) = 6.06 \times {10}^{- 7}$ $\text{m}$