Question

visible light spectrum
ultraviolet infrared
400 500 600 700
figure 1 wavelength (nanometers)

1. what is the relationship between frequency and wavelength?
2. what is the relationship between wavelength and energy?
3. what is the relationship between frequency and energy?
4. what is the speed of all electromagnetic waves in a vacuum?

Explanation:

Step1: Recall wave - speed formula

The speed of a wave $v$ is given by $v = f\lambda$, where $f$ is the frequency and $\lambda$ is the wavelength. For electromagnetic waves in a vacuum, $v = c$ (the speed of light, a constant). So, $c=f\lambda$, and $f=\frac{c}{\lambda}$. This shows an inverse relationship.

Step2: Recall energy - wavelength relationship

The energy of a photon is given by $E = hf$, and since $f=\frac{c}{\lambda}$, we can substitute to get $E=\frac{hc}{\lambda}$. This shows an inverse relationship between energy $E$ and wavelength $\lambda$.

Step3: Recall energy - frequency relationship

The energy of a photon is $E = hf$, where $h$ is Planck's constant. So, energy $E$ is directly proportional to frequency $f$.

Step4: Recall speed of EM waves in vacuum

The speed of all electromagnetic waves in a vacuum is a constant, approximately $c = 3\times10^{8}\ m/s$.

Answer:

1. Inverse relationship ($f=\frac{c}{\lambda}$, where $c$ is the speed of light in vacuum).
2. Inverse relationship ($E=\frac{hc}{\lambda}$, where $h$ is Planck's constant and $c$ is the speed of light in vacuum).
3. Direct relationship ($E = hf$, where $h$ is Planck's constant).
4. $3\times10^{8}\ m/s$.