Questions: A local FM radio station broadcasts at a frequency of 89.8 MHz .(1 MHz=10^6 s^-1) Calculate the energy of the frequency at which it is broadcasting. Energy = kJ

A local FM radio station broadcasts at a frequency of 89.8 MHz .(1 MHz=10^6 s^-1) Calculate the energy of the frequency at which it is broadcasting.

Energy = kJ

Transcript text: A local FM radio station broadcasts at a frequency of $89.8 \mathrm{MHz} .\left(1 \mathrm{MHz}=10^{6} \mathrm{~s}^{-1}\right)$ Calculate the energy of the frequency at which it is broadcasting. Energy = $\square$ kJ

Solution

Solution Steps

Step 1: Convert Frequency to Hertz

The frequency of the radio station is given as $89.8 \, \text{MHz}$. To convert this to hertz (Hz), we use the conversion factor $1 \, \text{MHz} = 10^6 \, \text{Hz}$.

\[ f = 89.8 \times 10^6 \, \text{Hz} \]

Step 2: Use Planck's Equation to Calculate Energy

The energy of a photon can be calculated using Planck's equation:

\[ E = h \cdot f \]

where $E$ is the energy in joules, $h$ is Planck's constant ($6.6261 \times 10^{-34} \, \text{J} \cdot \text{s}$), and $f$ is the frequency in hertz.

Substituting the values:

\[ E = 6.6261 \times 10^{-34} \, \text{J} \cdot \text{s} \times 89.8 \times 10^6 \, \text{Hz} \]

\[ E = 5.9481 \times 10^{-26} \, \text{J} \]

Step 3: Convert Energy from Joules to Kilojoules

To convert the energy from joules to kilojoules, divide by $1000$.

\[ E = \frac{5.9481 \times 10^{-26} \, \text{J}}{1000} = 5.9481 \times 10^{-29} \, \text{kJ} \]