The cyclotron frequency of an electron gyrating in a magnetic field of `1T` is approximately:

Options

A. 28 MHz
B. 280 MHz
C. 2.8 GHz
D. 28 GHz

Answer

To find the cyclotron frequency of an electron gyrating in a magnetic field of 1 Tesla, we can use the formula for cyclotron frequency:

\[ f = \frac{qB}{2\pi m} \]

Where:
- \( f \) is the cyclotron frequency,
- \( q \) is the charge of the electron,
- \( B \) is the magnetic field strength,
- \( m \) is the mass of the electron.

### Step 1: Identify the values
- The charge of an electron \( q = 1.6 \times 10^{-19} \, \text{C} \)
- The magnetic field \( B = 1 \, \text{T} \)
- The mass of an electron \( m = 9.1 \times 10^{-31} \, \text{kg} \)

### Step 2: Substitute the values into the formula
Substituting the values into the cyclotron frequency formula:

\[ f = \frac{(1.6 \times 10^{-19} \, \text{C})(1 \, \text{T})}{2\pi (9.1 \times 10^{-31} \, \text{kg})} \]

### Step 3: Calculate the denominator
First, calculate the denominator:

\[ 2\pi m = 2 \times 3.14 \times (9.1 \times 10^{-31}) \approx 5.65 \times 10^{-30} \, \text{kg} \]

### Step 4: Calculate the frequency
Now substitute this back into the equation for frequency:

\[ f = \frac{1.6 \times 10^{-19}}{5.65 \times 10^{-30}} \approx 2.83 \times 10^{10} \, \text{Hz} \]

### Step 5: Convert to gigahertz
To convert to gigahertz (GHz), we divide by \( 10^9 \):

\[ f \approx 28.3 \, \text{GHz} \]

### Final Answer
Thus, the cyclotron frequency of an electron gyrating in a magnetic field of 1 Tesla is approximately:

\[ f \approx 28 \, \text{GHz} \]