Calculate the gyro magnetic ratio of electron
(given 1.6`xx10^(-19)` C, `m_r=9.1xx10^(-3)` kg )

Answer

To calculate the gyromagnetic ratio of an electron, we can use the formula:

\[ \text{Gyromagnetic Ratio} (\gamma) = \frac{E}{2m} \]

where:
- \(E\) is the charge of the electron,
- \(m\) is the mass of the electron.

Given:
- Charge of electron, \(E = 1.6 \times 10^{-19} \, \text{C}\)
- Mass of electron, \(m = 9.1 \times 10^{-31} \, \text{kg}\)

Now, let's substitute the values into the formula:

\[ \gamma = \frac{1.6 \times 10^{-19}}{2 \times 9.1 \times 10^{-31}} \]

Step 1: Calculate the denominator.

\[ 2m = 2 \times 9.1 \times 10^{-31} = 1.82 \times 10^{-30} \, \text{kg} \]

Step 2: Substitute the denominator back into the gyromagnetic ratio formula.

\[ \gamma = \frac{1.6 \times 10^{-19}}{1.82 \times 10^{-30}} \]

Step 3: Perform the division.

\[ \gamma = \frac{1.6}{1.82} \times 10^{(-19 + 30)} = \frac{1.6}{1.82} \times 10^{11} \]

Step 4: Calculate the numerical value.

\[ \frac{1.6}{1.82} \approx 0.879 \]

Thus,

\[ \gamma \approx 0.879 \times 10^{11} \approx 8.79 \times 10^{10} \, \text{C/kg} \]

So, the gyromagnetic ratio of the electron is approximately:

\[ \gamma \approx 8.79 \times 10^{10} \, \text{C/kg} \]