Find the radius of a circular orbit of an electron of energy 5 keV in a field of `10^(-7)T`.

Answer

To find the radius of a circular orbit of an electron with a given energy in a magnetic field, we can follow these steps:

### Step 1: Convert Energy from keV to Joules
The energy of the electron is given as 5 keV. We need to convert this energy into joules using the conversion factor \(1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}\).

\[ E = 5 \text{ keV} = 5 \times 10^3 \text{ eV} = 5 \times 10^3 \times 1.6 \times 10^{-19} \text{ J} = 8 \times 10^{-16} \text{ J} \]

### Step 2: Calculate the Kinetic Energy
The kinetic energy (KE) of the electron can be expressed as:

\[ KE = \frac{1}{2} mv^2 \]

From the energy we calculated, we can equate:

\[ \frac{1}{2} mv^2 = 8 \times 10^{-16} \text{ J} \]

### Step 3: Solve for Velocity (v)
Rearranging the equation for \(v\):

\[ v^2 = \frac{2 \times 8 \times 10^{-16}}{m} \]

Substituting the mass of the electron \(m = 9.1 \times 10^{-31} \text{ kg}\):

\[ v^2 = \frac{16 \times 10^{-16}}{9.1 \times 10^{-31}} = \frac{16}{9.1} \times 10^{15} \]

Calculating \(v\):

\[ v = \sqrt{\frac{16}{9.1} \times 10^{15}} \approx \sqrt{1.758 \times 10^{15}} \approx 1.32 \times 10^7 \text{ m/s} \]

### Step 4: Use the Formula for Radius of Circular Motion
The radius \(R\) of the circular orbit in a magnetic field is given by:

\[ R = \frac{mv}{eB} \]

Where:
- \(e = 1.6 \times 10^{-19} \text{ C}\) (charge of the electron)
- \(B = 10^{-7} \text{ T}\) (magnetic field strength)

### Step 5: Substitute Values into the Radius Formula
Substituting the known values into the radius formula:

\[ R = \frac{(9.1 \times 10^{-31} \text{ kg})(1.32 \times 10^7 \text{ m/s})}{(1.6 \times 10^{-19} \text{ C})(10^{-7} \text{ T})} \]

Calculating the numerator:

\[ = 1.2 \times 10^{-23} \text{ kg m/s} \]

Calculating the denominator:

\[ = 1.6 \times 10^{-26} \text{ C T} \]

Now, substituting back:

\[ R = \frac{1.2 \times 10^{-23}}{1.6 \times 10^{-26}} = 7.5 \times 10^{2} \text{ m} = 750 \text{ m} \]

### Final Answer
The radius of the circular orbit of the electron is approximately:

\[ R \approx 750 \text{ m} \]