To solve the problem of finding the orbital radius of a 10 eV electron circulating in a magnetic field of induction \(10^{-4} \, \text{Wb/m}^2\) (or 1.0 Gauss), we can follow these steps:
### Step 1: Convert Energy from eV to Joules
The energy of the electron is given as 10 eV. To convert this energy into joules, we use the conversion factor:
\[
1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}
\]
Thus, the energy in joules is:
\[
E = 10 \, \text{eV} = 10 \times 1.6 \times 10^{-19} \, \text{J} = 1.6 \times 10^{-18} \, \text{J}
\]
### Step 2: Use the Kinetic Energy Formula
The kinetic energy (KE) of the electron can be expressed as:
\[
KE = \frac{1}{2} mv^2
\]
From this, we can express the momentum \(p\) as:
\[
p = mv = \sqrt{2m \cdot KE}
\]
### Step 3: Substitute KE into the Radius Formula
The formula for the radius \(r\) of the circular motion of a charged particle in a magnetic field is given by:
\[
r = \frac{mv}{qB}
\]
Substituting \(v\) from the kinetic energy expression, we have:
\[
r = \frac{p}{qB} = \frac{\sqrt{2m \cdot KE}}{qB}
\]
### Step 4: Substitute Known Values
We know:
- Mass of the electron, \(m = 9.1 \times 10^{-31} \, \text{kg}\)
- Charge of the electron, \(q = 1.6 \times 10^{-19} \, \text{C}\)
- Magnetic field strength, \(B = 10^{-4} \, \text{T}\)
- Kinetic energy, \(KE = 1.6 \times 10^{-18} \, \text{J}\)
Now substituting these values into the radius formula:
\[
r = \frac{\sqrt{2 \cdot (9.1 \times 10^{-31}) \cdot (1.6 \times 10^{-18})}}{(1.6 \times 10^{-19}) \cdot (10^{-4})}
\]
### Step 5: Calculate the Radius
Calculating the numerator:
\[
\sqrt{2 \cdot (9.1 \times 10^{-31}) \cdot (1.6 \times 10^{-18})} = \sqrt{2.912 \times 10^{-48}} \approx 5.39 \times 10^{-24}
\]
Calculating the denominator:
\[
(1.6 \times 10^{-19}) \cdot (10^{-4}) = 1.6 \times 10^{-23}
\]
Now substituting back into the radius formula:
\[
r = \frac{5.39 \times 10^{-24}}{1.6 \times 10^{-23}} \approx 0.33625 \, \text{m}
\]
### Step 6: Convert to Centimeters
To convert meters to centimeters:
\[
r \approx 33.6 \, \text{cm}
\]
### Final Result
The orbital radius of the electron is approximately:
\[
r \approx 33.6 \, \text{cm}
\]
