To solve the problem, we will follow these steps:
### Step 1: Understand the Problem
We need to find the binding energy of the K shell electrons when X-rays of energy 24.8 keV strike a material and cause photoelectrons to be emitted. The emitted photoelectrons move in a circular path in a magnetic field.
### Step 2: Write Down the Given Information
- Energy of the incident X-ray photon, \( E = 24.8 \, \text{keV} = 24.8 \times 10^3 \, \text{eV} \)
- Radius of the circular path, \( r = 23 \, \text{mm} = 23 \times 10^{-3} \, \text{m} \)
- Magnetic field strength, \( B = 2 \times 10^{-2} \, \text{T} \)
### Step 3: Relate the Forces
The magnetic force acting on the photoelectron is given by:
\[
F = qvB
\]
where \( q \) is the charge of the electron, \( v \) is the velocity of the electron, and \( B \) is the magnetic field strength.
This force also provides the centripetal force required for circular motion:
\[
F = \frac{mv^2}{r}
\]
where \( m \) is the mass of the electron.
### Step 4: Equate the Forces
Setting the two expressions for force equal gives:
\[
qvB = \frac{mv^2}{r}
\]
### Step 5: Solve for Velocity
Rearranging gives:
\[
v = \frac{qBr}{m}
\]
### Step 6: Calculate the Kinetic Energy
The kinetic energy (\( KE \)) of the photoelectron can be expressed as:
\[
KE = \frac{1}{2} mv^2
\]
Substituting for \( v \):
\[
KE = \frac{1}{2} m \left(\frac{qBr}{m}\right)^2 = \frac{1}{2} \frac{q^2 B^2 r^2}{m}
\]
### Step 7: Substitute Values
Substituting the known values:
- Charge of electron, \( q = 1.6 \times 10^{-19} \, \text{C} \)
- Mass of electron, \( m = 9.1 \times 10^{-31} \, \text{kg} \)
\[
KE = \frac{1}{2} \cdot \frac{(1.6 \times 10^{-19})^2 \cdot (2 \times 10^{-2})^2 \cdot (23 \times 10^{-3})^2}{9.1 \times 10^{-31}}
\]
### Step 8: Calculate the Kinetic Energy
Calculating the above expression:
\[
KE \approx 2.97 \times 10^{-15} \, \text{J}
\]
### Step 9: Convert Kinetic Energy to Electron Volts
To convert joules to electron volts:
\[
KE \text{ (in eV)} = \frac{KE \text{ (in J)}}{1.6 \times 10^{-19}} \approx 18.36 \, \text{keV}
\]
### Step 10: Calculate the Binding Energy
The binding energy (\( BE \)) of the K shell electrons is given by:
\[
BE = E - KE
\]
Substituting the values:
\[
BE = 24.8 \, \text{keV} - 18.36 \, \text{keV} \approx 6.44 \, \text{keV}
\]
### Step 11: Final Answer
The binding energy of the K shell electrons is approximately \( 6.44 \, \text{keV} \). The nearest answer is \( 6.2 \, \text{keV} \), which corresponds to option A.
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