Sure, let’s solve this problem step by step.
1. Understanding the given data:
- Number of electrons per second, [tex]\( n = 6 \times 10^6 \)[/tex] electrons/s
- Cross-sectional area of the electron beam, [tex]\( A = 10^{-6} \, \text{m}^2 \)[/tex]
- Charge of one electron, [tex]\( e = 1.6 \times 10^{-19} \, \text{C} \)[/tex]
2. Calculate the total current (I):
The current ([tex]\( I \)[/tex]) can be calculated by multiplying the number of electrons passing per second by the charge of one electron.
[tex]\[
I = n \times e
\][/tex]
Plugging in the values:
[tex]\[
I = (6 \times 10^6) \, \text{electrons/s} \times 1.6 \times 10^{-19} \, \text{C/electron}
\][/tex]
[tex]\[
I = 9.6 \times 10^{-13} \, \text{A}
\][/tex]
3. Calculate the current density (J):
Current density ([tex]\( J \)[/tex]) is defined as the current per unit area. It can be calculated by dividing the current by the cross-sectional area.
[tex]\[
J = \frac{I}{A}
\][/tex]
Plugging in the values:
[tex]\[
J = \frac{9.6 \times 10^{-13} \, \text{A}}{10^{-6} \, \text{m}^2}
\][/tex]
[tex]\[
J = 9.6 \times 10^{-7} \, \text{A/m}^2
\][/tex]
4. Conclusion:
The current in the electron beam is [tex]\( 9.6 \times 10^{-13} \)[/tex] amperes, and the current density is [tex]\( 9.6 \times 10^{-7} \)[/tex] amperes per square meter.
Thus, the solution states that the current density in the electron beam is [tex]\( 9.6 \times 10^{-7} \, \text{A/m}^2 \)[/tex].
