An electron beam has an aperture of `2mm^(2)`. A total of `7xx10^(16)` electrons flow through any perpendicular cross-section per second. Calculate the current density in the electron beam.

Options

A. `2.4 xx 10^3 A//m^3`
B. `3.9 xx 10^3 A//m^3`
C. `5.6 xx 10^3 A//m^3`
D. None of these

Answer

To calculate the current density in the electron beam, we will follow these steps:

### Step 1: Understand the formula for current density
The current density \( J \) is defined as the current \( I \) flowing per unit area \( A \):
\[ J = \frac{I}{A} \]

### Step 2: Find the total charge \( Q \)
The total charge \( Q \) can be calculated using the number of electrons \( n \) and the charge of a single electron \( e \):
\[ Q = n \cdot e \]
Where:
- \( n = 7 \times 10^{16} \) electrons
- \( e = 1.6 \times 10^{-19} \) coulombs (the charge of an electron)

### Step 3: Calculate the total charge \( Q \)
Substituting the values into the equation:
\[ Q = (7 \times 10^{16}) \cdot (1.6 \times 10^{-19}) = 11.2 \times 10^{-3} \text{ coulombs} \]

### Step 4: Determine the time period \( t \)
Since the number of electrons flowing through the cross-section is given per second, we can take \( t = 1 \) second.

### Step 5: Calculate the current \( I \)
The current \( I \) can be calculated as:
\[ I = \frac{Q}{t} = \frac{11.2 \times 10^{-3}}{1} = 11.2 \times 10^{-3} \text{ amperes} \]

### Step 6: Convert the area \( A \) from mm² to m²
The aperture area is given as \( 2 \text{ mm}^2 \). To convert this to square meters:
\[ A = 2 \text{ mm}^2 = 2 \times 10^{-6} \text{ m}^2 \]

### Step 7: Calculate the current density \( J \)
Now we can substitute \( I \) and \( A \) into the current density formula:
\[ J = \frac{I}{A} = \frac{11.2 \times 10^{-3}}{2 \times 10^{-6}} = 5.6 \times 10^{3} \text{ A/m}^2 \]

### Final Answer
The current density in the electron beam is:
\[ J = 5.6 \times 10^{3} \text{ A/m}^2 \]

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