A beam of electrons moving at a speed of `10^(6)m//s` along a line produces a current of `1.6xx10^(-6)` A. The number of electrons in the 1 metre of the beam is

Options

A. `10^(6)`
B. `10^(7)`
C. `10^(13)`
D. `10^(19)`

Explanation

To find the number of electrons in a 1-meter beam of electrons moving at a speed of \(10^6 \, \text{m/s}\) that produces a current of \(1.6 \times 10^{-6} \, \text{A}\), we can follow these steps:

### Step 1: Calculate the time taken to travel 1 meter
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Given that the distance is 1 meter and the speed is \(10^6 \, \text{m/s}\):
\[ \text{Time} = \frac{1 \, \text{m}}{10^6 \, \text{m/s}} = 1 \times 10^{-6} \, \text{s} \]

### Step 2: Relate current to charge and time
The current \(I\) can be expressed in terms of charge \(Q\) and time \(t\):
\[ I = \frac{Q}{t} \]
Rearranging this gives:
\[ Q = I \cdot t \]
Substituting the values of current and time:
\[ Q = (1.6 \times 10^{-6} \, \text{A}) \cdot (1 \times 10^{-6} \, \text{s}) = 1.6 \times 10^{-12} \, \text{C} \]

### Step 3: Calculate the number of electrons
The charge of a single electron \(e\) is approximately \(1.6 \times 10^{-19} \, \text{C}\). The total charge \(Q\) can also be expressed as:
\[ Q = n \cdot e \]
where \(n\) is the number of electrons. Rearranging gives:
\[ n = \frac{Q}{e} \]
Substituting the values:
\[ n = \frac{1.6 \times 10^{-12} \, \text{C}}{1.6 \times 10^{-19} \, \text{C}} = 10^7 \]

### Conclusion
The number of electrons in 1 meter of the beam is:
\[ n = 10^7 \]

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