The power of an X-ray tube is `16 W`. If the potential difference applied across the tube is `51 kV` , then the number of electrons striking the target per second is

Options

A. `8.4 xx 10^(16)`
B. `5 xx 10^(17)`
C. `2 xx 10^(16)`
D. `2 xx 10^(19)`

Explanation

To find the number of electrons striking the target per second in an X-ray tube, we can use the relationship between power, voltage, and the charge of the electrons. Here’s a step-by-step solution:

### Step 1: Understand the relationship between power, voltage, and current
The power (P) of the X-ray tube can be expressed as:
\[ P = V \times I \]
where:
- \( P \) is the power in watts (W),
- \( V \) is the potential difference in volts (V),
- \( I \) is the current in amperes (A).

### Step 2: Express current in terms of the number of electrons
The current (I) can also be expressed in terms of the number of electrons (n) striking the target per second:
\[ I = \frac{n \cdot e}{t} \]
where:
- \( n \) is the number of electrons,
- \( e \) is the charge of an electron (\( e \approx 1.6 \times 10^{-19} \) coulombs),
- \( t \) is the time in seconds (for our calculation, we can take \( t = 1 \) second).

### Step 3: Substitute current in the power equation
Substituting the expression for current into the power equation gives:
\[ P = V \cdot \frac{n \cdot e}{t} \]
For \( t = 1 \) second, this simplifies to:
\[ P = V \cdot n \cdot e \]

### Step 4: Rearrange to find the number of electrons
Rearranging the equation to solve for \( n \):
\[ n = \frac{P}{V \cdot e} \]

### Step 5: Plug in the values
Now we can substitute the given values into the equation:
- \( P = 16 \, \text{W} \)
- \( V = 51 \, \text{kV} = 51 \times 10^3 \, \text{V} \)
- \( e = 1.6 \times 10^{-19} \, \text{C} \)

Substituting these values:
\[ n = \frac{16}{51 \times 10^3 \times 1.6 \times 10^{-19}} \]

### Step 6: Calculate the value of n
Calculating the denominator:
\[ 51 \times 10^3 \times 1.6 \times 10^{-19} = 8.16 \times 10^{-15} \]

Now substituting this back into the equation for \( n \):
\[ n = \frac{16}{8.16 \times 10^{-15}} \]

Calculating \( n \):
\[ n \approx 1.96 \times 10^{16} \]

### Step 7: Round to significant figures
Rounding to two significant figures, we get:
\[ n \approx 2 \times 10^{16} \]

Thus, the number of electrons striking the target per second is:
\[ \boxed{2 \times 10^{16}} \]