To find the drift velocity of electrons in the metal when a current of 16 A passes through, we can use the formula for current in terms of drift velocity:
\[ I = n \cdot A \cdot e \cdot v_d \]
Where:
- \( I \) = current (in Amperes)
- \( n \) = number of free electrons per unit volume (in m\(^{-3}\))
- \( A \) = cross-sectional area (in m\(^2\))
- \( e \) = charge of an electron (approximately \( 1.6 \times 10^{-19} \) C)
- \( v_d \) = drift velocity (in m/s)
### Step 1: Calculate the number of free electrons per unit volume \( n \)
To find \( n \), we can use the formula:
\[ n = \frac{\text{Density} \times N_A}{\text{Molar Mass}} \]
Where:
- Density = \( 5 \times 10^3 \, \text{kg/m}^3 \)
- \( N_A \) (Avogadro's number) = \( 6 \times 10^{23} \, \text{mol}^{-1} \)
- Molar Mass = \( 60 \, \text{g/mol} = 0.06 \, \text{kg/mol} \)
Plugging in the values:
\[
n = \frac{(5 \times 10^3 \, \text{kg/m}^3) \times (6 \times 10^{23} \, \text{mol}^{-1})}{0.06 \, \text{kg/mol}}
\]
Calculating \( n \):
\[
n = \frac{(5 \times 10^3) \times (6 \times 10^{23})}{0.06}
\]
\[
n = \frac{30 \times 10^{26}}{0.06} = 5 \times 10^{28} \, \text{m}^{-3}
\]
### Step 2: Rearrange the current formula to solve for drift velocity \( v_d \)
Now, we can rearrange the formula for current to solve for \( v_d \):
\[
v_d = \frac{I}{n \cdot A \cdot e}
\]
### Step 3: Substitute the known values into the drift velocity formula
Substituting the known values:
- \( I = 16 \, \text{A} \)
- \( n = 5 \times 10^{28} \, \text{m}^{-3} \)
- \( A = 10^{-6} \, \text{m}^2 \)
- \( e = 1.6 \times 10^{-19} \, \text{C} \)
\[
v_d = \frac{16}{(5 \times 10^{28}) \cdot (10^{-6}) \cdot (1.6 \times 10^{-19})}
\]
Calculating the denominator:
\[
(5 \times 10^{28}) \cdot (10^{-6}) \cdot (1.6 \times 10^{-19}) = 8 \times 10^{4}
\]
Now substituting back into the equation for \( v_d \):
\[
v_d = \frac{16}{8 \times 10^{4}} = 0.0002 \, \text{m/s}
\]
### Step 4: Convert \( v_d \) to mm/s
To convert \( v_d \) to mm/s, we multiply by 1000:
\[
v_d = 0.0002 \, \text{m/s} \times 1000 = 0.2 \, \text{mm/s}
\]
### Final Answer
The drift velocity of electrons in the metal is \( 0.2 \, \text{mm/s} \).
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