How long does it take electrons to get from a car battery to the starting motor? Assume the current is 285 A and the electrons travel through a copper wire with cross-sectional area `0.17 cm^(2)` and length 0.43 m. The number of charge carriers per unit volume is `8.49 xx 10^(28) m^(-3).`

Answer

To solve the problem of how long it takes for electrons to travel from a car battery to the starting motor, we can follow these steps:

### Step 1: Convert the Cross-Sectional Area to Square Meters
The cross-sectional area \( A \) is given as \( 0.17 \, \text{cm}^2 \). We need to convert this to square meters.

\[ A = 0.17 \, \text{cm}^2 \times \left( \frac{1 \, \text{m}}{100 \, \text{cm}} \right)^2 = 0.17 \times 10^{-4} \, \text{m}^2 = 1.7 \times 10^{-5} \, \text{m}^2 \]

### Step 2: Identify Given Values
- Current \( I = 285 \, \text{A} \)
- Length of wire \( L = 0.43 \, \text{m} \)
- Number of charge carriers per unit volume \( n = 8.49 \times 10^{28} \, \text{m}^{-3} \)
- Charge of an electron \( e = 1.6 \times 10^{-19} \, \text{C} \)

### Step 3: Calculate the Drift Velocity \( v_d \)
The drift velocity \( v_d \) can be calculated using the formula:

\[ I = n \cdot e \cdot A \cdot v_d \]

Rearranging this gives:

\[ v_d = \frac{I}{n \cdot e \cdot A} \]

Substituting the values:

\[ v_d = \frac{285}{(8.49 \times 10^{28}) \cdot (1.6 \times 10^{-19}) \cdot (1.7 \times 10^{-5})} \]

Calculating the denominator:

\[ n \cdot e \cdot A = (8.49 \times 10^{28}) \cdot (1.6 \times 10^{-19}) \cdot (1.7 \times 10^{-5}) \approx 2.41 \times 10^{5} \]

Now substituting back into the equation for \( v_d \):

\[ v_d = \frac{285}{2.41 \times 10^{5}} \approx 1.18 \times 10^{-3} \, \text{m/s} \]

### Step 4: Calculate the Time \( T \)
The time \( T \) taken for the electrons to travel the length of the wire can be calculated using the formula:

\[ T = \frac{L}{v_d} \]

Substituting the values:

\[ T = \frac{0.43}{1.18 \times 10^{-3}} \approx 364.41 \, \text{s} \]

### Step 5: Convert Time to Minutes
To convert seconds to minutes:

\[ T \approx \frac{364.41}{60} \approx 6.07 \, \text{minutes} \]

### Final Answer
It takes approximately **6.07 minutes** for electrons to travel from the car battery to the starting motor.

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