The starter motor of a car engine draws an electric current of 110 A from the battery. The copper wire to the motor is 4.20 mm in diameter and 1.73 m long. The starter motor runs for 0.95 s before the car engine starts up. How much electric charge passes through the starter motor?

What is the current density in the wire?

How far does an electron travel along the wire while the starter motor is on? (The density of conduction electrons in copper is n = 8.50×1028 1/m3.)

a. 104.5 C b. 7.94 × 10⁶ A/m² c. 5.83 × 10⁻⁴ m/s

Explanation:

a. How much electric charge passes through the starter motor?

Using Q = It where Q = electric charge passing through the starter motor, I = current = 110 A and t = time = 0.95 s

So, Q = It = 110 A × 0.95 s = 104.5 C

b. What is the current density in the wire?

The current density, J = I/A where I = current = 110 A and A = cross-sectional area = πd²/4 where d = diameter of copper wire = 4.20 mm = 4.20 × 10⁻³ m

So, J = I/A

= I/πd²/4

= 4I/πd²

= 4 × 110 A/π(4.20 × 10⁻³ m)²

= 440 A/55.42 × 10⁻⁶ m²

= 7.94 × 10⁶ A/m²

c. How far does an electron travel along the wire while the starter motor is on? (The density of conduction electrons in copper is n = 8.50×1028 1/m3.)

To find how far the electron travels, we need to find the electron drift velocity from

J = nev where J = current density = 7.94 × 10⁶ A/m², n = electron density = 8.50 × 10²⁸ m⁻³, e = electron charge = 1.602 × 10⁻¹⁹ C, v = drift velocity of electrons and A = cross-sectional area of wire = πd²/4 where d = diameter of copper wire = 4.20 mm = 4.20 × 10⁻³ m

So, v = J/ne

Substituting the values of the variables into the equation, we have

v = 7.94 × 10⁶ A/m² ÷ (8.50 × 10²⁸ m⁻³ × 1.602 × 10⁻¹⁹ C)

v = 7.94 × 10⁶ A/m² ÷ (13.617 × 10⁹ Cm⁻³)

v = 0.583 × 10⁻³ m/s

v = 5.83 × 10⁻⁴ m/s

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