A current of 3.14 amp is flowing through a wire of radius 1cm. If the number of free electrons per unit volume is 10^24 per m^3, then calculate the drift velocity

To find the drift velocity of electrons in a wire, we can use the formula that relates current, charge, number of charge carriers, and their drift velocity. The formula is given by:

Formula for Drift Velocity

The drift velocity (v_d) can be calculated using the equation:

I = n × A × e × v_d

Where:

• I = current (in amperes)
• n = number of free electrons per unit volume (in m^-3)
• A = cross-sectional area of the wire (in m²)
• e = charge of an electron (approximately 1.6 × 10^-19 coulombs)
• v_d = drift velocity (in m/s)

Step-by-Step Calculation

Let's break down the calculation into manageable steps:

1. Calculate the Cross-Sectional Area (A)

The cross-sectional area of a wire can be calculated using the formula for the area of a circle:

A = πr²

Given that the radius (r) is 1 cm, we first convert it to meters:

r = 1 cm = 0.01 m

Now, substituting the radius into the area formula:

A = π(0.01 m)² = π(0.0001 m²) ≈ 3.14 × 10^-4 m²

2. Substitute Values into the Drift Velocity Formula

Now that we have the area, we can rearrange the drift velocity formula to solve for v_d:

v_d = I / (n × A × e)

Substituting the known values:

• I = 3.14 A
• n = 10²⁴ m^-3
• A ≈ 3.14 × 10^-4 m²
• e ≈ 1.6 × 10^-19 C

3. Perform the Calculation

Now we can plug in the values:

v_d = 3.14 A / (10²⁴ m^-3 × 3.14 × 10^-4 m² × 1.6 × 10^-19 C)

Calculating the denominator:

n × A × e = 10²⁴ × 3.14 × 10^-4 × 1.6 × 10^-19 ≈ 5.024 × 10¹

Now substituting back into the drift velocity equation:

v_d = 3.14 / 5.024 ≈ 0.624 m/s

Final Result

The drift velocity of the electrons in the wire is approximately 0.624 m/s. This value indicates how slowly the electrons are moving through the conductor, despite the current being relatively high. The drift velocity is typically quite small compared to the speed of the electrical signal, which travels much faster due to the electromagnetic wave propagation in the wire.