a 10c charge flows through a wire for 5 min. radius of wire is 1mm. it contains 5*10 power 28. electons per centimetre cube. find drift velocity.

To find the drift velocity of electrons in a wire, we can use the formula that relates current, charge, and drift velocity. Let's break down the problem step by step.

Understanding the Problem

We have the following information:

• Charge (Q) = 10 coulombs
• Time (t) = 5 minutes = 5 × 60 = 300 seconds
• Radius of the wire (r) = 1 mm = 0.001 m
• Number density of electrons (n) = 5 × 10²⁸ electrons/cm³ = 5 × 10³⁴ electrons/m³ (since 1 cm³ = 10^-6 m³)

Calculating the Current

The first step is to calculate the current (I) flowing through the wire. Current is defined as the charge per unit time:

I = Q / t

Substituting the values we have:

I = 10 C / 300 s = 1/30 A ≈ 0.0333 A

Finding the Cross-Sectional Area of the Wire

Next, we need to determine the cross-sectional area (A) of the wire, which is circular in shape. The area can be calculated using the formula for the area of a circle:

A = πr²

Substituting the radius:

A = π(0.001 m)² ≈ 3.14 × 10^-6 m²

Calculating Drift Velocity

The drift velocity (v_d) can be found using the formula:

I = nAv_de

Where:

• I = current
• n = number density of electrons
• A = cross-sectional area
• e = charge of an electron ≈ 1.6 × 10^-19 C

Rearranging the formula to solve for drift velocity gives us:

v_d = I / (nAe)

Now, substituting the known values:

v_d = (0.0333 A) / (5 × 10³⁴ electrons/m³ × 3.14 × 10^-6 m² × 1.6 × 10^-19 C)

Calculating the Values

First, calculate the denominator:

nAe = (5 × 10³⁴) × (3.14 × 10^-6) × (1.6 × 10^-19)

nAe ≈ 2.51 × 10¹⁰ C/m²

Now, substituting this back into the drift velocity equation:

v_d = 0.0333 A / 2.51 × 10¹⁰ C/m² ≈ 1.33 × 10^-12 m/s

Final Result

The drift velocity of the electrons in the wire is approximately 1.33 × 10^-12 m/s. This value illustrates how slowly electrons drift through a conductor, even though the electrical signal travels at a much faster speed due to the electromagnetic wave propagation in the wire.