A length of wire has a radius of 3.00 × 10-3 m and a resistance of 0.200 Ω. When the potential difference across the wire is 10.0 V, the electron drift speed is found to be 2.98 × 10-4 m/s. On the basis of these data, calculate the density of free electrons in the wire.

Question

A length of wire has a radius of 3.00 × 10-3 m and a resistance of 0.200 Ω. When the potential difference across the wire is 10.0 V, the electron drift speed is found to be 2.98 × 10-4 m/s. On the basis of these data, calculate the density of free electrons in the wire.

Your help is much appreciated! :)

Answer 1

Compute the current I using Ohm's law.

The flux of free electrons F is the free electron density tomes the drift velocity v. If you multiply this flux by the electron charge and the cross section, you get the current. By equating this with what you get from Ohm's law, you can thus solve for the free electron density.

Answer 2

Call the free electron density N, m^-3

e = electron charge , Coulombs
V = electron drift speed
I = current, Amps
A = cross sectional area, pi r^2, m^2

I = N*A*e*V

Solve for N in this case.
Get the current I from ohms law. I = V/R

Answer 3

To calculate the density of free electrons in the wire, we need to use the formula:

n = (I * A) / (q * V * v_d)

Where:
n = density of free electrons
I = current flowing through the wire
A = cross-sectional area of the wire
q = charge of an electron
V = potential difference across the wire
v_d = drift velocity of electrons in the wire

First, let's find the current flowing through the wire using Ohm's Law:

I = V / R

Here, V is the potential difference (10.0 V) and R is the resistance (0.200 Ω). Plugging the values into the formula, we get:

I = 10.0 V / 0.200 Ω
= 50.0 A

Now, we need to calculate the cross-sectional area of the wire. The formula to find the area of a circle is:

A = π * r^2

Here, r is the radius of the wire (3.00 × 10^-3 m). Plugging the value into the formula, we get:

A = π * (3.00 × 10^-3 m)^2
= 2.83 × 10^-5 m^2

Next, let's calculate the charge of an electron. The charge of an electron is equal to the elementary charge (e), which is approximately 1.60 × 10^-19 C.

q = 1.60 × 10^-19 C

Finally, we can calculate the density of free electrons using the formula mentioned earlier:

n = (I * A) / (q * V * v_d)

Plugging in the values we calculated:

n = (50.0 A * 2.83 × 10^-5 m^2) / (1.60 × 10^-19 C * 10.0 V * 2.98 × 10^-4 m/s)
= 5.89 × 10^28 m^-3

Therefore, the density of free electrons in the wire is approximately 5.89 × 10^28 electrons per cubic meter.