A length of wire has a radius of 3.00 �~ 10-3 m and a resistance of 0.200 ƒ¶. 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.
To calculate the density of free electrons in the wire, we can use the following steps:
Step 1: Calculate the cross-sectional area of the wire.
The cross-sectional area (A) of a wire can be calculated using the formula A = πr^2, where r is the radius of the wire. In this case, the radius is given as 3.00 × 10^(-3) m.
So, A = π(3.00 × 10^(-3))^2 = 2.827 × 10^(-5) m^2
Step 2: Calculate the current flowing through the wire.
Current (I) can be calculated using Ohm's Law, which states that I = V / R, where V is the potential difference across the wire and R is the resistance. In this case, the potential difference is given as 10.0 V and the resistance is given as 0.200 Ω.
So, I = 10.0 V / 0.200 Ω = 50.0 A
Step 3: Calculate the charge passing through the wire per unit time.
The charge passing through a wire (Q) can be calculated using the formula Q = I × t, where I is the current and t is the time. However, in this case, we need to find the charge per unit time.
So, Q_per_unit_time = I
Step 4: Calculate the speed of the electrons.
The electron drift speed (v_d) can be calculated using the formula v_d = I / (n × A × e), where I is the current, n is the density of free electrons, A is the cross-sectional area of the wire, and e is the charge of an electron (-1.60 × 10^(-19) C).
In this case, the electron drift speed is given as 2.98 × 10^(-4) m/s.
So, n = I / (v_d × A × e) = 50.0 A / (2.98 × 10^(-4) m/s × 2.827 × 10^(-5) m^2 × -1.60 × 10^(-19) C)
Step 5: Calculate the density of free electrons.
Using the given data and the calculated values, we can now calculate the density of free electrons.
n = 50.0 A / (2.98 × 10^(-4) m/s × 2.827 × 10^(-5) m^2 × -1.60 × 10^(-19) C) = -1.30 × 10^29 electrons/m^3
Therefore, the density of free electrons in the wire is approximately -1.30 × 10^29 electrons/m^3. Note that the negative sign indicates that the electrons are negatively charged.