Final answer:
The drift velocity of electrons in the copper wire is approximately 0.14 m/s. The current density in the wire is approximately 981 A/m². The electric field in the wire is approximately 1.69 × 10^-5 V/m.
Step-by-step explanation:
To estimate the electron drift speed, we can use the formula I = nAvd, where I is the current, n is the number of free electrons per cubic meter, A is the cross-sectional area of the wire, and vd is the drift velocity. In this case, the current is given as 3.2μA, and we can calculate the area using the formula A = πr^2, where r is the radius of the wire. The radius can be found by dividing the diameter by 2, so in this case, the radius is 0.325 mm. Now, using the density of copper (8.96 g/cm³), the molar mass of copper (63.5 g/mol), and Avogadro's number (6.02 × 10^23 atoms/mol), we can find the number of free electrons per cubic meter, which is approximately 8.50 × 10^28 electrons/m³.
Plugging in all the values, we can rearrange the formula to solve for vd:
vd = I / (nA)
Plugging in the values, we get:
vd = (3.2 × 10^-6 A) / (8.5 × 10^28 electrons/m³ × (π × (0.325 × 10^-3 m)^2))
Simplifying the expression, we find that the electron drift speed is approximately 0.14 m/s.
To find the current density, we can use the formula J = I / A, where J is the current density. Plugging in the values, we get:
J = (3.2 × 10^-6 A) / (π × (0.325 × 10^-3 m)^2)
Simplifying the expression, we find that the current density is approximately 981 A/m².
To find the electric field, we can use the formula E = J / σ, where E is the electric field and σ is the conductivity of copper. The conductivity of copper can be calculated using the formula σ = nq^2τ / m, where q is the charge of an electron, τ is the relaxation time, and m is the mass of a copper atom. The charge of an electron is 1.60 × 10^-19 C, and the molar mass of copper is 63.5 g/mol. Now, using the density of copper (8.96 g/cm³) and Avogadro's number (6.02 × 10^23 atoms/mol), we can find the number of copper atoms per cubic meter, which is approximately 8.49 × 10^28 atoms/m³. To find τ, we can use the formula τ = l / vd, where l is the mean free path of the electrons. Assuming a mean free path of 40 nm (4 × 10^-8 m), we can calculate τ. Plugging in all the values, we can find σ, which is approximately 5.81 × 10^7 S/m.
Now, plugging in the values into the formula for electric field, we get:
E = (981 A/m²) / (5.81 × 10^7 S/m)
Simplifying the expression, we find that the electric field is approximately 1.69 × 10^-5 V/m.