An 18 gauge copper wire with a diameter of 1.02 mm carries a constant current of 1.65 A to a 200 W lamp. The free electron density in the wire is 8.5 X 1028 per cubic meter. Find (a) the current density and (b) the drift speed.

To find the current density, we can use the formula:

Current density (J) = Current (I) / Cross-sectional area (A)

First, let's calculate the cross-sectional area of the wire. The diameter of the wire is given as 1.02 mm, so the radius (r) can be calculated as half of the diameter:

r = 1.02 mm / 2 = 0.51 mm = 0.00051 m

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

A = π * r^2

A = 3.14 * (0.00051 m)^2

A = 8.18 x 10^-7 m^2

Now, we can calculate the current density (J) using the given current (I) and the cross-sectional area (A):

J = 1.65 A / 8.18 x 10^-7 m^2

J ≈ 2.01 x 10^6 A/m^2

Therefore, the current density is approximately 2.01 x 10^6 A/m^2.

To find the drift speed (v), we can use the formula:

v = I / (n * A * q)

where: I = current (1.65 A) n = free electron density (8.5 x 10^28 electrons/m^3) A = cross-sectional area (8.18 x 10^-7 m^2) q = charge of an electron (1.6 x 10^-19 C)

Substituting the values into the formula:

v = 1.65 A / (8.5 x 10^28 electrons/m^3 * 8.18 x 10^-7 m^2 * 1.6 x 10^-19 C)

v ≈ 1.16 x 10^-4 m/s

Therefore, the drift speed is approximately 1.16 x 10^-4 m/s.