What current would it need to carry in order to experience a very modest 1.0*10-3n⋅m torque?

By using the formula:

Tau = (I) (A) (B) Sinθ

Get the area first of the circle:

A = πr^2

A = π (0.08m) ^2

A = 0.02011 m^2

Then solve,

B (earth) = 5x10^-5 T

θ = 0, since the angle of the wire is oriented for maximum torque in the earth's field. Which means the angle is 0.

Reorder the formula to solve for the charge (I):

I = (tau) / (A) (B) Sinθ

I = (tau) / (A) (B)

I = (1.0x10^-3) / (0.02011) (5x10^-5)

I = 994 A