Final answer:
The magnitude of the current density in a wire with a circular cross-section can be calculated using the equation J(r) = Cr². To find the current just outside the wire, evaluate the current density at a radius slightly larger than the wire's radius. In this case, the magnitude of the current density just outside the wire is 10 A/m².
Step-by-step explanation:
The magnitude of the current density in a wire with a circular cross-section can be calculated using the equation J(r) = Cr², where J is the current density in amps per square meter and r is the radius in meters. The constant C represents the current flowing through the wire. To find the current just outside the wire, we need to evaluate the current density at a radius slightly larger than the radius of the wire.
In this case, the current density is given as J(r) = Cr² with C = 10³ A/m⁴. We are asked to find the current just outside the wire, which has a radius of 1.0 mm. Converting the radius to meters (1.0 mm = 0.001 m), we plug it into the equation:
J(0.001 m) = (10³ A/m⁴)(0.001 m)²
J(0.001 m) = 10 A/m²
Therefore, the magnitude of the current density just outside the wire is 10 A/m².