Final answer:
To find the value of b in terms of a if the magnetic field is zero outside the coaxial cable, we can use the fact that the net current crossing the area bounded by a circular path where r > b is zero. Therefore, the current passing through the inner conductor's cross-sectional area must be equal to the current passing through the outer conductor's cross-sectional area. Using the current density of the inner conductor and assuming uniform current density for the outer conductor, we can solve for b in terms of a.
Step-by-step explanation:
To find the value of b in terms of a if the magnetic field is zero outside the coaxial cable, we need to consider the magnetic fields inside the cable. Since the magnetic field is zero outside the cable, we can conclude that the net current crossing the area bounded by a circular path where r > b is zero. Therefore, the outer conductor must carry an equal and opposite current to balance the current in the inner conductor. This means that the current passing through the inner conductor's cross-sectional area is equal to the current passing through the outer conductor's cross-sectional area.
Using the current density αr³ for the inner conductor and assuming the current density is uniform for the outer conductor, we can set up an equation:
αa³ * π * a² = I * π * (b² - a²)
Here, α is the constant current density for the inner conductor and I is the current passing through both conductors. Simplifying the equation, we get:
αa⁵ = I(b² - a²)
Now, solving for b, we can rearrange the equation:
b² - a² = (αa⁵) / I
b² = (αa⁵) / I + a²
b = sqrt((αa⁵) / I + a²)