A coaxial cable has an inside conductor of radius 'a' carrying a current 'I'.The current returns as a uniform current through the outside conductor with a inside radius 'b' and outside radius 'c'.What is the magnetic field at a distance 'r' from the center, for the following cases.
r<a, a<r<b, b<r<c, c<r
You seem to have posted this twice
To find the magnetic field at different distances from the center of a coaxial cable, you can use Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space.
1. For the case where r < a:
Since the point of interest is inside the inner conductor, the current enclosed by the loop is equal to the current carried by the inner conductor, I. Applying Ampere's law, the integral of the magnetic field around the loop is equal to μ₀I, where μ₀ is the permeability of free space (μ₀ ≈ 4π × 10⁻⁷ T m/A).
2. For the case where a < r < b:
In this case, the current enclosed by the loop is equal to the total current, I, as both the inner and outer conductors contribute to the current at this distance. By applying Ampere's law, the integral of the magnetic field around the loop is again equal to μ₀I.
3. For the case where b < r < c:
Similarly, the current enclosed by the loop is equal to the total current, I, as both the inner and outer conductors contribute to the current at this distance. Using Ampere's law, the integral of the magnetic field around the loop is μ₀I.
4. For the case where r > c:
Since the point of interest is outside the outer conductor, the current enclosed by the loop is again equal to the total current, I. Applying Ampere's law, the integral of the magnetic field around the loop is μ₀I.
In summary:
- For r < a: Magnetic field B = μ₀I.
- For a < r < b: Magnetic field B = μ₀I.
- For b < r < c: Magnetic field B = μ₀I.
- For r > c: Magnetic field B = μ₀I.
It is important to note that the magnetic field from a coaxial cable is independent of the distance from the center within the respective ranges.