Final answer:
To calculate the magnetic energy stored in a coaxial transmission line, first, use Ampère's law to find the magnetic field. Then integrate the magnetic energy density over the volume between the conductors to find the total energy stored.
Step-by-step explanation:
To determine the magnetic energy stored in the insulating medium of a coaxial transmission line, we first need to calculate the magnetic field in the region between the two conductors. Ampère's law is used for this purpose. The magnetic field inside a coaxial cable is given by the equation B = (µ₀I)/(2πr) for the region between the inner and outer conductors, where µ₀ is the permeability of free space, I is the current flowing through the cable, and r is the radial distance from the center.
The magnetic energy per unit volume (energy density) is then given by the equation u = B²/(2µ₀). To obtain the total magnetic energy stored in a length 'l' of the cable, we integrate this energy density over the volume between the two conductors:
∫ B²/(2µ₀) * dV,
where dV is the differential volume element for the cylindrical shell between the inner radius r1 and outer radius r2 of the cable. The integral simplifies to:
U = l(µ₀I²)/(4π²)∫₂₁ᵢᵒ dr/r,
which evaluates to:
U = (lµ₀I²)/(4π²)ln(r2/r1).
For the given values, where l = 3 m, I = current, r1 = 5 cm, and r2 = 10 cm, the magnetic energy stored is:
U = (3m * µ₀ * I²)/(4π²) * ln(10 cm / 5 cm).