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Understanding how to calculate the magnetic field within a coaxial cable is essential when tackling problems in electromagnetism. Ampere's law, a fundamental principle in electromagnetism, relates the integrated magnetic field around a closed loop to the electric current passing through the loop. For a coaxial cable with currents flowing in opposite directions in the inner and outer conductors, the situation simplifies significantly.

To calculate the magnetic field at any point between the two conductors, we apply Ampere's law in the form \( B = \frac{\mu_{0} i}{2 \pi r} \) where \( \mu_{0} \) is the permeability of free space, \( i \) is the current, and \( r \) is the radial distance from the cable's axis. This formula holds true in the region between the conductors and gives us the magnitude of the magnetic field generated by the current within that region.

This concept is paramount for understanding various phenomena such as the transmission of electrical signals through cables and the design of electrical circuits.

Magnetic flux provides us with a measure of the total magnetic field passing through a given area. When dealing with coaxial cables, it is particularly useful to determine the link between the magnetic field and the geometry of the cable.

The infinitesimal magnetic flux \(d \Phi_{B}\) through a narrow strip inside a coaxial cable can be written as \( d \Phi_{B} = B \cdot da \), where \( da \) is the area of the strip, which can be expressed as \( l dr \) with \( l \) being the length of the cable and \( dr \) a small change in radius. Utilizing the previous calculation for \( B \), the expression becomes \( d \Phi_{B} = \frac{\mu_{0} i l dr} {2 \pi r} \). This integral will reveal the total magnetic flux between the inner and outer conductors which is important for further calculations such as finding the self-inductance of the coaxial cable.

Self-inductance is a property of a conductor that quantifies its ability to oppose changes in the electric current passing through it, a concept known as Lenz's law. When current flows through a coaxial cable, a magnetic field is created around it, and a change in this current will induce a changing magnetic field which, in turn, induces an electromotive force (EMF).

The self-inductance \( L \) of a coaxial cable can be directly calculated by the magnetic flux through the cable for a given current. According to the exercise, the inductance per unit length \( l \) can be expressed as \( L_\cdot = l \frac{\mu_{0}}{2 \pi} \ln \left(\frac{b}{a}\right) \). The ratio \( \frac{b}{a} \) compares the radii of the coaxial cable’s outer and inner conductors, and \( \mu_{0} \) is again the permeability of free space.

Knowing the self-inductance is critical in electronics to predict how a circuit will respond to changing currents, which affects the performance of devices such as transformers and inductors.

The energy stored in the magnetic field is a result of electrical energy being converted into magnetic energy when a current flows through a conductor. For a coaxial cable, this energy is directly proportional to the self-inductance of the cable as well as the square of the current passing through it.

The expression for the energy \( E_{m} \) stored in the magnetic field for a length \( l \) of the coaxial cable is \( E_{m}=\frac{1}{2}L_\cdot i^{2} \), where \( i \) is the current and \( L_\cdot \) is the inductance per unit length. Substituting the formula for \( L_\cdot \) yields the equation \( E_{m}= \frac{1}{2} \left[l \frac{\mu_{0}}{2 \pi} \ln \left(\frac{b}{a}\right)\right]i^{2} \). This energy represents the potential to do work and it is what enables inductors in electric circuits to release energy when needed, for instance, to smooth out electrical currents or to provide a surge of power.