The cylindrical shell, \(1 \mathrm{~cm}<\rho<1.2 \mathrm{~cm}\), is composed of a conducting material for which \(\sigma=10^{6} \mathrm{~S} / \mathrm{m}\). The external and internal regions are nonconducting. Let \(H_{\phi}=2000 \mathrm{~A} / \mathrm{m}\) at \(\rho=1.2 \mathrm{~cm}\). Find \((a) \mathbf{H}\) everywhere; \((b) \mathbf{E}\) everywhere; \((c)\langle\mathbf{S}\rangle\) everywhere.

Ampere's Law is a fundamental principle in electromagnetic field theory that relates an electric current or a changing electric field to the magnetic field it produces. The law is encapsulated by the mathematical expression \[\oint \mathbf{H} \cdot d\mathbf{l} = I_{enc} + \epsilon_0 \frac{d\Phi_E}{dt}\]where \(\mathbf{H}\) is the magnetic field intensity, \(d\mathbf{l}\) is the differential vector element of the closed path surrounding the current, \(I_{enc}\) is the electric current enclosed within the path, \(\epsilon_0\) is the vacuum permittivity, and \(\frac{d\Phi_E}{dt}\) is the change in electric flux through the enclosed area. In the context of the textbook solution, the cylindrical shell generates a magnetic field due to the current flowing through the conductor. However, since only the shell itself has conductivity and the inside and outside regions are nonconducting, Ampere's Law indicates that the magnetic field \(\mathbf{H}\) will only be present in the shell where current exists.

The Poynting vector is a vector quantity used in the field of electromagnetics to represent the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. Mathematically, it is given by the expression \[\langle\mathbf{S}\rangle = \frac{1}{\text{\mu}_0} \mathbf{E}\times \mathbf{H}\]where \(\mathbf{E}\) is the electric field, \(\mathbf{H}\) is the magnetic field, and \(\text{\mu}_0\) is the permeability of free space. The Poynting vector has both magnitude and direction, with the direction being perpendicular to both the electric and magnetic field vectors. In the solution provided, the Poynting vector \(\langle\mathbf{S}\rangle\) measures the rate at which electromagnetic energy is conveyed outward from the cylindrical shell.

Ohm's Law is one of the most well-known principles in the field of electrical engineering and relates the voltage across a conductor to the current through it, proportional to the resistance of the conductor. The classical form of the law is given as \(V = IR\), where \(V\) is voltage, \(I\) is current, and \(R\) is resistance. Ohm's Law can also be expressed in terms of the electric field \(\mathbf{E}\) and the current density \(\mathbf{J}\), especially in the study of electromagnetic fields, through the relation \(\mathbf{J} = \sigma \mathbf{E}\), where \(\sigma\) is the material's electrical conductivity. This implies that for a given electric field, a higher conductivity material would allow more current to flow. In our exercise, this relationship lets us calculate the electric field within the conducting material of the cylindrical shell based on the identified value of the magnetic field.

A magnetic field is a vector field represented by the symbol \(\mathbf{B}\) or \(\mathbf{H}\) that exerts a magnetic force on moving electric charges and magnetic dipoles. Magnetic fields are generated by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The direction of the magnetic field is indicated by the alignment of iron filings or the orientation of a compass needle. In the context of this exercise, the magnetic field inside and outside the cylindrical shell is zero because there are no currents in these regions. Within the conducting shell, however, the magnetic field is due to the current, and Ampere's Law is used to determine its magnitude and direction.

The electric field is a concept in physics that represents the force that would be exerted on a stationary charged particle at any given point in space. Represented by the symbol \(\mathbf{E}\), its direction and magnitude describe the force per unit charge that would be felt by a positive test charge placed in the field. Electric fields are created by electric charges or by time-varying magnetic fields. Within the solution to our exercise, the electric field \(\mathbf{E}\) in the nonconducting regions is zero due to the absence of any current or charge distribution. In the conducting region of the shell, we calculate the electric field from the knowledge of the current density and conductivity, making use of Ohm's Law.

Electromagnetic conductivity, usually referred to simply as electrical conductivity, is a property of a material that quantifies how easily it can conduct an electric current. Conductivity is denoted by the Greek letter \(\sigma\), and its SI unit is the siemens per meter (S/m). Materials with high conductivity, such as metals, offer low resistance to the flow of charge, while insulators, with low conductivity, oppose such flow. In electromagnetic field analysis, conductivity is key to understanding how electric fields behave within different materials. In our exercise, the conductivity of the cylindrical shell is given, allowing us to calculate the electric field within the shell when combined with the magnetic field information using the principles of electromagnetics such as Ohm's Law and the concept of current density.