Figure 33-74 shows a cylindrical resistor of length l,...

Figure $33-74$ shows a cylindrical resistor of length $l,$ radius $a$, and resistivity $\rho,$ carrying current $i$ (a) Show that the Poynting vector $\vec{S}$ at the surface of the resistor is every where directed normal to the surface, as shown. (b) Show that the rate $P$ at which energy flows into the resistor through its cylindrical surface, calculated by integrating the Poynting vector over this surface, is cqual to the rate at which thermal energy is produced: $$\int \vec{S} \cdot d \vec{A}=i^{2} R$$ where $d \vec{A}$ is an element of area on the cylindrical surface and $R$ is the resistance.

Figure $33-74$ shows a cylindrical resistor of length $l,$ radius $a$, and resistivity $\rho,$ carrying current $i$ (a) Show that the Poynting vector $\vec{S}$ at the surface of the resistor is every where directed normal to the surface, as shown. (b) Show that the rate $P$ at which energy flows into the resistor through its cylindrical surface, calculated by integrating the Poynting vector over this surface, is cqual to the rate at which thermal energy is produced:
$$\int \vec{S} \cdot d \vec{A}=i^{2} R$$
where $d \vec{A}$ is an element of area on the cylindrical surface and $R$ is the resistance.

Key Concepts

Ohm's Law and Resistivity

Ohm's Law relates the electric field within a material to the current density through it, mediated by the material's resistivity. The concept of resistivity provides a measure of how strongly a material opposes the flow of electric current, and, along with the geometrical dimensions of a conductor, determines the overall resistance. This relationship is key to linking the applied electric fields and the resulting energy dissipation in electrical components.

Joule Heating

Joule heating, or resistive heating, is the process by which electrical energy is converted into thermal energy in a resistor. It is quantitatively described by the expression i²R, which relates the current through the resistor and its resistance to the rate at which heat is generated. This concept is crucial for understanding energy dissipation in electrical circuits.

Electromagnetic Energy Conservation

This principle, often derived from the Poynting theorem, states that the decrease in electromagnetic energy within a volume is accounted for by the net flow of electromagnetic energy across the boundary and by energy conversion within the material. It provides the framework for equating the energy entering a system through its boundaries to the energy dissipated or stored within that system.

Poynting Vector

The Poynting vector is a fundamental concept in electromagnetism representing the rate of energy flow per unit area, defined as the cross product of the electric and magnetic fields. It encapsulates how electromagnetic energy is transported through space and is central to understanding how energy is delivered to systems, such as resistors, where it may be converted into other forms like heat.

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