91Ó°ÊÓ

A cylindrical conductor with a circular cross section has a radius \(a\) and a resistivity \(\rho\) and carries a constant current \(I\). (a) What are the magnitude and direction of the electric-field vector \(\overrightarrow{\boldsymbol{E}}\) at a point just inside the wire at a distance \(a\) from the axis? (b) What are the magnitude and direction of the magneticfield vector \(\overrightarrow{\boldsymbol{B}}\) at the same point? (c) What are the magnitude and direction of the Poynting vector \(\overrightarrow{\boldsymbol{S}}\) at the same point? (The direction of \(\vec{S}\) is the direction in which electromagnetic energy flows into or out of the conductor.) (d) Use the result in part (c) to find the rate of flow of energy into the volume occupied by a length \(l\) of the conductor. (Hint: Integrate \(\vec{S}\) over the surface of this volume.) Compare your result to the rate of generation of thermal energy in the same volume. Discuss why the energy dissipated in a currentcarrying conductor, due to its resistance, can be thought of as entering through the cylindrical sides of the conductor.

Step by step solution

01

Find electric-field magnitude

Using Ohm's Law, the electric field inside the wire is related to the current density \( J \) and resistivity \( \rho \). The current density is given by \( J = \frac{I}{A} \), where \( A = \pi a^2 \) is the cross-sectional area. Hence, \( E = \rho J = \rho \frac{I}{\pi a^2} \).

02

Determine electric-field direction

The direction of the electric field is along the axis of the cylindrical conductor, which is the same direction as the current \( I \).

03

Calculate magnetic-field magnitude

Using Ampère's Law, the magnetic field at a distance \( a \) from the axis of a cylindrical wire is given by \( B = \frac{\mu_0 I}{2\pi a} \), where \( \mu_0 \) is the permeability of free space.

04

Determine magnetic-field direction

The direction of the magnetic field follows the right-hand rule: if the thumb points in the direction of the current, the magnetic field curls around the conductor.

05

Calculate Poynting vector magnitude

The Poynting vector magnitude is given by \( S = \frac{1}{\mu_0} (E \cdot B) \). Substitute \( E = \rho \frac{I}{\pi a^2} \) and \( B = \frac{\mu_0 I}{2\pi a} \) to get \( S = \frac{\rho I^2}{2 (\pi a)^3} \).

06

Determine Poynting vector direction

The Poynting vector is directed radially inward, indicating the flow of electromagnetic energy into the conductor.

07

Integrate Poynting vector over surface

Integrate \( S \) over the surface of the cylinder to find the total power: \[ P = \int S \cdot dA = (\frac{\rho I^2}{2 (\pi a)^3}) \cdot (2 \pi a l) = \frac{\rho I^2 l}{\pi a^2} \].

08

Compare power to thermal energy generation

The rate of thermal energy generation in the conductor due to resistance \( R \) is \( P = I^2 R \), and since \( R = \rho \frac{l}{\pi a^2} \), both expressions for power match, verifying the flow of energy inward corresponds to thermal generation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Field in a Cylindrical Conductor

The electric field inside a cylindrical conductor is crucial to understand, especially when dealing with electrically charged systems. Imagine a wire carrying a constant current; inside, an electric field is established to drive this flow of charge. According to Ohm's Law, the electric field

• is proportional to the current density;
• depends on the material's resistivity (\(\rho\)).

Ohm's Law can be expressed as \(E = \rho J\), where \(J = \frac{I}{A}\) is the current density and \(A = \pi a^2\) (being the cross-sectional area of the wire). Thus, the magnitude of the electric field at a point just inside the wire is given by \(E = \rho \frac{I}{\pi a^2}\). The direction of this electric field is along the length of the wire, in the same direction as the current flow.
Understanding the electric field inside the wire helps explain how charges move and how electric energy is transformed within the conductor.

Ohm's Law and its Implications

Ohm's Law is a fundamental principle in understanding electric circuits and materials. It defines the linear relationship between voltage, current, and resistance in a conductor: \( V = IR \).
This law implies several important characteristics of a system:

• A higher resistivity material (like a less conductive wire) will have a larger electric field for the same amount of current compared to a lower resistivity material.
• The resistance \( R \) itself is a physical property given by \( R = \rho \frac{l}{A} \), where \( l \) is the length and \( A \) is the cross-sectional area of the material.

This equation illustrates why using longer wires or those with smaller cross sections results in larger resistances. Ohm's Law helps in designing electrical systems by selecting the proper materials and configurations to control and optimize the electric current flow. Understanding how resistance and other factors influence current and voltage is key in many technological applications, from simple light circuits to complex communication systems.

Magnetic Field Due to a Current Carrying Conductor

Whenever current flows through a conductor, it generates a magnetic field around it, which is described by Ampère's Law. In the case of a cylindrical conductor, we use Ampère's Law to calculate the magnetic field at a point \( a \) distance from the middle axis of the conductor. The magnetic field \( B \) has a magnitude of \( B = \frac{\mu_0 I}{2\pi a} \), where\( \mu_0 \)is the permeability of free space.
The direction of this magnetic field is determined by the right-hand rule. If you curl your hand around the conductor with your thumb pointing in the direction of the current flow, your fingers show the direction the magnetic field lines wrap around the conductor. This magnetic field can be visualized as concentric circles around the conductor.
Understanding the behavior of these fields provides insights into electromagnetic interactions, essential for applications like motors, transformers, and inductors within electric circuits.

Poynting Vector: Energy Flow in an Electromagnetic Field

The Poynting vector represents the power per unit area carried by an electromagnetic wave, and it's crucial for understanding energy flow in fields surrounding conductors. It is given by\( \vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B}) \).
Inside a cylindrical conductor:

• We find its magnitude by calculating \( S = \frac{\rho I^2}{2 (\pi a)^3} \).
• The direction of the Poynting vector indicates the direction of energy flow, which is inward into the conductor at our point of interest.

This inward energy flow corresponds to how thermal energy builds up due to resistance when a current passes through. Integrating the Poynting vector over a surface gives the total power flowing into the cylinder. It identifies how much exerted energy converts to thermal energy within the conductive material, confirming how electromagnetic energy leads to the warming of the conductor due to resistance.