Question

A resistor consists of a solid cylinder of radius $r$ and length $L$. The resistor has resistance $R$ and is carrying current $i$. Use the Poynting vector to calculate the power radiated out of the surface of the resistor.

Short answer

Based on the given solution, answer the following question: **Question**: Calculate the power radiated out of the surface of a cylindrical resistor with current $i$, length $L$, radius $r$, and conductivity $\sigma$. **Answer**: The power radiated out of the surface of the cylindrical resistor can be calculated by integrating the Poynting vector over the entire surface of the resistor. Using the provided steps in the solution, we obtain the simplified expression: $P={\int }_0^L{\int }_0^{2\pi }Srd\varphi dz$ Substitute the expression for $\overrightarrow{S}=\frac{1}{{\mu }_0}(\frac{J}{\sigma }\times \frac{i}{2\pi r})$ and then evaluate the integral to find the power radiated out of the surface of the cylindrical resistor.

Step-by-step solution

Understand the Poynting vector

The Poynting vector ($\overrightarrow{S}$) is a vector which represents the power flow density (power per unit area) in an electromagnetic field. It is equal to the cross product of the electric field ($\overrightarrow{E}$) and the magnetic field ($\overrightarrow{H}$), divided by the permeability of free space (${\mu }_0$). The formula for the Poynting vector is: $\overrightarrow{S}=\frac{1}{{\mu }_0}(\overrightarrow{E}\times \overrightarrow{H})$ The Poynting vector is useful for determining the power flow through a given area in space.

Determine the electric field in the resistor

The electric field ($\overrightarrow{E}$) inside the resistor can be found using Ohm's law for a cylindrical resistor. Ohm's law for a cylindrical resistor is defined as: $\overrightarrow{E}=\frac{J}{\sigma }$ Where $J$ is the current density in the resistor, and $\sigma$ is the conductivity of the resistor. The current density can be calculated as the current ($i$) divided by the cross-sectional area of the resistor ($A=\pi r^2$): $J=\frac{i}{\pi r^2}$

Determine the magnetic field around the resistor

The magnetic field ($\overrightarrow{H}$) around the resistor can be calculated using Ampere's law, which states that the line integral of the magnetic field ($\overrightarrow{H}$) around a closed path is equal to the total current enclosed by the path. For a cylindrical resistor, the magnetic field is concentric circles around the resistor, and its magnitude can be computed as follows: $H=\frac{i}{2\pi r}$

Calculate the Poynting vector

Now that we have both the electric field and the magnetic field, we can calculate the Poynting vector ($\overrightarrow{S}$) using their cross product, as mentioned in step 1: $\overrightarrow{S}=\frac{1}{{\mu }_0}(\overrightarrow{E}\times \overrightarrow{H})$ Substitute the expressions for $\overrightarrow{E}$ and $\overrightarrow{H}$ found in steps 2 and 3: $\overrightarrow{S}=\frac{1}{{\mu }_0}(\frac{J}{\sigma }\times \frac{i}{2\pi r})$

Calculate power radiated out of the surface of the resistor

Finally, we can compute the power radiated out of the surface of the resistor by integrating the Poynting vector over the entire surface of the resistor: $P=\int \overrightarrow{S}\cdot d\overrightarrow{A}$ Considering the cylindrical symmetry, we can simplify the integral and solve for the power radiated: $P={\int }_0^L{\int }_0^{2\pi }Srd\varphi dz$ Now, plug in the expression for $\overrightarrow{S}$ from step 4 and evaluate the integral. Simplify the expression and the power radiated out of the surface of the resistor will be obtained.

Key concepts

Resistance

The concept of resistance is crucial in understanding how electrical devices manage currents. In essence, it measures the degree to which a material opposes the flow of electric current. Imagine water flowing through a pipe: if the pipe is narrow, it presents more resistance to the flow of water. Similarly, electrons moving through a conductor encounter resistance due to interactions within the material.

Resistance in a material is quantified by the resistivity, often represented by the Greek letter \rho (rho). The formula for resistance R of a uniform cylindrical conductor, like the resistor in our exercise, is given by:
$$R=\frac{\rho L}{A}$$
where L is the length, and A is the cross-sectional area ($\frac{\rho L}{\text{A}}$). The units of resistance are Ohms (Ω).

Factors Influencing Resistance

Several factors affect the resistance of a conductor:

• Material: Materials with higher resistivity ($\rho$) have higher resistance.
• Length: Longer conductors have greater resistance as there's more material to oppose electron flow.
• Area: A larger cross-sectional area reduces resistance by allowing more paths for electron flow.
• Temperature: Higher temperatures usually increase resistance as atoms vibrate more and therefore scatter electrons more.

Ohm's Law

Ohm's Law is a foundational principle in electrical engineering and physics that describes the relationship between voltage (V), current (I), and resistance (R). It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance of the conductor. The law can be expressed as:
$$V=IR$$
This implies that if we know any two of the three quantities (voltage, current, resistance), we can calculate the third. In the context of the exercise, Ohm's Law helps us to understand the electric field inside the resistor. The electric field E, which is the force experienced by a charge due to the voltage, relates to current density J and conductivity σ (the inverse of resistivity) as follows:
$$E=\frac{J}{\text{σ}}$$
When the resistance, current, and material properties of the resistor are known, we can apply Ohm's Law to deduce how the electric field behaves inside the solid cylinder, which is essential in finding the power radiated.

Electromagnetic Fields

Electromagnetic fields (EMFs) are fundamental forces of nature, consisting of electric and magnetic fields that propagate as waves through space and matter. These fields are intrinsic to the operation of numerous technologies, including the flow of electricity in conductors.

Electric fields arise from electric charges, which can be static or moving. When charges move, as in a current-carrying wire, they also create magnetic fields. According to electromagnetism theory, these fields are interdependent; a time-varying magnetic field generates an electric field, and vice versa, a phenomenon described by Faraday's Law and Maxwell's equations.

Applications of EMFs

EMFs are pivotal in a wide array of applications:

• Communication systems: Radio waves are EMFs utilized for broadcasting and communication devices.
• Medical imaging: Magnetic fields play a crucial role in MRI machines, allowing us to visualize internal body structures.
• Induction cooking: Changing magnetic fields can generate heat in conductive pans.
• Power generation and transformation: EMFs are essential in the generation of electricity in power plants and its transformation in transformers.

Ampere's Law

Ampere's Law is one of the basic equations used in the study of electromagnetism, relating the magnetic field B in space to the current I that produces it. The law states that the magnetic field around a closed loop is proportional to the current passing through the loop. Mathematically, Ampere's Law can be written as:
$$\oint \overrightarrow{B}\bullet d\text{→l}={\text{μ}}_0{I}_{\text{enc}}$$
Here, μ₀ is the permeability of free space and I_enc represents the net current enclosed by the loop. In the context of our cylindrical resistor, Ampere's Law allows us to calculate the magnetic field H generated by the current flowing through the resistor.

Magnetic fields created by currents are what enable electric motors and generators to function. These fields can be visualized as concentric circles around the wire, with the direction given by the right-hand rule. Applying Ampere's Law makes it possible to predict the strength and direction of this magnetic field, which is essential when calculating the Poynting vector as part of determining the power radiated out of the resistor's surface.