Question

A material is said to be ohmic if an electric field, \(\vec{E}\), in the material gives rise to current density \(\vec{J}=\sigma \vec{E},\) where the conductivity, \(\sigma\), is a constant independent of \(\vec{E}\) or \(\vec{J}\). (This is the precise form of Ohm's Law.) Suppose in some material an electric field, \(\vec{E}\), produces current density, \(\vec{J},\) not necessarily related by Ohm's Law; that is, the material may or may not be ohmic. a) Calculate the rate of energy dissipation (sometimes called ohmic heating or joule heating) per unit volume in this material, in terms of \(\vec{E}\) and \(\vec{J}\). b) Express the result of part (a) in terms of \(\vec{E}\) alone and \(\vec{J}\) alone, for \(\vec{E}\) and \(\vec{J}\) related via Ohm's Law, that is, in an ohmic material with conductivity \(\sigma\) or resistivity \(\rho .\)

Short answer

Answer: The expressions for the rate of energy dissipation per unit volume (W) are: - In terms of electric field (E) and current density (J): \(W = \frac{\vec{J} \cdot \vec{E}}{V}\) - In terms of electric field (E) alone for an ohmic material: \(W = \frac{\sigma |\vec{E}|^2}{V}\) - In terms of current density (J) alone for an ohmic material: \(W = \frac{|\vec{J}|^2 \rho}{V}\)

Step-by-step solution

(Step 1: Calculate the rate of energy dissipation)

(Recall the formula for power in terms of electric field and current density: \(P = \vec{J} \cdot \vec{E}\). The rate of energy dissipation per unit volume (\(W\)) is power per unit volume, so we simply divide the power by the volume: \(W = \frac{P}{V}\). Now, substituting the formula for power into the expression, we get \(W = \frac{\vec{J} \cdot \vec{E}}{V}\). Knowing that power has the unit of W (watt) and volume has the unit of m^3, we can deduce the unit of rate of energy dissipation: W/m^3.)

(Step 2: Express the result in terms of \(\vec{E}\) alone for Ohm's Law)

(For an ohmic material, we have \(\vec{J} = \sigma \vec{E}\). Plugging this into the expression for rate of energy dissipation, we get \(W = \frac{(\sigma \vec{E}) \cdot \vec{E}}{V} = \frac{\sigma \vec{E} \cdot \vec{E}}{V}\). Now, simplifying the expression gives \(W = \frac{\sigma |\vec{E}|^2}{V}\). With the rate of energy dissipation expressed in terms of the electric field alone, the unit remains the same: W/m^3.)

(Step 3: Express the result in terms of \(\vec{J}\) alone for Ohm's Law)

(To express the result in terms of the current density alone, we need to find how \(\vec{E}\) is related to \(\vec{J}\) in an ohmic material. We have \(\vec{E} = \rho \vec{J}\) for an ohmic material, where \(\rho\) is the resistivity of the material. Now, plugging this into the expression for rate of energy dissipation and replacing \(\sigma\) by \(\frac{1}{\rho}\), we have \(W = \frac{(\frac{1}{\rho} \rho \vec{J}) \cdot (\rho \vec{J})}{V} = \frac{|\vec{J}|^2 \rho}{V}\). With the rate of energy dissipation expressed in terms of the current density alone, the unit remains the same: W/m^3.) So, the rate of energy dissipation per unit volume \(W\) can be written in terms of: - \(\vec{E}\) and \(\vec{J}\): \(W = \frac{\vec{J} \cdot \vec{E}}{V}\) - \(\vec{E}\) alone (ohmic material): \(W = \frac{\sigma |\vec{E}|^2}{V}\) - \(\vec{J}\) alone (ohmic material): \(W = \frac{|\vec{J}|^2 \rho}{V}\)

Key concepts

Current Density

Current density is a core aspect of understanding electrical flows and is denoted by the symbol \( \vec{J} \). It describes how much electric current flows through a specific area of a material. Think of it like the number of cars crossing a toll booth lane, but instead of cars, we consider electric charges.

In mathematical terms, current density is expressed in amperes per square meter \( (A/m^2) \). This quantity is crucial because it helps physicists and engineers determine how a material will behave when an electric field \( \vec{E} \) is applied. The relationship between current density and electric field in an ohmic material is direct, following Ohm's Law, where \( \vec{J} = \sigma \vec{E} \). Here, \( \sigma \) represents the conductivity of the material.

Understanding current density is important for designing efficient and safe electrical systems, as it impacts energy dissipation and overall performance.

Electric Field

An electric field \( \vec{E} \) is a force field created around charged particles. It acts on other charges in the field, causing them to move if free to do so. The electric field is measured in volts per meter \( (V/m) \), showing that it is a determinant of how a charge will behave when placed within this field.

When a material is placed in an electric field, charges within it feel a force. This is the underlying concept of how electricity is conducted through materials. For ohmic materials, there is a linear relationship between the electric field and current density. As the electrical field strength increases, so does the current density, assuming constant temperature and material properties.

• The concept of an electric field is fundamental to understanding how electrical circuits function.
• It's like the invisible force pushing charged particles, determined both by the charges creating it and the charge experiencing it.

Energy Dissipation

Energy dissipation in electrical materials is often referred to as ohmic or Joule heating. It describes how electrical energy is converted into heat as current flows through a material. This process is critical in both understanding how materials behave under electrical stress and in practical applications, like heaters and motors.

According to the step-by-step solution, the rate of energy dissipation \( W \), is calculated as power per unit volume. For non-ohmic materials, it is given by \( W = \frac{\vec{J} \cdot \vec{E}}{V} \), whereas for ohmic materials, two distinct expressions can be used:

• In terms of electric field: \( W = \frac{\sigma |\vec{E}|^2}{V} \)
• In terms of current density: \( W = \frac{|\vec{J}|^2 \rho}{V} \)

These formulas highlight how the material's characteristics, like conductivity \( \sigma \) and resistivity \( \rho \), influence how much electrical energy is turned into heat. Understanding this helps in designing systems that manage heat effectively, ensuring devices operate safely and efficiently.

Conductivity

Conductivity is a property of materials that indicates how well they can conduct electric current. Represented by \( \sigma \), it measures the ease with which electrical charges move through a material when an electric field is applied. The units of conductivity are siemens per meter \( (S/m) \).

In ohmic materials, the higher the conductivity, the lower the resistance, allowing more current to flow for a given electric field. The relationship \( \vec{J} = \sigma \vec{E} \) indicates that current density is directly proportional to both conductivity and the electric field.

Conductivity is an important factor in material selection for electrical applications. Different materials have different conductivities. For instance:

• Metals typically have high conductivity, making them ideal for wires and circuits.
• Insulating materials have low conductivity, used to prevent unauthorized flow of electricity.

With this understanding, engineers can make informed decisions in the design of electrical systems.

Resistivity

Resistivity is the measure of how strongly a material opposes the flow of electric current. It is represented by the symbol \( \rho \) and is the reciprocal of conductivity (\( \rho = \frac{1}{\sigma} \)). The units of resistivity are ohm-meters \( (\Omega \, m) \).

In ohmic materials, resistivity remains constant with changing electric fields which corresponds to a linear relationship in Ohm's Law, \( \vec{E} = \rho \vec{J} \).

Higher resistivity means that the material is less conductive, making it suitable for applications like resistors, which are used to control current flow in circuits. This is analogous to bottlenecks in traffic; a higher resistivity means a bigger traffic jam for electric charges.

• This concept aids in selecting appropriate materials based on desired electrical resistance.
• Special materials with tailored resistivity are used extensively in electronic components to achieve specified electrical characteristics.

Understanding resistivity helps us grasp how different materials will respond to electrical currents, a vital part of planning and designing electrical circuits.