Question

In a uniform, non-dielectric, conducting medium with unit relative permittivity, charge density \(\rho\), current density \(\mathbf{J}\), electric field \(\mathbf{E}\) and magnetic field \(\mathbf{B}\), Maxwell's electromagnetic equations take the form (with \(\left.\mu_{0} \epsilon_{0}=c^{-2}\right)\) (i) \(\nabla \cdot \mathbf{B}=0\), (ii) \(\nabla \cdot \mathbf{E}=\rho / \epsilon_{0}\), (iii) \(\nabla \times \mathbf{E}+\mathbf{B}=\mathbf{0}\), (iv) \(\nabla \times \mathbf{B}-\left(\hat{\mathbf{E}} / c^{2}\right)=\mu_{0} \mathbf{J}\), The density of stored energy in the medium is given by \(\frac{1}{2}\left(\epsilon_{0} E^{2}+\mu_{0}^{-1} B^{2}\right)\). Show that the rate of change of the total stored energy in a volume \(V\) is equal to $$ -\int_{V} \mathbf{J} \cdot \mathbf{E} d V-\frac{1}{\mu_{0}} \oint_{S}(\mathbf{E} \times \mathbf{B}) \cdot d \mathbf{S} $$ where \(S\) is the surface bounding \(V\). (The first integral gives the ohmic heating loss, whilst the second gives the electromagnetic energy flux out of the bounding surface. The vector \(\mu_{0}^{-1}(\mathbf{E} \times \mathbf{B})\) is known as the Poynting vector.)

Short answer

The rate of change of the total stored energy is \( -\int_V \mathbf{J} \cdot \mathbf{E} dV - \frac{1}{\mu_0} \oint_S (\mathbf{E} \times \mathbf{B}) \cdot d\mathbf{S} \) representing ohmic heating loss and the Poynting vector.

Step-by-step solution

Rate of Change of Stored Energy

Start by identifying the total stored electromagnetic energy in the volume. Given the energy density (u) as \[ u = \frac{1}{2} ( \epsilon_0 E^2 + \mu_0^{-1} B^2 ) \] the total energy in volume V is \[ W = \int_V u \, dV \]

Time Derivative of Total Stored Energy

Take the time derivative of the total stored energy: \[ \frac{dW}{dt} = \frac{d}{dt} \int_V u \, dV = \int_V \frac{\partial u}{\partial t} \, dV \]

Expressing the Time Derivative of Energy Density

Substitute the expression for the energy density into this integral: \[ \frac{\partial u}{\partial t} = \frac{1}{2} \epsilon_0 \frac{\partial}{\partial t}(E^2) + \frac{1}{2 \mu_0} \frac{\partial}{\partial t}(B^2) \]

Using Maxwell's Equations

Maxwell's equations in the given form can help transform these time derivatives. Note: \[ \frac{\partial E^2}{\partial t} = 2 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} \] and \[ \frac{\partial B^2}{\partial t} = 2 \mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t} \]. Replace these expressions accordingly in step 3.

Evaluate Time Derivatives via Maxwell’s Equations

Utilize Maxwell's equations (iii) and (iv): \[ abla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} \] and \[ abla \times \mathbf{B} = \mu_0 \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]. Use these to replace time derivatives of E and B fields.

Combine Derivatives

Insert into the integrand: \[ \frac{dW}{dt} = \int_V \left( \epsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} + \frac{1}{\mu_0} \mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t} \right) dV \]. Substitute for \[ \frac{\partial \mathbf{E}}{\partial t} = c^2 (abla \times \mathbf{B} - \mu_0 \mathbf{J}) \] and \[ \frac{\partial \mathbf{B}}{\partial t} = -abla \times \mathbf{E} \] to get:

Simplifying the Derivatives

\[ \epsilon_0 \mathbf{E} \cdot (abla \times \mathbf{B} - \mu_0 \mathbf{J}) + \frac{1}{\mu_0} \mathbf{B} \cdot (- abla \times \mathbf{E}) \]. Break up the integrals and simplify using vector identities. The terms simplify to \[ - \mu_0 \int_V \mathbf{J} \cdot \mathbf{E} dV - \epsilon_0 \int_V abla \cdot ( \mathbf{E} \times \mathbf{B}) dV \]

Apply Divergence Theorem

Apply the divergence theorem to convert the volume integral to a surface integral. The expression becomes: \[ -\int_V \mathbf{J} \cdot \mathbf{E} dV - \frac{1}{\mu_0} \oint_S ( \mathbf{E} \times \mathbf{B}) \cdot d \mathbf{S} \]

Conclusion

The rate of change of the total stored energy consists of parts corresponding to ohmic heating loss and electromagnetic energy flux: \[ -\int_V \mathbf{J} \cdot \mathbf{E} dV - \frac{1}{\mu_0} \oint_S (\mathbf{E} \times \mathbf{B}) \cdot d \mathbf{S} \]

Key concepts

Electromagnetic Energy

Electromagnetic energy is the energy stored in the electric and magnetic fields. In a given medium, this is represented by the energy density formula:
$$ u = \frac{1}{2} \big( \text{𝜖}_0 E^2 + \text{𝜇}_0^{-1} B^2 \big) $$
Here:

• \text{𝜖}_0 denotes the permittivity of free space.
• \text{𝜇}_0 stands for the permeability of free space.
• E and B are the magnitudes of the electric and magnetic fields, respectively.

To analyze the stored electromagnetic energy in a volume V, we integrate its density over that volume:
$$ W = \text{∫}_V u \text{ dV} $$
Here, W represents the total energy stored. This energy changes over time due to various interactions, which we quantify using Maxwell's Equations.

Poynting Vector

The Poynting vector $$ \textbf{S} = \frac{1}{\text{𝜇}_0} (\textbf{E} \times \textbf{B}) $$ shows the direction and magnitude of the electromagnetic energy flow. Here,

• The cross product \( \textbf{E} \times \textbf{B} \) combines the electric and magnetic fields to show energy flow perpendicular to both fields.
• The term $$ \frac{1}{\text{𝜇}_0} $$ scales this product by the permeability of free space, giving us the accurate energy flux.

By evaluating this vector over a bounding surface (S), we can determine how electromagnetic energy enters or leaves a given volume (V). The surface integral captures this flux:
$$ \frac{1}{\text{𝜇}_0} \text{∮}_S (\textbf{E} \times \textbf{B}) \text{ ⋅ dS} $$ This integral helps to illustrate electromagnetic energy transfer and is crucial in energy conservation analyses.

Energy Density

Energy density (u) describes the amount of electromagnetic energy stored in a unit volume. It combines contributions from both electric and magnetic fields: $$ u = \frac{1}{2} (\text{𝜖}_0 E^2 + \text{𝜇}_0^{-1} B^2) $$
To get the total energy in any volume (V), we integrate this energy density over the volume: $$ W = \text{∫}_V u \text{ dV} $$
The time derivative of the total energy gives the rate at which electromagnetic energy is changing: $$ \frac{dW}{dt} = \frac{d}{dt} \text{∫}_V u \text{ dV} = \text{∫}_V \frac{∂u}{∂t} \text{ dV} $$
For \text{∂u/∂t}, we decompose the energy density terms by taking their respective time derivatives: $$ \frac{∂u}{∂t} = \frac{1}{2} \text{𝜖}_0 \frac{∂}{∂t}(E^2) + \frac{1}{2} \frac{1}{\text{𝜇}_0} \frac{∂}{∂t}(B^2) $$ Using these expressions, we can further explore how electromagnetic fields contribute to energy changes.

Ohmic Heating

Ohmic heating, also known as resistive heating, arises from the dissipative interaction between current density $$ \textbf{J} $$ and electric field $$ \textbf{E} $$.

• The power loss due to Ohmic heating per unit volume is given by< $$ \text{𝜎E}^2 $$ where \text{𝜎} is the electrical conductivity.
• Integrating this power loss over a volume gives the total ohmic heating loss:

$$ \text{∫}_V \textbf{J} \text{ ⋅ E dV } $$ This term represents the rate of energy dissipated as heat within the volume.

Ohmic heating is a key component in the overall energy balance of electromagnetic systems, as it quantifies the energy lost to thermal processes.

Divergence Theorem

The Divergence Theorem is a critical mathematical tool linking volume integrals to surface integrals. For a vector field \textbf{F}, it states:

$$ \text{∫}_V \textbf{ ∇ ⋅ F dV } = \text{∮}_S \textbf{F} \text{ ⋅ dS} $$ Here, \text{∇ ⋅ F} is the divergence of \textbf{F}.
In the context of electromagnetic energy, the divergence theorem allows us to convert the volume integral of the energy flow into a surface integral:
$$ \text{∫}_V \textbf{ ∇ ⋅ (E × B) dV } = \text{∮}_S (\textbf{E} × \textbf{B}) \text{⋅ dS} $$ This conversion simplifies calculations and lets us focus on the net flux of energy through a surface. By applying this theorem, we can clearly separate the contributions of internal and surface energy changes, enabling a better understanding of electromagnetic energy dynamics.