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Chapter 6: Problem 17

Give reasons why in a disordered (i.e. random) distribution of pure radiation (a 'photon gas') the electromagnetic field components will satisfy the following relations on the (time) average: (i) \(e_{1}^{2}=e_{2}^{2}=e_{3}^{2}, \quad b_{1}^{2}=b_{2}^{2}=b_{3}^{2}\), (ii) \(e_{1} e_{2}=e_{2} e_{3}=e_{3} e_{1}=0, \quad b_{1} b_{2}=b_{2} b_{3}=b_{3} b_{1}=0\), (iii) \(e_{2} b_{3}-e_{3} b_{2}=e_{3} b_{1}-e_{1} b_{3}=e_{1} b_{2}-e_{2} b_{1}=0\). Deduce that the only non-zero components of the averaged energy tensor can then be written as \(M^{00}=\sigma_{0}, M^{11}=M^{22}=M^{33}=p\), where \(3 p=\sigma_{n}\).

Short Answer

Expert verified
Answer: The given relations are satisfied on average for the electromagnetic field components because the components of the electric and magnetic fields are isotropic and of the same magnitude in a random distribution. They are also uncorrelated and orthogonal, and the electric and magnetic fields are mutually orthogonal on average. This leads to the given relations being satisfied, allowing us to deduce the components of the averaged energy tensor.

Step by step solution

01

Understanding the Electromagnetic Field Components

In this step, let's first understand each of the components of the electromagnetic field described in the given relations: - \(e_{1}, e_{2}, e_{3}\) are the components of the electric field - \(b_{1}, b_{2}, b_{3}\) are the components of the magnetic field Now we will go through each of the given relations and provide a reason why they are satisfied in a disordered distribution.
02

Verifying relation (i)

In a random distribution of pure radiation, the components of the electric and magnetic fields are isotropic and of the same magnitude on average. Thus, we have: $$e_{1}^{2}=e_{2}^{2}=e_{3}^{2}, \quad b_{1}^{2}=b_{2}^{2}=b_{3}^{2}$$
03

Verifying relation (ii)

Since the electric and magnetic fields are randomly distributed, their components are uncorrelated and orthogonal on average. This gives us: $$e_{1} e_{2}=e_{2} e_{3}=e_{3} e_{1}=0, \quad b_{1} b_{2}=b_{2} b_{3}=b_{3} b_{1}=0$$
04

Verifying relation (iii)

The electric and magnetic fields are mutually orthogonal on average, which results in the following relation: $$e_{2} b_{3}-e_{3} b_{2}=e_{3} b_{1}-e_{1} b_{3}=e_{1} b_{2}-e_{2} b_{1}=0$$
05

Deduce the Averaged Energy Tensor Components

Now that we have verified the relations, we can deduce the components of the averaged energy tensor. The electromagnetic energy tensor, \(M^{\mu\nu}\) is given by: $$M^{\mu\nu}=\sum_{i}\big[e_{i}^{\mu}e_{i}^{\nu} + b_{i}^{\mu}b_{i}^{\nu}\big]$$ On average, the following components can be written as: $$M^{00} = \sigma_{0} \quad M^{11} = M^{22} = M^{33} = p$$ where \(\sigma_0\) represents the average energy density, \(p\) represents the average pressure, and we are given that: $$3p = \sigma_{n}$$

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Key Concepts

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

Understanding Special Relativity
Special relativity is a fundamental theory in physics developed by Albert Einstein, which describes the laws of physics from the standpoint of observers in different inertial frames of reference. One of the key insights of special relativity is that the laws of physics are the same for all non-accelerating observers, and the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.

Within the context of special relativity, time and space are not absolute; instead, they are intertwined into a four-dimensional spacetime continuum. The theory has profound implications for the concepts of time dilation, length contraction, and mass-energy equivalence, which is epitomized by the famous equation \(E=mc^2\). In dealing with a photon gas and its interaction with electromagnetic fields, special relativity plays a crucial role, as it treats the propagation of light and its interaction with matter in a consistent way across different reference frames.

This theory forms the bedrock for the description of the properties of a photon gas, including isotropy, which is essential for understanding the equalization of electromagnetic field components and the formulation of the energy tensor in these contexts.
Electromagnetic Field Components
The electromagnetic field consists of two vector components: the electric field \(\vec{E}\) and the magnetic field \(\vec{B}\). The electric field is produced by stationary or moving electric charges, and the magnetic field arises from moving charges (currents) or magnetic materials.

The electric field components, denoted as \(e_1, e_2, e_3\), represent the field's strength and direction in a three-dimensional space, typically aligned with the x, y, and z axes, respectively. Similarly, the magnetic field components, \(b_1, b_2, b_3\), describe the magnetic field's magnitude and orientation along those axes.

In a photon gas, these field components can fluctuate randomly due to the constant motion and interaction of photons. However, when averaged over time, the random orientations and magnitudes tend to balance out, resulting in a uniform distribution of the squared magnitudes of both the electric and magnetic fields, as described in relation (i) from the exercise. Moreover, the uncorrelated nature of these components, as indicated in relation (ii), is characteristic of a disordered system where directional biases are absent.
The Energy Tensor in Special Relativity
In special relativity, the energy-momentum tensor (also known as the stress-energy tensor) is an important mathematical object that encapsulates the density and flow of energy and momentum in spacetime. Specifically, for electromagnetic fields, this tensor includes contributions from both the electric and magnetic components and reflects the energy density, momentum density, and stress associated with these fields.

The averaged energy tensor in a system of a photon gas, as described in the problem, has components \(M^{00}\) representing the energy density, and \(M^{11}, M^{22}, M^{33}\) representing pressures in the respective x, y, and z directions. Due to the symmetrical and isotropic nature of a photon gas, these pressures are equal, fulfilling the expression for isotropy. The tensor is crucial for understanding how energy and momentum are conserved and distributed within the context of special relativity. It is also key to predicting how a photon gas behaves under different physical circumstances, such as in the presence of gravitational fields.
Isotropy of Photon Gas
Isotropy is a term used to describe the property of being uniform in all orientations or directions. In the context of a photon gas, isotropy implies that the physical properties of the gas, like pressure and energy density, are the same in all directions.

This characteristic is crucial for the homogeneity of the cosmic microwave background radiation, a relic from the early universe that is consistent with an isotropic photon gas. It also supports the principle that, on a large scale, the universe is homogeneous and isotropic, a foundation of modern cosmology.

In the given exercise, isotropy underlies the reasons why the averaged electromagnetic field components exhibit the relationships indicated. The equality of the squared electric and magnetic field components (relation i), and the mutual orthogonality of each electric and magnetic field vector pair (relation iii), are all indicators of this isotropic behavior. These properties substantiate the deduction of the uniform energy tensor components for a photon gas in random distribution.

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Most popular questions from this chapter

The apparent brightness \(b\) of a distant luminous source (which can be idealized as a point source) is the rate at which radiant energy from it is received on a unit area perpendicular to the line of sight; the absolute brightness \(B\) is defined as the apparent brightness at unit distance from the source in its rest frame. Prove that for two momentarily coincident observers the apparent brightnesses of the same source are in the ratio of the squares of the observed frequencies: \(b^{\prime} / b=v^{\prime 2} / v^{2}\). Hence or otherwise prove that \(b=\left(B / R^{2}\right)\left(v^{4} / v_{0}^{4}\right)\). where \(R\) is the distance of the source from the oberserver in the observer's inertial frame at the time the light he measures was emitted, and \(v_{0}\) is its proper frequency. [Hint: for the two frames of Fig. \(10(b)\), \(\left.R \sin \alpha=R^{\prime} \sin \alpha^{\prime} .\right]\)

Prove, by any method, that the electric field e at a point P due to an infinite straight line distribution of static charge, i per unit length, is given by \(\mathbf{e}=\dot{\lambda} / 2 \pi \varepsilon_{0} r^{2}\), where \(\mathbf{r}\) is the perpendicular vectordistance of P from the line. Deduce, by transforming to a frame in which this line moves, that the magnetic field b at P due to an infinitely long straight current \(\mathbf{i}\) is given by \(\mathbf{b}=(\mathbf{i} \times \mathbf{r}) / 2 \pi \varepsilon_{0} c^{2} r^{2}\). Check this with (38 20) (ii)

(i) A particle of rest mass \(m\) and charge \(q\) is injected at velocity \(\mathbf{u}\) into a constant pure magnetic field \(\mathbf{b}\) at right angles to the field lines. Use the Lorentz force law \((38.16)\) to establish that the particle will trace out a circle of radius \(m u \gamma(u) / q b\) with period \(2 \pi m \gamma(u) / q b\). [It was the y-factor in the period that necessitated the development of synchrotrons from cyclotrons, at whose energies the \(\gamma\) was still negligible.] (ii) If the particle is injected into the field with the same velocity but at an angle \(\theta \neq \pi / 2\) to the field lines, prove that the path is a helix, of smaller radius, but that the period for one complete cycle is the same as before.

14\. Prove that the electromagnetic energy tensor satisfies the following two identities: $$ M_{\mu}^{\mu}=0, \quad M_{\sigma}^{\mu} M_{v}^{a}=\left(I \varepsilon_{0} / 2\right)^{2} \delta_{v}^{\mu}, $$ where \(I^{2}=\left(c^{2} b^{2}-e^{2}\right)^{2}+4 c^{2}(e \cdot b)^{2}\). [Hint: It may be easiest to establish the second identity in a particular frame, e.g. if \(I \neq 0\), in one in which e is parallel to b. (We regard the vanishing of e or b as a particular case of this.) For the case \(I=0\), appeal to continuity. \(]\)

Prove the 'zero component' lemma for an antisymmetric tensor \(T^{\mu v}\) : if any one of its off-diagonal components is zero in all inertial frames, then the entire tensor is zero.
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