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mysearch
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On the basis that Larmor’s formula specifies the energy/power radiated from an accelerated charge, provided v<<c, I was assuming that this equation must also characterised the energy power transferred to the EM waves/photons emitted by the accelerated charge. I know Larmor’s work predates the idea of a photon and quantised energy, but looking at the derivation of Larmor’s formula raised some questions, which I was hoping somebody might be able to clarify.
First, the derivation of Larmor’s formula I have been studying exists within section 4.2/page 31 of Daniel Schroeder’s pdf document, which can be access via the following link: http://physics.weber.edu/schroeder/mrr/MRRnotes.pdf . While I am not totally sure whether this approach is generally accepted, it seems to be based on the idea that an accelerating charge causes a ‘kink’ in the electric field lines, such that the net electric field (E) has both a radial (Er) and transverse (Et) component. The strength of these components is given on page 33 as follows:
[1] [tex]Er=\frac{q}{ 4 \pi \epsilon_0 R^2}[/tex]
[2] [tex]Et=\frac{qa*sin \theta}{ 4 \pi \epsilon_0 c^2R}[/tex]
These equations show different proportionality with respect to the radius [R] such that (Et) starts to dominate as [R] increases. So my first question is whether these equations are reflective of the idea of ‘near’ and ‘far’ field definitions?
Over the next few page (p34/35) of the Daniel Schroeder’s document, he derives Larmor’s formula starting with:
[3] [tex]Energy-per-unit-volume=\frac{\epsilon_0 E^2}{2}[/tex]
Equation [2] is substituted into [3] to give:
[4] [tex]Energy-per-unit-volume=\frac{q^2a^2 sin^2 \theta}{32 \pi \epsilon_0 c^4 R^2}[/tex]
This equation is only accounting for the electric field, i.e. no initial consideration of the magnetic field, which eventually doubles the energy. However, the point I want to clarify is that the energy/unit volume appears to be still subject to the inverse square law. Only after multiplying equation [4] by the volume of the spherical shell, i.e. [tex]4 \pi R^2 * thickness[/tex], in which (Et) exists, do we eventually arrive at Larmor’s formula.
[5] [tex]Power-radiated=\frac{q^2a^2}{ 6 \pi \epsilon_0 c^3}[/tex]
What is unclear to me is whether this equation is representative of energy/power radiated by EM wave(s) in all directions, while noting the directionality associated with the [tex]sin^2 \theta[/tex] term in [4]. For while the total radiated energy in the spherical shell remains constant, the energy density associated with (Et), as defined by [4], would seem to diminish as the (Et) kink propagates outwards to a larger radius [R]. As such, I don’t see how this equation really reflects the energy of a EM wave or photon propagating without loss in a vacuum.
A photon as a discrete quantum of energy [E=hf] could define a lossless container of propagating energy, but presumably requires the distribution spread of the EM wave to be aggregated by a statistical distribution, which I am assuming is the subject of quantum theory rather than classical fields. So how did classical theory explain how light travels from the stars without loss?
Thanks
First, the derivation of Larmor’s formula I have been studying exists within section 4.2/page 31 of Daniel Schroeder’s pdf document, which can be access via the following link: http://physics.weber.edu/schroeder/mrr/MRRnotes.pdf . While I am not totally sure whether this approach is generally accepted, it seems to be based on the idea that an accelerating charge causes a ‘kink’ in the electric field lines, such that the net electric field (E) has both a radial (Er) and transverse (Et) component. The strength of these components is given on page 33 as follows:
[1] [tex]Er=\frac{q}{ 4 \pi \epsilon_0 R^2}[/tex]
[2] [tex]Et=\frac{qa*sin \theta}{ 4 \pi \epsilon_0 c^2R}[/tex]
These equations show different proportionality with respect to the radius [R] such that (Et) starts to dominate as [R] increases. So my first question is whether these equations are reflective of the idea of ‘near’ and ‘far’ field definitions?
Over the next few page (p34/35) of the Daniel Schroeder’s document, he derives Larmor’s formula starting with:
[3] [tex]Energy-per-unit-volume=\frac{\epsilon_0 E^2}{2}[/tex]
Equation [2] is substituted into [3] to give:
[4] [tex]Energy-per-unit-volume=\frac{q^2a^2 sin^2 \theta}{32 \pi \epsilon_0 c^4 R^2}[/tex]
This equation is only accounting for the electric field, i.e. no initial consideration of the magnetic field, which eventually doubles the energy. However, the point I want to clarify is that the energy/unit volume appears to be still subject to the inverse square law. Only after multiplying equation [4] by the volume of the spherical shell, i.e. [tex]4 \pi R^2 * thickness[/tex], in which (Et) exists, do we eventually arrive at Larmor’s formula.
[5] [tex]Power-radiated=\frac{q^2a^2}{ 6 \pi \epsilon_0 c^3}[/tex]
What is unclear to me is whether this equation is representative of energy/power radiated by EM wave(s) in all directions, while noting the directionality associated with the [tex]sin^2 \theta[/tex] term in [4]. For while the total radiated energy in the spherical shell remains constant, the energy density associated with (Et), as defined by [4], would seem to diminish as the (Et) kink propagates outwards to a larger radius [R]. As such, I don’t see how this equation really reflects the energy of a EM wave or photon propagating without loss in a vacuum.
A photon as a discrete quantum of energy [E=hf] could define a lossless container of propagating energy, but presumably requires the distribution spread of the EM wave to be aggregated by a statistical distribution, which I am assuming is the subject of quantum theory rather than classical fields. So how did classical theory explain how light travels from the stars without loss?
Thanks