Larmor formula

In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by Joseph Larmor in 1897, Formula is mentioned in the text on the last page. in the context of the wave theory of light.

When any charged particle (such as an electron, a proton, or an ion) accelerates, energy is radiated in the form of electromagnetic waves. For a particle whose velocity is small relative to the speed of light (i.e., nonrelativistic), the total power that the particle radiates (when considered as a point charge) can be calculated by the Larmor formula: where $\backslash dot\; v$ or $a$ is the proper acceleration, $q$ is the charge, and $c$ is the speed of light. A relativistic generalization is given by the Liénard–Wiechert potentials.

In either unit system, the power radiated by a single electron can be expressed in terms of the classical electron radius and electron mass as: $$P\; =\; \backslash frac\{2\}\{3\}\; \backslash frac\{m\_e\; r\_e\; a^2\}\{c\}$$

One implication is that an electron orbiting around a nucleus, as in the Bohr model, should lose energy, fall to the nucleus and the atom should collapse. This puzzle was not solved until quantum theory was introduced.

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Derivation To calculate the power radiated by a point charge $q$ at a position $\backslash mathbf\{r\}(t)$, with a velocity, $\backslash mathbf\{v\}(t),$ we integrate the Poynting vector over the surface of a sphere of radius R, to get: $$P\; =\; \backslash frac\{c\; R^2\}\{4\backslash pi\}\backslash oint\; \backslash mathbf\{\backslash hat\{r\}\}\backslash cdot\backslash mathbfd\backslash Omega.$$ The electric and magnetic fields are given by the Liénard–Wiechert field equations,

+ \frac{q}{c} \left\{\frac{\mathbf\hat{r}_r \times \left[(\mathbf{\hat{r}}_r - \boldsymbol{\beta}_r) \times \dot\boldsymbol{\beta}_r\right]} {r_r {\left(1 - \mathbf{\hat{r}}_r \cdot \boldsymbol{\beta}_r\right)}^3}\right\}
     

\\1ex \mathbf{B}(\mathbf{r},t) &= \mathbf\times\mathbf{E(\mathbf{r},t)}. \end{align} The radius vector, $\{\backslash mathbf\; r\}\_r$, is the distance from the charged particle's position at the retarded time to the point of observation of the electromagnetic fields at the present time, $\backslash boldsymbol\{\backslash beta\}$ is the charge's velocity divided by $c$, $\backslash dot\{\backslash boldsymbol\{\backslash beta\}\}$ is the charge's acceleration divided by $c$, and $\backslash gamma\; =\; (1\; -\; \backslash beta^2\; )^\{-1/2\}$. The variables, $\{\backslash mathbf\; r\}\_r$, $\backslash boldsymbol\{\backslash beta\}\_r$, $\backslash gamma\_r$, and $\{\backslash mathbf\; a\}\_r$ are all evaluated at the retarded time, $t\_r\; =\; t\; -\; r\_r/c$.

We make a Lorentz transformation to the rest frame of the point charge where $\backslash mathbf\; v\text{'}=0$, and Here, $\{\backslash mathbf\; a\text{'}\}\_\backslash parallel$ is the rest frame acceleration parallel to $\backslash mathbf\; v$, and $\backslash mathbf\; a\text{'}\_\backslash perp$ is the rest frame acceleration perpendicular to $\backslash mathbf\; v$.

We integrate the rest frame Poynting vector over the surface of a sphere of radius R', to get. $$P\text{'}\; =\; \backslash frac\{c\; R\text{'}^2\}\{4\backslash pi\}\; \backslash oint\; \backslash mathbf\{\backslash hat\; r\}\text{'}\; \backslash cdot\; \backslash left\backslash mathbf\{E\}\text{'}(\backslash mathbf\{r\}\text{'},\; d\backslash Omega\text{'},$$ We take the limit $R\text{'}\; \backslash to\; 0.$ In this limit, $t\text{'}\_r\; =\; t\text{'}$, and $\{\backslash mathbf\; a\text{'}\}\_r=\{\backslash mathbf\; a\text{'}\},$ so the electric field is given by $$\backslash mathbf\{E\}\text{'}(\backslash mathbf\{r\}\text{'},t\text{'})\; =\; \backslash frac\{q\; \backslash mathbf\{\backslash hat\{r\}\}\text{'}\}\{r\text{'}^2\}\; +\; \backslash frac\{q\; \backslash left\backslash mathbf\{\backslash hat\{r\}\}\text{'}\}\{c^2\; r\text{'}\},$$ with all variables evaluated at the present time.

Then, the surface integral for the radiated power reduces to $$P\text{'}\; =\; \backslash frac\{q^2\}\{4\backslash pi\; c^3\}\; \backslash oint\{\backslash mathbf\backslash hat\; r\text{'}\}\; \backslash cdot\; \backslash left\; d\backslash Omega\text{'}\; =\; \backslash frac\{2\; q^2\; a\text{'}^2\}\{3c^3\}.$$ The radiated power can be put back in terms of the original acceleration in the moving frame, to give $$P\text{'}\; =\; \backslash frac\{2q^2\}\{3c^3\}\; \backslash left(a^2\_\backslash parallel\; \backslash gamma^6\; +\; a^2\_\backslash perp\; \backslash gamma^4\backslash right)\; =\; \backslash frac\{2q^2\; \backslash gamma^6\}\{3c^3\}\; \backslash lefta^2.$$ The variables in this equation are in the original moving frame, but the rate of energy emission on the left hand side of the equation is still given in terms of the rest frame variables. However, the right-hand side will be shown below to be a Lorentz invariant, so radiated power can be Lorentz transformed to the moving frame, finally giving $$P\; =\; \backslash frac\{2\; q^2\; \backslash gamma^4\}\{3c^3\}\; \backslash left(a^2\_\backslash parallel\backslash gamma^2\; +\; a\_\{\backslash perp\}^2\backslash right)\; =\; \backslash frac\{2q^2\; \backslash gamma^6\}\{3c^3\}\; \backslash lefta^2.$$ This result (in two forms) is the same as Liénard's relativistic extension of Larmor's formula, and is given here with all variables at the present time. Its nonrelativistic reduction reduces to Larmor's original formula.

For high-energies, it appears that the power radiated for acceleration parallel to the velocity is a factor $\backslash gamma^2$ larger than that for perpendicular acceleration. However, writing the Liénard formula in terms of the velocity gives a misleading implication. In terms of momentum instead of velocity, the Liénard formula becomes

$$P\; =\; \backslash frac\{2q^2\}\{3c^3m^2\}\backslash left\backslash left(\backslash frac\{d\backslash mathbf.$$

This shows that the power emitted for $d\backslash mathbf\{p\}/dt$ perpendicular to the velocity is larger by a factor of $\backslash gamma^2$ than the power for $d\backslash mathbf\{p\}/dt$ parallel to the velocity. This results in radiation damping being negligible for linear accelerators, but a limiting factor for circular accelerators.

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Covariant form The radiated power is actually a Lorentz scalar, given in covariant form as

$$P\; =\; -\backslash frac\{2\}\{3\}\; \backslash frac\{q^2\}\{m^2c^3\}\; \backslash frac\{dp\_\{\backslash mu\}\}\{d\backslash tau\}\; \backslash frac\{dp^\{\backslash mu\}\}\{d\backslash tau\}.$$

To show this, we reduce the four-vector scalar product to vector notation. We start with $$\backslash frac\{dp\_\{\backslash mu\}\}\{d\backslash tau\}\; \backslash frac\{dp^\{\backslash mu\}\}\{d\backslash tau\}\; =\; \backslash gamma^2\; \backslash left[\backslash left(\backslash frac\{dp\_0\}\{dt\}\backslash right)^2\; -\; \backslash left(\backslash frac\{d\backslash mathbf\; p\}\{dt\}\backslash right)^2\backslash right].$$

The time derivatives are

$$\backslash frac\{dp\_0\}\{dt\}\; =\; \backslash frac\{d\}\{dt\}(m\backslash gamma)\; =\; m\backslash gamma^3\; v\; a\_\backslash parallel$$ $$\backslash frac\{d\backslash mathbf\; p\}\{dt\}\; =\; \backslash frac\{d\}\{dt\}\; \backslash left(m\; \backslash mathbf\{v\}\; \backslash gamma\backslash right)\; =\; m\backslash gamma^3\; \backslash left(\backslash mathbf\{a\}\_\backslash parallel\; +\; \backslash mathbf\{a\}\_\backslash perp/\backslash gamma^2\backslash right).$$ When these derivatives are used, we get$$\backslash frac\{dp\_\{\backslash mu\}\}\{d\backslash tau\}\; \backslash frac\{dp^\{\backslash mu\}\}\{d\backslash tau\}\; =\; -\; m^2\backslash gamma^4\backslash left(\backslash gamma^2\; \backslash mathbf\{a\}\_\backslash parallel^2\; +\; \{\backslash mathbf\{a\}\}\_\backslash perp^2\backslash right).$$

With this expression for the scalar product, the manifestly invariant form for the power agrees with the vector form above, demonstrating that the radiated power is a Lorentz scalar

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Angular distribution The angular distribution of radiated power is given by a general formula, applicable whether or not the particle is relativistic. In CGS units, this formula is

John D. Jackson (1998). 047130932X, Wiley. 047130932X

(Section 14.2ff) $$P\; =\; \backslash frac\{2\}\{3\}\; \backslash frac\{q^2\}\{m^2\; c^3\}\; \{\backslash left|\backslash dot\{\backslash mathbf\{p\}\}\backslash right|\}^2.$$ $$\backslash frac\{dP\}\{d\backslash Omega\}\; =\; \backslash frac\{q^2\}\{4\backslash pi\; c\}\; \backslash frac\{\backslash left|\backslash mathbf\{\backslash hat\{n\}\}\; \backslash times\; \backslash left\; \backslash right|^2\}\; \backslash cdot\; \backslash boldsymbol\{\backslash beta\}\backslash right)\}^5\},$$ where $\backslash mathbf\{\backslash hat\{n\}\}$ is a unit vector pointing from the particle towards the observer. In the case of linear motion (velocity parallel to acceleration), this simplifies to $$\backslash frac\{dP\}\{d\backslash Omega\}\; =\; \backslash frac\{q^2a^2\}\{4\backslash pi\; c^3\}\; \backslash frac\{\backslash sin^2\; \backslash theta\}$$

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