91Ó°ÊÓ

The Larmor formula for the power radiated by a point charge is$P=\frac{{\mathrm{μ}}_0{q}^2{a}^2}{6\mathrm{π}c}$,.

Here,$q$ is the charge of the particle,$a$ is the acceleration of the particle at the retarded time$\mathrm{τ}$ , $c$is the speed of light and${\mathrm{μ}}_0$ is the permittivity of vacuum.

The Lienard formula for the power radiated by a point charge is,.

$P=\frac{{\mathrm{μ}}_0{q}^2}{6\mathrm{Ï€}c}{\mathrm{γ}}^6[a^2−{\left|\frac{\mathrm{Ï\dots }×a}{c}\right|}^2]$

Here, $\mathrm{γ}≡\sqrt{1−\frac{{\mathrm{Ï\dots }}^2}{c^2}}$is the factor of time dilation and $\mathrm{Ï\dots }$is the velocity of the charge at the retarded time.

Consider a point charge of certain magnitude, moves in a uniform electric and magnetic field then the power radiated by the point charge is represented using the Larmor formula.

The value of the power radiated by a point charge increases with the increase in the acceleration of the point charge.

Using the Minkowski version of Newton’s second law, the proper power as the zeroth component of a 4-vector can be written as,

$\begin{array}{c}\frac{dW}{d\mathrm{τ}}=K^0×c\\ K^0=\frac{1}{c}\frac{dW}{d\mathrm{τ}}\end{array}$

Here,$K$ is the proper force.

According to the Larmor formula, for $\mathrm{Ï\dots }=0$, the rate at which energy left the charge is given by,

$\begin{array}{l}\frac{dW}{d\mathrm{Ï„}}=\frac{\frac{dW}{dt}}{\frac{∂\mathrm{Ï„}}{∂t}}\\ \frac{dW}{d\mathrm{Ï„}}=\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{2}{3}\frac{q^2}{c^3}{a}^2\end{array}$

When $\mathrm{Ï\dots }=0$, the value of acceleration $a^2$can be written as,.$a^2={\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}$ Also,the value of a 4-vector$K^{\mathrm{μ}}$ in terms of${\mathrm{η}}^{\mathrm{μ}}$, whose zeroth component is just $c$is given by,

$K^{\mathrm{μ}}=\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{2}{3}\frac{q^2}{c^3}\left({\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}\right){\mathrm{η}}^{\mathrm{μ}}$

Reducing the above expression to the Larmor formula when$\mathrm{Ï\dots }=0$ ,

$\begin{array}{c}\frac{dW}{dt}=\frac{1}{\mathrm{γ}}\frac{dW}{d\mathrm{Ï„}}\\ \frac{dW}{dt}=\frac{1}{\mathrm{γ}}c{K}^0\\ \frac{dW}{dt}=\frac{1}{\mathrm{γ}}c\frac{{\mathrm{μ}}_0{q}^2}{6\mathrm{Ï€}{c}^3}\left({\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}\right){\mathrm{η}}^0\end{array}$

Here, ,${\mathrm{η}}^0=c\mathrm{γ}$ so,

$\frac{dW}{dt}=\frac{{\mathrm{μ}}_0{q}^2}{6\mathrm{Ï€}c}\left({\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}\right)$ ….. (1)

The value of the term$\left({\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}\right)$ in terms of ordinary velocity and acceleration is given by

role="math" localid="1658915579537" $\begin{array}{c}{\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}={\mathrm{γ}}^4[a^2+\frac{{(\mathrm{Ï\dots }â‹\dots a)}^2}{(c^2−{\mathrm{Ï\dots }}^2)}]\\ {\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}={\mathrm{γ}}^6[a^2{\mathrm{γ}}^{−2}+\frac{1}{c^2}{(\mathrm{Ï\dots }â‹\dots a)}^2]\\ {\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}={\mathrm{γ}}^6[a^2(1−\frac{{\mathrm{Ï\dots }}^2}{c^2})+\frac{1}{c^2}{(\mathrm{Ï\dots }â‹\dots a)}^2]\\ {\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}={\mathrm{γ}}^6[a^2−\frac{1}{c^2}\{{\mathrm{Ï\dots }}^2{a}^2−{(\mathrm{Ï\dots }â‹\dots a)}^2\}]\end{array}$

Here,$\mathrm{Ï\dots }â‹\dots a=\mathrm{Ï\dots }a\cos \mathrm{θ}$, where$\mathrm{θ}$ is the angle between $\mathrm{Ï\dots }$and $a$, so,

$\begin{array}{c}{\mathrm{Ï\dots }}^2{a}^2−{(\mathrm{Ï\dots }â‹\dots a)}^2={\mathrm{Ï\dots }}^2{a}^2(1−{\cos }^2\mathrm{θ})\\ {\mathrm{Ï\dots }}^2{a}^2−{(\mathrm{Ï\dots }â‹\dots a)}^2={\mathrm{Ï\dots }}^2{a}^2{\sin }^2\mathrm{θ}\\ {\mathrm{Ï\dots }}^2{a}^2−{(\mathrm{Ï\dots }â‹\dots a)}^2={|\mathrm{Ï\dots }×a|}^2\end{array}$

Substituting this in the expression,

$\begin{array}{c}{\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}={\mathrm{γ}}^6[a^2−\frac{1}{c^2}{|\mathrm{Ï\dots }×a|}^2]\\ {\mathrm{Î\pm }}^{\mathrm{ν}}{\mathrm{Î\pm }}_{\mathrm{ν}}={\mathrm{γ}}^6[a^2−{\left|\frac{\mathrm{Ï\dots }×a}{c}\right|}^2]\end{array}$

From equation (1),

$\begin{array}{c}\frac{dW}{dt}=\frac{{\mathrm{μ}}_0{q}^2}{6\mathrm{Ï€}c}{\mathrm{γ}}^6[a^2−{\left|\frac{\mathrm{Ï\dots }×a}{c}\right|}^2]\\ P=\frac{{\mathrm{μ}}_0{q}^2}{6\mathrm{Ï€}c}{\mathrm{γ}}^6[a^2−{\left|\frac{\mathrm{Ï\dots }×a}{c}\right|}^2]\end{array}$

Hence, the above expressionrepresents the Lienard’s generalization of the Larmor formula.