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Suppose that an electron may be considered to be a ball of radius a carrying an electric charge e uniformly tributed over its surface. It may be shown that outside this ball and on its surface the field will be the same as that of an equal point charge, inside the ball the field is zero. From these considerations find the energy of the electron's field. Assuming it to be equal to the electron's rest energy estimate the radius of this ball (double this quantity is called the classical radius of the electron). Compare with Problem 14.23 |
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Answer» Solution :A ball of radius a carrying a surface charge may be regarded as a spherical capacitor whose external sphere is infinitely FAR away (i.e. `R=a,R_(1) to oo)` . Making use of the result of the previous problem, we obtain `C=lim_(R_(1) to oo) (4pi epsi_(0)aR_(1))/(R_(1)-a)= lim_(R_(1) to oo) (4pi epsi_(0)a)/(1-(a//R_(1)))= 4pi epsi_(0)a` The energy of the field is `W=(e^(B))/(2C)=(e^(2))/(8 pi epsi_(0)a)` Equating it to the rest energy of an electron `delta_(0)=m_(e)c^(2)`, we obtaine `a=(e^(2))/(8 pi epsi_(0) m_(e)c^(2))=1.4xx10^(-15)m` As is shown in $72.5, the term "classical electron radius" usually applies to a quantity twice as large `r_(el)=2a=2.8xx10^(-16)m`. Comparing this result with the result obtained in Problem 14.23 we see that the latter was TWO orders of magnitude greater. This implies the incorrectness of the solutions of the two problems. In MODERN science the problem of the dimensions of elementary particles, including the electron, is far from being SOLVED. |
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