Physics

Physics · electromagnetism

Problem

d. Roughly estimate the total energy stored in the magnetic field of the electron, in terms of

E_{0}

and

\alpha

. (Hint: one way to do this is to suppose the magnetic field has roughly constant magnitude inside the sphere and is negligible outside of it, then estimate the field inside the sphere.) Given constants: ## Electroneering An electron is a particle with charge

-q

, mass

m

, and magnetic moment

\mu

. In this problem we will explore whether a classical model consistent with these properties can also explain the rest energy

E_{0}=m c^{2}

of the electron. Let us describe the electron as a thin spherical shell with uniformly distributed charge and radius

R

. Recall that the magnetic moment of a closed, planar loop of current is always equal to the product of the current and the area of the loop. For the electron, a magnetic moment can be created by making the sphere rotate around an axis passing through its center. Context question: a. If no point on the sphere's surface can travel faster than the speed of light (in the frame of the sphere's center of mass), what is the maximum magnetic moment that the sphere can have? You may use the integral:

\int_{0}^{\pi} \sin ^{3} \theta d \theta=\frac{4}{3}

Context answer:

\mu=q c R / 3

Context question: b. The electron's magnetic moment is known to be

\mu=q \hbar / 2 m

, where

\hbar

is the reduced Planck constant. In this model, what is the minimum possible radius of the electron? Express your answer in terms of

m

and fundamental constants. Context answer:

R=\frac{3}{2} \frac{\hbar}{m c}

Context question: c. Assuming the radius is the value you found in part (b), how much energy is stored in the electric field of the electron? Express your answer in terms of

E_{0}=m c^{2}

and the fine structure constant,

\alpha=\frac{q^{2}}{4 \pi \epsilon_{0} \hbar c} \approx \frac{1}{137}

Context answer:

U_{E}=\frac{1}{3} \alpha E_{0}

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