 Charged circular elastic ring in a uniform magnetic field

ID: charged-circular-elastic-ring-in-a-uniform-magnetic-field

A uniformly charged circular ring has

L = \frac{m}{2} (\overset{r}{˙}^{2} + r^{2} \dot{ϕ}^{2}) - 2 π^{2} k (r - r_{0})^{2} + \frac{qB r ^{2}}{2} \dot{ϕ} .

Here

k

is the spring constant for extension of its full circumference and the symmetric gauge is used. The canonical angular momentum

J = m r^{2} \dot{ϕ} + qB r^{2} /2

is conserved. Eliminating the angle gives

m \overset{r}{¨} = J^{2} / (m r^{3}) - q^{2} B^{2} r / (4 m) - 4 π^{2} k (r - r_{0})

. This model neglects electric self-interaction and noncircular deformation.

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