Figure below illustrates a simplified electron lens formed by a circular loop of wire with radius \( a \) carrying a current \( I \). For \( \rho \ll a \), the vector potential is approximately given by:
\( A_\phi = \frac{\mu_0 I a^2 \rho}{(a^2 + z^2)^{3/2}} \)
The Lagrangian in cylindrical coordinates \( (\rho, \phi, z) \) for a particle of charge \( q \) moving in this field is
1
\( L = \frac{1}{2} m (\dot{\rho}^2 + \rho^2 \dot{\phi}^2 + \dot{z}^2) + q \frac{\mu_0 I a^2 \rho^2 \dot{\phi}}{(a^2 + z^2)^{3/2}} \)
2
\( L = \frac{1}{2} m (\dot{\rho}^2 + \rho^2 \dot{\phi}^2 + \dot{z}^2) -q \frac{\mu_0 I a^2 \rho^2 \dot{\phi}}{(a^2 + z^2)^{3/2}} \)
3
\( L = \frac{1}{2} m (\dot{\rho}^2 + \rho^2 \dot{\phi}^2 + \dot{z}^2) + q \frac{\mu_0 I a^3 \rho \dot{\phi}}{(a^2 + z^2)^{3/2}} \)
4
\( L = \frac{1}{2} m (\dot{\rho}^2 + \rho^2 \dot{\phi}^2 + \dot{z}^2) - q \frac{\mu_0 I a^3 \rho \dot{\phi}}{(a^2 + z^2)^{3/2}} \)