System of charges in an external field

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Presentation transcript:

System of charges in an external field LL2 Section 42

External E-field, whose potential is f(r) A system of charges

The potential energy of the charges in the external field is Assume E changes slowly near the charges. Expand U in powers of ra. Value of the potential at the origin. Zeroth approximation assumes all charges are located at the origin.

This is the net force on the charge distribution as a whole. Dipole moment of the system of charges Field at the origin Force = This is the net force on the charge distribution as a whole. Determined by how the potential and field change near the origin.

If the system is electrically neutral Then, HW For there to be a force on a neutral system, there must be a dipole moment, and the electric field must be non-uniform Torque

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Neutral molecule must have a dipole moment and the electric field must be non-uniform. For a neutral molecule to experience a torque in an external electric field, do we still need a dipole moment and field non-uniformity?

The lowest non-zero contribution to their fields is the dipole term. Suppose we have two systems of charges, each with zero total charge. There separation R is large compared to their dimensions. The lowest non-zero contribution to their fields is the dipole term. (system 2 is in the field from system 1) Potential energy Evaluated at the origin of system 2 (40.8) for dipole

Note the choice of direction for R Two systems of charges, one with non-zero total charge e, well separated Note the choice of direction for R The monopole filed is the lowest order contribution

satisfies Laplace’s equation Coordinates of the ath charge satisfies Laplace’s equation So we also have We can add this zero to U(2) without changing it.

Quadrupole moment tensor

Some coefficients characterizing the external field satisfies Laplace’s equation. In spherical coordinates, sepatarion of variables can be applied The solution R(r) to the radial equation that is finite at the origin is rl. General solution (exact) Potential of the externally applied field at the position r within the system of charges. Spherical harmonics Some coefficients characterizing the external field

2l-pole moment of the system of charges (41.13) lth order term in the expansion of U in powers of ra.