Electrostatic Energy

Electrostatic Energy

1. 1. The electrostatic energy of charges. A uniform sphere

   • The total electrostatic energy U is the sum of the energies of all possible pairs of charges
   • The energy of a uniform sphere of charge can be computed by imagining that it is assembled from successive spherical shells.
   • Or if we wish to express the result in terms of the total charge Q of the sphere

2. 2. The energy of a condenser. Forces on charged conductors

   • The field at the surface of a conductor varies from zero to E0=σ/ϵ0, as one passes through the layer of surface charg
   • The charge on the condenser was constant, so the total charge could not change
     • The force between the plates of a parallel-plate condenser
     • The energy of the condenser
   • The condenser was held at a constant potential difference as we made the virtual displacement

3. 3. The electrostatic energy of an ionic crystal

   • Cross section of a salt crystal on an atomic scale
   • The energy of dissociation can also be given as
   • The center-to-center spacing between ions
   • The whole lattice is held together by electrical Coulomb

4. 4. Electrostatic energy in nuclei

   • The force between two protons depends on every possible parameter
   • Properties of the main forces
     • A. The force is not a simple function of the distance between the two protons. At large distances there is an attraction, but at closer distances there is a repulsion
     • B. The force depends on the orientation of the protons’ spin, is different when the spins are parallel from what it is when they are antiparallel
     • C. the force is considerably different when the separation of the two protons is in the direction parallel to their spins, than it is when the separation is in a direction perpendicular to the spins
     • D. The force depends, as it does in magnetism, on the velocity of the protons, only much more strongly than in magnetism. And this velocity-dependent force is not a relativistic effect; it is strong even at speeds much less than the speed of light.
   • If we consider that a nucleus is like a sphere with uniform charge density, we would expect the electrostatic energy
   • It is found that its radius is given by
   • Is the energy needed to add one more proton to the outside of the core
   • The energy levels of B11 and C11 (energies in MeV)

5. 5. Energy in the electrostatic field

   • When an electric field is present, there is located in space an energy whose density (energy per unit volume) is
   • it is possible for us to represent the energy of any charge distribution as being the integral over an energy density located in the field.
   • Each volume element dV=dxdydz in an electric field contains the energy (ϵ0/2)E2dV

6. 6. The energy of a point charge

   • Even a single point charge q will have some electrostatic energy
   • So the energy density at the distance r from the charge is
   • We can take for an element of volume a spherical shell of thickness dr and area 4πr2. The total energy is
   • We must conclude that the idea of locating the energy in the field is inconsistent with the assumption of the existence of point charges. One way out of the difficulty would be to say that elementary charges, such as an electron, are not points but are really small distributions of charge

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