On the Non-Relativistic Electrodynamics of an Isolated Spherical Shell of Charge G.R.Dixon, 12/6/2003 1. Electromagnetic Reaction Forces. In Maxwellian theory a spherical shell of charge can persist in time only if an agent non-electromagnetically holds the charge together. Indeed this is true of all distributions of charge. Every increment of charge in a distribution repels every other increment, and an external agent must counteract each tiny repulsive force if the distribution is to maintain its shape. Although often not explicitly mentioned, these counteracting agent forces are always implied. The good news is that, like the electromagnetic forces they counteract, they sum to zero provided the distribution is at rest or moves with constant velocity. Lorentz and others derived the sum of the internal, interactive Lorentz (electromagnetic) forces when a spherical shell of charge’s velocity is (1.1) For shells with radii R<<1 meter, the net internal force can be simplified to . (1.2) Here again in order for the accelerated shell to maintain its shape, an external agent must counteract each infinitesimal internal force; the . (1.3) It seems clear in Eqs. 1.2 and 1.3 that F 2. The Equation of Motion for Charged Particles.
Now there are certain interesting differences between charge and "neutral matter" so far as momentum and kinetic energy are concerned. The mechanical momentum and kinetic energy of an uncharged particle appear to be local attributes of the particle. The electromagnetic momentum and kinetic energy of a , (2.1) and . (2.2) Obviously there is motivation to define the shell’s "electromagnetic mass" to be . (2.3) Eq. 1.3 can then be more compactly stated as . (2.4) Or, since all known charged particles appear to have both mechanical and electromagnetic mass, . (2.5) In the case of Quite often Eq. 2.5 is . (2.6) To the extent the da 3. Energy Conservation and Agent-Particle/Field Dynamics.
The da As discussed in the previous article, the agent’s counteraction to the radiation reaction force provides a dynamic basis for the energy that flows to/from the particle’s . (3.1) Similarly the agent’s counteraction to the electromagnetic inertial reaction force does the same for the energy that flows to/from the particle’s electromagnetic "inertial" field: . (3.2) Of particular interest is the agent’s expended work per cycle in the case of . (3.3) Clearly the net work per cycle, expended to counteract the mechanical and electromagnetic inertial reaction forces, is zero. In the case of the electromagnetic part, "inertial" energy flows from agent to field part of each cycle, and back from field to agent the remainder of the cycle. The work expended each cycle to counteract the . (3.4) 4. Newton’s Inertial Reaction Force … Also an Electric Force?
Although the (mechanical) inertial reaction force is somewhat mysterious in the case of uncharged particles (such as atoms), it is interesting to speculate whether these reaction forces might also be electromagnetic. For example, the electromagnetic field outside an atom is nominally zero. But there are significant fields within. Might the atom’s resistance to being accelerated ultimately be explained by electric forces on the nucleus and orbital electrons? It is a question worthy of investigation, perhaps in a follow-on article dealing with the interactive forces between multiple charge distributions. Suffice it to say here that an external agent will also generally have to counteract such interactive forces in order to maintain a given motion. (Special case: two charge distributions at rest or moving with a common, constant velocity. Clearly some agent must counteract the attractive/repulsive forces between the distributions.) 5. In Conclusion.
The equation of motion for a tiny (R<<1 m) spherical shell of charge is, in one dimension, . (5.1) If the charge’s motion is given, then the agent force needed to drive the charge is readily calculated. Solving for the motion when F Eq. 5.1 (the spherical shell of charge version of Newton’s second law) applies only to The energy flow out of/into a surface that contains a dimensionally small oscillating charge is readily computed using the point charge field solutions. The flow per cycle is always positive (away into infinite space), and it equates to the agent work per cycle expended to counteract the radiation reaction force. In the case of uncharged particles, Newton’s second law applies. Among other things, the agent work per cycle, expended to counteract the mechanical inertial reaction force, is zero. Oscillating uncharged particles do not radiate. It is important to bear in mind what the net internal electromagnetic force determined by Lorentz et al (Eq. 1.1) acts upon. It is a reaction force that acts upon the external agent which maintains the charge’s shape. The external agent’s role does not vanish if and when the In Maxwellian theory a spherical shell of charge can persist in time only if an agent non-electromagnetically holds the charge together. Indeed this is true of all distributions of charge. Every increment of charge in a distribution repels every other increment, and an external agent must counteract each tiny repulsive force if the distribution is to maintain its shape. Although often not explicitly mentioned, these counteracting agent forces are always implied. The good news is that, like the electromagnetic forces they counteract, they sum to zero provided the distribution is at rest or moves with constant velocity. Lorentz and others derived the sum of the internal, interactive Lorentz (electromagnetic) forces when a spherical shell of charge’s velocity is (1.1) For shells with radii R<<1 meter, the net internal force can be simplified to . (1.2) Here again in order for the accelerated shell to maintain its shape, an external agent must counteract each infinitesimal internal force; the . (1.3) It seems clear in Eqs. 1.2 and 1.3 that F 2. The Equation of Motion for Charged Particles.
Now there are certain interesting differences between charge and "neutral matter" so far as momentum and kinetic energy are concerned. The mechanical momentum and kinetic energy of an uncharged particle appear to be local attributes of the particle. The electromagnetic momentum and kinetic energy of a , (2.1) and . (2.2) Obviously there is motivation to define the shell’s "electromagnetic mass" to be . (2.3) Eq. 1.3 can then be more compactly stated as . (2.4) Or, since all known charged particles appear to have both mechanical and electromagnetic mass, . (2.5) In the case of Quite often Eq. 2.5 is . (2.6) To the extent the da 3. Energy Conservation and Agent-Particle/Field Dynamics.
The da As discussed in the previous article, the agent’s counteraction to the radiation reaction force provides a dynamic basis for the energy that flows to/from the particle’s . (3.1) Similarly the agent’s counteraction to the electromagnetic inertial reaction force does the same for the energy that flows to/from the particle’s electromagnetic "inertial" field: . (3.2) Of particular interest is the agent’s expended work per cycle in the case of . (3.3) Clearly the net work per cycle, expended to counteract the mechanical and electromagnetic inertial reaction forces, is zero. In the case of the electromagnetic part, "inertial" energy flows from agent to field part of each cycle, and back from field to agent the remainder of the cycle. The work expended each cycle to counteract the . (3.4) 4. Newton’s Inertial Reaction Force … Also an Electric Force?
Although the (mechanical) inertial reaction force is somewhat mysterious in the case of uncharged particles (such as atoms), it is interesting to speculate whether these reaction forces might also be electromagnetic. For example, the electromagnetic field outside an atom is nominally zero. But there are significant fields within. Might the atom’s resistance to being accelerated ultimately be explained by electric forces on the nucleus and orbital electrons? It is a question worthy of investigation, perhaps in a follow-on article dealing with the interactive forces between multiple charge distributions. Suffice it to say here that an external agent will also generally have to counteract such interactive forces in order to maintain a given motion. (Special case: two charge distributions at rest or moving with a common, constant velocity. Clearly some agent must counteract the attractive/repulsive forces between the distributions.) 5. In Conclusion.
The equation of motion for a tiny (R<<1 m) spherical shell of charge is, in one dimension, . (5.1) If the charge’s motion is given, then the agent force needed to drive the charge is readily calculated. Solving for the motion when F Eq. 5.1 (the spherical shell of charge version of Newton’s second law) applies only to The energy flow out of/into a surface that contains a dimensionally small oscillating charge is readily computed using the point charge field solutions. The flow per cycle is always positive (away into infinite space), and it equates to the agent work per cycle expended to counteract the radiation reaction force. In the case of uncharged particles, Newton’s second law applies. Among other things, the agent work per cycle, expended to counteract the mechanical inertial reaction force, is zero. Oscillating uncharged particles do not radiate. It is important to bear in mind what the net internal electromagnetic force determined by Lorentz et al (Eq. 1.1) acts upon. It is a reaction force that acts upon the external agent which maintains the charge’s shape. The external agent’s role does not vanish if and when the In Maxwellian theory a spherical shell of charge can persist in time only if an agent non-electromagnetically holds the charge together. Indeed this is true of all distributions of charge. Every increment of charge in a distribution repels every other increment, and an external agent must counteract each tiny repulsive force if the distribution is to maintain its shape. Although often not explicitly mentioned, these counteracting agent forces are always implied. The good news is that, like the electromagnetic forces they counteract, they sum to zero provided the distribution is at rest or moves with constant velocity. Lorentz and others derived the sum of the internal, interactive Lorentz (electromagnetic) forces when a spherical shell of charge’s velocity is (1.1) For shells with radii R<<1 meter, the net internal force can be simplified to . (1.2) Here again in order for the accelerated shell to maintain its shape, an external agent must counteract each infinitesimal internal force; the . (1.3) It seems clear in Eqs. 1.2 and 1.3 that F |