While reading an Introduction to Electrodynamics by D.J. Griffiths if became confused about what happens in electrodynamics when a macroscopic charged body, could be either a conductor or dielectric, is moving with respect to an inertial reference frame. I'm experiencing difficulty understanding this with Maxwell's equations, because in the book all problems deal with continuous charge and current distributions that do not change position in time, mainly circuits. My reasoning is that each charge element $dq_i$ on the moving body, such as a moving sphere, has a position $\vec{r_i}(t)$ and velocity $\vec{v_i}(t)$ creating a velocity vector field $\vec{v}(x,y,z,t)$ that gives a current volume density $\vec{dJ}(x,y,z,t)=\vec{v}(x,y,z,t)dq$ at each position where there is a moving charge element. Now since the body moves all charge elements $dq$ change position, hence the local charge density $\rho(x,y,z,t)$ and current volume density $\vec{J}(x,y,z,t)$ must both change. Therefore, since $\rho$ and $J$ are changing with time I suspect the electric and magnetic fields at the points where the body used to be a little time $dt$ earlier must also change. I believe my goal here is to understand the relation between what happens to the dynamics of a body under the influence of electrodynamic forces. Anyhow, how would one analyze (via Maxwell's equations and Lorentz-force law) the E-fields and B-fields when charged rigid bodies, for example, move and accelerate?
1 Answers
My reasoning is that each charge element $dq_i$ on the moving body, such as a moving sphere, has a position $\vec{r_i}(t)$ and velocity $\vec{v_i}(t)$ creating a velocity vector field $\vec{v}(x,y,z,t)$ that gives a current volume density $\vec{dJ}(x,y,z,t)=\vec{v}(x,y,z,t)dq$ at each position where there is a moving charge element. Now since the body moves all charge elements $dq$ change position, hence the local charge density $\rho(x,y,z,t)$ and current volume density $\vec{J}(x,y,z,t)$ must both change. Therefore, since $\rho$ and $J$ are changing with time I suspect the electric and magnetic fields at the points where the body used to be a little time $dt$ earlier must also change.
All of this is correct.
I believe my goal here is to understand the relation between what happens to the dynamics of a body under the influence of electrodynamic forces.
That will generally be a pretty difficult problem because it typically turns the problem into a highly coupled set of equations. Most of the time this will require a numerical solution, often using techniques like finite element modeling.
Anyhow, how would one analyze (via Maxwell's equations and Lorentz-force law) the E-fields and B-fields when charged rigid bodies, for example, move and accelerate?
For a point charge you can use the Liénard–Wiechert fields. For a more general charge and current distribution you can use the retarded potentials which become particularly simple in the Lorenz gauge. Specifically, in the Lorenz gauge the potentials are: $$ \phi(t,\vec r) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho(t_r,\vec R)}{|\vec r - \vec R|}d^3 \vec R$$$$ \vec A(t,\vec r) = \frac{\mu_0}{4\pi}\int \frac{\vec J(t_r,\vec R)}{|\vec r - \vec R|}d^3 \vec R$$ where $t_R = t-|\vec r - \vec R|/c$ is the retarded time.
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