Let a bunch of charge move with a constant velocity $\mathbf v\;.$ Since, the charges are moving, they would create magnetic field $\bf B$ as it is current that produces magnetic field.
Now, would $\bf B$ affect these moving charges which themselves create $\mathbf B\;?$
My thoughts:
For electrostatics, electric field is given by the formula $$\mathbf E = \frac{q}{r^2}\hat {\mathbf {r}};$$ since $r= 0$ for the same charge that created the field, that would imply the electric field is exerting an infinite force on the charge which is not possible.
Now, if the bunch of charge is moving to constitute a steady-state current $I$ through a small-section $d\mathbf l,$ then magnetic field due to them is given by Biot-Savart Law:
$$d\mathbf B= \frac{I\;d\mathbf l\times \hat{\mathbf{r}}}{r^2}\;;$$ force on each charge moving through $d\mathbf l$ is given by $$\mathbf F= \mathbf v\times\frac{I\;d\mathbf l\times \hat{\mathbf{r}}}{r^2} \;;$$ since $d\mathbf l$ & $\hat{\mathbf r}$ are in the same direction, the numerator is zero & in the denominator $r$ is zero which would again make the force infinite. This is not possible.
However, when the current is non-steady, the moving charges produce time-dependent magnetic field which results in self-induction.
The self-induced emf caused due to change in flux of the magnetic field created by the charges does work on the charges; each charge feels force $\mathbf F= q\; \mathbf v\times \mathbf B $ where $\mathbf B$ is created by themselves.
So, that means, charges are moving & they are creating a magnetic field which again acts on the charges which created the field.
At steady state, charges are not affected by their own magnetic field whereas on the other hand, at non-steady state, the magnetic field, that those moving charges produced, imparts force on each charge leading to the production of self-emf.
But why is it so?
Fields are independent entities; they can share momentum & energy with charges as said by Timaeus.
Charges are indeed affected by their field when they read. But up to my reading, I've not found, at non-steady state of current, charges radiate. Still they are affected by the changing flux of their own created magnetic field. This produces self-emf; which means the magnetic field impart force on those moving charges which create the field.
However, why is it that moving charges are not affected by their own magnetic field when the current is at steady state but at non-steady state, charges are affected by the force from the magnetic field they create?
If charges are indeed affected by the magnetic field they create, is there any way to mathematically prove that during self-induction, the moving charges are forced by their own created magnetic field?
I'm quite confused; am I missing something here? Please help.
Best Answer
Neither Coulomb nor Biot-Savart are correct equations for the electromagnetic field except in statics. There are time dependent generalizations, such as Jefimenko's equations.
$$\vec E(\vec r,t)=\frac{1}{4\pi\epsilon_0}\int\left[\frac{\rho(\vec r',t_r)}{|\vec r -\vec r'|}+\frac{\partial \rho(\vec r',t_r)}{c\partial t}\right]\frac{\vec r -\vec r'}{|\vec r -\vec r'|^2}\; \mathrm{d}^3\vec{r}' -\frac{1}{4\pi\epsilon_0c^2}\int\frac{1}{|\vec r-\vec r'|}\frac{\partial \vec J(\vec r',t_r)}{\partial t}\mathbb{d}^3\vec r'$$ and $$\vec B(\vec r,t)=\frac{\mu_0}{4\pi}\int\left[\frac{\vec J(\vec r',t_r)}{|\vec r -\vec r'|^3}+\frac{1}{|\vec r -\vec r'|^2}\frac{\partial \vec J(\vec r',t_r)}{c\partial t}\right]\times(\vec r -\vec r')\mathbb{d}^3\vec r'$$ where $t_r$ is actually a function of $\vec r'$, specifically $t_r=t-\frac{|\vec r-\vec r'|}{c}.$
These reduce to Coulomb and Biot-Savart only when those time derivatives are exactly zero, which is statics. So Jefimenko is an example of proper time dependent laws for the electromagnetic field. Note that both the electric and the magnetic part of the electromagnetic field have parts that depend on the time variation of current. Faraday's law relates the two together.
And the next fact is also key: the EMF around a stationary wire is 100% due to the electric field. And if you have a universe with neutral objects everywhere then Jefimenko predicts the electric field is solely and 100% caused by time variation of current. And I do mean cause, as in cause and effect. The equations are causal since the charge (and change in charge) and the current (and change in current) are things in the past ($t_r\leq t$) that affect the present.
So lets review. Coulomb and Biot-Savart are incorrect when not in statics. Self induction of a stationary wire is caused by electric fields making a nonzero EMF. These electric fields are caused by time varying currents. And yes, these time varying currents also cause (and yes I do mean cause) magnetic fields.
And when people say that changing electric fields cause magnetic fields and vice versa, they are not this kind of causality where one causes the other, they are just equality since they related two things that happen at the same time. Causality relates the present (the effects) to the past (the causes). To quote Jefimenko:
Now let's get to self fields. It is easy to imagine that each charge only reacts to the fields of the other charges. But do not over simplify when considering radiation reaction and other such complications.
In reality you have charges and fields. And you have to specify both. Each has energy, each has momentum, and they exchange energy and momentum with each other. Fields are not just mathematical fictions to compute forces between particles, they are real things with their own degrees of freedom, their own energy, momentum, and even stress.
Then you can find out the energy and momentum of the fields and the charges and then find out how they mutually exchange energy and momentum amongst themselves.
That is not what everyone says. Self force is much more general than radiation reaction forces. The most common case of harmonic motion had the particle gain as much energy as it loses from the inductive fields (Schott fields, that fall off faster than the radiation fields but still store energy) and only on average loses energy to the radiative fields.
Lots of people study this. Stephen Lyle wrote an entire book on just a uniformly accelerating charged particle and an entire book on self forces. Herbert Spohn wrote an entire book on the mutual coupled dynamics of electromagnetic fields and charges. And Fritz Rohrlich wrote a classic book on Classical Charged Particles. And those are just books, there are probably at least a dozen articles published every year for at least the last hundred years on the topic.
So. There are lots of self forces. Debate continues to this day about whether it is changing forces, jerk (changing acceleration), or acceleration itself that causes self forces. Or whether it's something completely different.
Keep in mind that a textbook case of a classical sized wire is usually discussing macroscopic fields, where you average over a large enough number of atoms that the fields and charge and current become smoother fields that don't jump around due to thermal effects or based on where in the lattice of the solid you are. And when someone does care about those effects, they use statistical physics and/or quantum physics. Not just pure Maxwell.