# Generation of back emf with relativistic approach to EM Waves

I learnt that Magnetic force around a current carrying conductor is nothing but relativistic electrostatic force observed by a charge particle from its frame of reference. But I am not able to understand through this approach, how back emf is produced when the current through the conductor is stopped.

How do I explain it?

Maxwell by itself doesn't say that there is a particular electric or magnetic field due to charges or currents. Maxwell requires you fix boundary conditions as well as charges and currents before you can solve for electric or magnetic fields.

Now if you have initial fields and you have currents, then you could use Maxwell to find out how the fields evolve in time. In which case things change by:

The back EMF through a stationary thin conducting loop, is not due to a magnetic force. It's caused by an electric field.

The EMF due to the electric field around a loop happens to equal the flux of $-\partial\vec B/\partial t$ because they have a common cause.

And there are two fundamental ways they caused. Either electric and magnetic fields each exist, and you take their initial values and from them get how they change over time from Maxwell, and get:

$$\frac{\partial B}{\partial t}=-\vec \nabla \times \vec E$$ and

$$\frac{\partial E}{\partial t}=\frac{1}{\epsilon_0}\left(-\vec J +\frac{1}{\mu_0}\vec \nabla \times \vec B\right).$$

And with this approach, both the fields (electric and magnetic) are real and have their own values and the electric field tells you how the magnetic field changes, and the imbalance between the magnetic field and the current tells you how the electric field changes.

If you don't know the current, this isn't directly applicable. But you seemed to ask about the result of a changing current. So this works fine if you already know the initial fields and you know the current. but of course this has the causality indicate that the flux of $-\partial\vec B/\partial t$ is literally caused by the electric EMF and so it doesn't answer why you have an electric EMF. But the electric EMF happened because of the electric field now, which itself depends on the electric field in the past and how the electric field changes. And it changed when the current wasn't perfectly balanced by the magnetic field, and that imbalance happened when you changed the current.

So really, when the electric current doesn't match up with the magnetic field perfectly then the electric field changes. And when it changes it can change into something that makes magnetic fields change.

So just look at the two equations of Maxwell as telling you how things evolve.

And it works even for electromagnetic waves propagating in empty space, and this is simply what Maxwell says.

But if your goal was to directly assign fields due to charges and currents, there are examples such as Jefimenko's equations:

$$\vec E(\vec r,t)=\frac{1}{4\pi\epsilon_0}\int\left[\frac{\rho(\vec r',t_r)}{|\vec r -\vec r'|}+\frac{\partial \rho(\vec r',t_r)}{c\partial t}\right]\frac{\vec r -\vec r'}{|\vec r -\vec r'|^2} -\frac{1}{|\vec r-\vec r'|c}\frac{\partial \vec J(\vec r',t_r)}{c\partial t}\mathbb{d}^3\vec r'$$ 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.

So when the current changes at place-time $(\vec r_1,t_1)$, there is an electric and a magnetic field. But the field exists only at place-times $(\vec r_2,t_2)$ where $t_2=t_1+\frac{|\vec r_2-\vec r_1|}{c}$.

So when the current changes, there is a spherical shell that expands at the speed $c$ and on that shell there is both an electric and a magnetic field.

And sure, if you want forces, and you use the Lorentz Force then all you need is the electric field in the instantaneously comoving frame of the charge. So you can ignore the magnetic field to compute forces with the Lorentz Force if you are willing to change frames every instant.

But you don't need to ignore the magnetic forces for a stationary loop becasue the magnetic forces contribute zero EMF in a stationary loop.