# Can the inducing dynamic magnetic field exerce a lorentz force on eddy currents?

Suppose I have a loop placed above a conducting material, I run an alternative current through the loop, this will creat a varying magnetic field, which will creat eddy currents inside the conducting material, my question is :

(a) To make it easier, I will state something first (what I think) : the Lorentz force is the force that pushed the charges inside the material, creating eddy currents.

(b) My question : Can I look at the dynamic magnetic inducing field (the one created in the loop) and then look at eddy currents, and say I have a lorentz force that is going to act on the charges (eddy current) ???

Currect me if I am wrong, but I think Lorentz force can not be used twice in (a), where it pushes the charges in the eddy current direction. And in (b) where it will push them in a different direction (perpendicular to eddy current direction)

I feel very pedantic writing this, but I think it is an important part of the answer to your question -

The Lorentz Equation

$${\mathbf F} = q({\mathbf E} + {\mathbf v} \times {\mathbf B})$$

describes the force, ${\mathbf F}$, on a particle with charge, $q$, travelling with velocity, ${\mathbf v}$, in an electric and magnetic field, ${\mathbf E}$ and ${\mathbf B}$

Whereas Maxwell's equation for induction

$$\nabla \times {\mathbf E} = -{\partial {\mathbf B} \over \partial t}$$

is more relevant for the generation of an eddy current due to a change in the magnetic field.

This may be too simple an answer, but it seems to me that the change in magnetic field, $\partial {\mathbf B} \over \partial t$, generates an electric field, ${\mathbf E}$ inside the metal, which makes the eddy current flow.

So ${\mathbf E}$ is supplied by the changing magnetic field, but as I think your question implies, we also need to include the magnetic field $\mathbf B$ in the metal when we calculate the force on the electrons with the Lorentz equation.

So to calculate how the electrons move in the eddy current we first need to calculate the electric field driving them ${\mathbf E}$ from the changing magnetic field and then we can use the Lorentz equation to calculate the force on the electrons, but we cannot discount the ${\mathbf B}$ field in the magnetic field, which, of course, is changing.

How would you calculate this in practice?

If we know at $t=0$ values for ${\mathbf v}$, ${\mathbf E}$, ${\mathbf B}$ and $\partial {\mathbf B} \over \partial t$ the for some small time step $dt$ we can numerically solve the differential equations to find at $t=dt$ new values for ${\mathbf v}$, ${\mathbf E}$, ${\mathbf B}$ and $\partial {\mathbf B} \over \partial t$. The process can then be repeated for $t = 2dt$ etc. up until any time $t$ provided we are careful and make sure we don't introduce too many errors. Alternatively, in certain special circumstances we may be able to analytically solve for ${\mathbf v}$, ${\mathbf E}$, ${\mathbf B}$ and $\partial {\mathbf B} \over \partial t$ to get exact expressions for them as a function of time.

Finally, a much simpler example of this sort of calculation is simple harmonic motion for a mass on a spring where the force on the mass depends on the position of the mass, but the position of the mass is changing with time as the mass has a certain velocity. The force on the mass depends on the position of the mass, but the mass is moving with a velocity which depends on the acceleration, which depends on the position. Ultimately in that simple system all the physics is contained in the differential equation and we can get a nice analytical solution and understand what is happening. For your problem there are two partial differential equations (at least) and it is much more complicated, but ultimately the same principle applies that if we solve the differential equations (numerically or analytically) we can find out how the system behaves.

• Thank you for you answer. What I am saying is : the definition of Lorentz force (as you state it) :describes the force, F, on a particle with charge, q, travelling with velocity, v, in an electric and magnetic field, E and B, The charge has already a velocity inside an already existing fields (E and B). However in my case, it is the dynamic fields (E and B) that are creating the motion of the charge (meaning v in the definition must depend on B &E). I hope you understood what I wrote (I am not always clear :) Mar 14 '18 at 8:23
• @EdwardoNewagte - I have put some edits in. A similar but much much simpler system is simple harmonic motion for a mass on a spring where the force on the mass depends on the position of the mass, but the mass is moving with a velocity which depends on the acceleration, which depends on the position. Ultimately in that simple system all the physics is contained in the differential equation and we can get a nice analytical solution and understand what is happening. For your problem there are two (partial) differential equations (at least) and it is much more complicated,....
– tom
Mar 14 '18 at 12:05
• @EdwardoNewagte ...., but the principles are pretty much the same. v depends on E and B, but E and B are changing - and the changing B value changes E... etc.
– tom
Mar 14 '18 at 12:05
• @tom-- Thank you again for your time. My issue is not in the mathematical computations. To simplify my question. When you wrote Lorentz force formula : is the E field, the one that verifies MaxFar equation ? or is it independent. Is the B field the some of the dynamic and induced fields or just the dynamic field ? Thank you in advance Mar 14 '18 at 16:45
• @EdwardoNewagte the ${\mathbf E}$ field is, of course, the Electric Field - normally it is zero inside a metal conductor (or very close to zero), but the MaxFar Equation means that a change in the magnetic field, ${\mathbf B}$, will induce an electric field inside the metal. ( We normally write the change in magnetic field as $\partial {\mathbf B} \over \partial t$). The induced ${\mathbf E}$ field causes electrons to move inside the metal, this is what makes eddy currents. The magnetic field is ${\mathbf B}$..... I hope this helps, but do come back with more questions.....
– tom
Mar 15 '18 at 0:15