Two electromotive forces existing at the same time how?

[PhiowPhi]

Something a bit boggling to think, which I accepted as a fact in nature, but might anyone explain as how is it possible to have to electromotive forces(EMF) at the same time in the same conductor?

Assume a circuit placed in a magnetic field, that is connected to a voltage source(##\epsilon_1##) that would allow current flow, assume there is there is a Lorentz force(##F_L##) on the wire with length ##L## with respect to the magnetic field, and it starts to move. Based on the laws of electromagnetic induction, there is an induced ##-\epsilon##, with the original ##\epsilon_1## in that same conductor with length ##L##? How can they exist at the same time? I can't picture it... Induced EMF(from the voltage source) and back EMF from the electromagnetic induction due to the change in flux over time.

It somewhat helps to imagine it as two force vectors acting against one another and influencing the free charges, I know this incorrect, but it's my interpretation which I hope someone could enlighten me with.

 

[nasu]

The "two" exist in our brain, in the way we analyse the nature by breaking it into parts and looking at the parts separately. We do this with all kind of phenomena, motion, forces, electric fields, etc. Of course that in the conductor there is "one" field producing the motion of the charge carriers.
But even that "one" is actually just a model we use to understand how things work.

 

[Jano L.]

...might anyone explain as how is it possible to have to electromotive forces(EMF) at the same time in the same conductor?

...Based on the laws of electromagnetic induction, there is an induced ##-\epsilon##, with the original ##\epsilon_1## in that same conductor with length ##L##? How can they exist at the same time?

There are more emfs at the same time because there are more bodies acting with forces on the current-carrying charges in the circuit.

Electromotive force due to body ##A## is integral of electromotive intensity ##\mathbf E_A^*## due to that body along the path ##\gamma## running through the circuit:

$$
emf_A = \oint_\gamma \mathbf E_A^*\cdot d\mathbf x.
$$

Electromotive intensity ##\mathbf E^*## is current driving force intensity (per unit charge), most often due to chemical gradient (battery) or temperature gradient (Seebeck effect). Electric strength ##\mathbf E## (it has the same units) may also be included as one kind of electromotive intensity, due to presence of electric field.

Particular emf just quantifies partial effect of particular ##\mathbf E^*## on the whole circuit in Volts.

Because emfs due to different bodies are integrals of independent force intensities, they add as real numbers.

Total electromotive intensity ##\mathbf E_{tot}^*## is a sum of electric intensity ##\mathbf E## and all other electromotive intensities present.

And naturally total electromotive force in a circuit is a sum of emfs due to electric strength and due to all other electromotive intensities present.

It somewhat helps to imagine it as two force vectors acting against one another and influencing the free charges, I know this incorrect, but it's my interpretation which I hope someone could enlighten me with.

It's not terribly incorrect - emfs can act sympathetically or they can oppose each other just as forces do. The only difference is that emf is a number assigned to one of two possible ways of how current may flow in the circuit. Force is a 3-component vector.