# Induced emf from a time varying magnetic field & motion of charges simultaneously?

Due to the current flow, supplied by the battery, and the production of the magnetic field($$-B\hat{k})$$, A Lorentz force($$f_L = IL\times B$$) will accelerate the rod.

My issue with this, is classical problems would have $$B$$-uniform and constant, but in reality it's neither. $$B$$ changes over time due to the addition of more current carrying elements $$dl$$ with the changes of $$x$$, leading to a sum $$B$$ that's greater at it's final position.

However, these problems it's always assumed that the induced $$\varepsilon$$ is due to the separation of charges caused by the magnetic force acting on them($$vBL$$), but isn't the an induced electric force due to the changed of B?

$$\varepsilon = - \frac{\delta\Phi}{\delta t} = - vBh$$

Wouldn't it be:

$$\varepsilon = - (vBh + \int \frac{\delta B}{\delta t} \cdot da)$$

You are right. Rigorously, one should write $${{e}_{ind}}=-\frac{d\Phi }{dt}$$ with $$\Phi =B(t)lx$$. It would give $${{e}_{ind}}=-lx\frac{dB}{dt}-Blv=-S\frac{dB}{dt}-Blv$$