Why is a constant magnetic field able to move individual charges, but must be varying if it is to generate current through a wire of constant area? Isn't current just moving charges? I was thinking about this, and thought, maybe it is because the electron drift velocity is zero, so it doesn't experience a force, but then I thought again, if we use qvb, no matter how much the field changes, the electrons will never move. So why does this phenomenon happen? A theoretical, intuitive explanation would be great. Thanks!
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$\begingroup$ What is the setup you are considering: Permanent magnet in the vicinity of a wire? $\endgroup$– SemoiApr 25, 2020 at 7:37
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A constant magnetic field is only able to move a particle carrying a charge if that particle is moving through the field. If the particle is moving through the field at some velocity $\bar{v}$ then the motion of the particle carrying the charge is governed by the Lorentz force equation
$$\bar{F}=q\left(\bar{E}+\bar{v}\times\bar{B}\right),\qquad\qquad\qquad(1.1)$$ where $q$ is the charge, $\bar{E}$ the external electric field, and $\bar{B}$ the magnetic field in question. We can see from Eq.(1.1) that the particle carrying some charge $q$ does indeed experience a force when under motion at velocity $\bar{v}$. Now, for the case of a wire loop with some constant magnetic field which is normal to the plane of the loop, charges in the wire (electrons) do indeed have a mean velocity of $\langle\bar{v}\rangle=\bar{u}=0$ which can also be expressed in terms of the drift velocity $\bar{u}=\mu\bar{E}$, where $\mu$ is the electron mobility of the material the wire loop is made from, considering there is no external electric field in your example there is no drift velocity.
Now, what happens in the case of a changing magnetic field? Here we turn to the differential form of the Maxwell-Faraday equation, $$\nabla\times\bar{E}=-\frac{\partial\bar{B}}{\partial t}.\qquad\qquad\qquad(1.2)$$ We see in Eq.(1.2) that a changing magnetic field in time generates an electric field which is circulating around the direction of the magnetic field. Such that, in the expression for the electron drift velocity or the Lorentz force Eq.(1.1) there is now at all points a component of the electric field parallel to the wire loop, thus creating an electric current.
