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I have a wheel (free to spin around the $z-$axis) with four spokes that is connected by sliding contacts to a circuit with $U_0 = 0,72V$. Also, there is a B-Field parallel to the $z-$axis Imgur

For the induced electric potential I have: $$U_i = - \frac{1}{2}(R_{outer}^2 - R_{inner}^2)\omega B$$

(with $\omega$ = angular velocity)

I'm now asked to find out the constant $\omega_0$ after the system is in a stationary state (moves with constant speed). The assignment points out to look at one mesh (with one spoke) on the wheel and to determine if Kirchhoffs second rule applies and if there is a current flowing in a mesh.

The most obvious way I can think of would be (since $U_0$ should be equal in any spoke): $$U_0 - U_i = 0$$ and to solve for $\omega$. But elsewhere I was told that Kirchhoffs rules don't apply in systems with changing magnetic fields. Also I'm not sure if there still would be an emf induced in stationary conditions since the flow wouldn't change anymore then.

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Let's explain the principle of rotation here from the scratch.

Say you have the circuit with 1T magnetic field perpendicular to it but the battery is intially switched off. Obviously the wheel will not rotate now as there is no moving charge and hence no Lorentz Force . Now when you switch on the battery, electrons will flow through the spokes from outer rim to inner rim.

Now due to this motion of the electrons, the wheel will start to rotate clockwisely( just apply Lorentz force $\vec{F}=-e(\vec{v}\times \vec{B} )$ ,where e is positive). Now the electrons have another velocity due to motion of the wheel in addition to the radial velocity due to the battery. Remember this velocity starts from zero as the wheel was initially not rotating.

Due to this new velocity (which is increasing from zero), the electrons in the spoke will feel a force along the outward direction( again apply Lorentz force due to this new velocity). See as the wheel rotational speed increases this force increases, which reduces the velocity of the electrons flowing from outer rim to inner rim.

After some time a situation comes when there is no flow of electrons from outer rim to inner rim, which you have described by $\ U_0=U_i$. At this situation there will be no Lorentz force perpendicular the spokes. As there is perpendicular force on the spokes, if you consider Torque( = $\vec{x}\times \vec{F}$),then that will be zero. Hence angular momentum $\vec{L}(=I\vec{\omega}$) is constant.

"But elsewhere I was told that Kirchhoff's rules don't apply in systems with changing magnetic fields."

But before answering this, at first let me answer the following:

"Also I'm not sure if there still would be an emf induced in stationary conditions since the flow wouldn't change anymore then."

Obviously, there will be induced emf in the stationary condition. See Lorentz force is still present along radially out on the electrons due to the wheel rotation.

See, I don't know why you want to use KVL or something. You have some induced emf in the spoke which is opposing the battery so that no current can flow in the stationary state. Nothing else.

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