# Flux cutting in a uniform magnetic field

Say we have a uniform magnetic field acting into the page, and we have a bicycle wheel which is spinning at a constant tangential velocity $v$. Each spoke has a equal length, $l$. Reading has told me that there is a constant EMF induced in the spokes, but I fail to see why - according to Faraday's law $$E = -N d\phi/dt$$ there shouldn't be a flux induced, as the flux doesn't change when the spokes cut across it? If the field is uniform, and spokes simply move across but are always perpendicular to the field, I don't understand why $d\phi$ isn't 0. For a similar reason, I don't understand why there is a force when a dc current is in a magnetic field - no emf would be induced, so why does Fleming's left hand rule say there will be a force acting?

Consider a single proton moving at velocity "v" from left to right through the uniform magnetic field. The force on the proton will be perpendicular to both the direction of motion and the direction of the magnetic field. This means that the proton will initially experience a force that is "up" in your drawing.

Now, as an extension of this example, assume that you have some type of "container" for a group of protons, and you orient this container in a "vertical" direction in your drawing, and move the container at velocity "v" from left to right through the uniform magnetic field. All of the protons in the container will feel an upwards force on them, and they will tend to accumulate at the top of the container.

As a further extension of this example, the spokes of the bicycle wheel are normally composed of a metal, which typically has one free electron per atom available to move within the metal. In reality, this makes the wheel's spokes the containers for the charged particles, which are actually electrons rather than protons, and those electrons moving through a magnetic field will also feel a force on them. Despite the fact that the bicycle wheel is turning in a circle, and despite the fact that the magnetic flux experienced by the bicycle wheel is constant, the electrons in the spokes are always moving perpendicular to the stated magnetic field, so they always feel a magnetic force on them. This means that electrons will move towards the center of the wheel, and there will be a net positive charge on the wheel's periphery, resulting in a voltage gradient between the rim of the wheel and the center of the wheel.

The logic behind this could be that when the spokes move each one of them traces an area per sec. There are free electrons in the spoke. Assume that the spoke is divided into small parts. Electrons of a part have a velocity which is equal to the velocity of that part of the spoke, since they are perpendicular to the magnetic field they experience a force. Each electron shifts to one side of the spoke due to force given by F = BeV. Thus creating a potential difference/EMF.

Consider a free electron near the outer edge of the spoke and another free electron near the centre. Since one is farther away from the centre it will have a velocity greater than the one nearer to the centre. Also the force acting on the electron will be directed towards the centre of the wheel. Since the force on the outer electrons (electrons with greater tangential velocity) is more, there will be a net current towards the centre of the wheel and cause the outer edge to gain a positive charge and the centre to gain a negative charge which will be responsible for the constant EMF. Current will stop flowing the moment this induced EMF is successful at stopping electron migration from outer edge to centre. Hope it clears up your confusion.