Difference between flux of wire loop and wire in a uniform magnetic field

How come when a wire loop sweeps out an area in a uniform magnetic field and moves with constant velocity there is no induced current because there is no change in flux. However, when a straight wire segment sweeps out an area by moving perpendicular to the field(same orientation as loop), there is a change in flux. Why is this? Why does the right hand rule for lenz' law work in this case?

• The loop area doesn't change, but the moving wire segment changes the enclosed flux area by $dA/dt=l*v$. – CuriousOne Jan 11 '16 at 22:49

1 Answer

You have a misunderstanding of flux. Recall that flux is given by $$\Phi=\iint \vec{B}\cdot d\vec{a}$$. Note that there is a necessary differential amount of area that must be considered. Therefore, since a straight wire has no area through which the flux can change, there will be no induced emf.

The reason that there is an emf in the wire is by motional emf. This is deduced from the lorentz force law $$\vec{F}=\vec{E}+q(\vec{v}\times \vec{B})$$. Note that there are free electrons in the wire, each one of which following this law. Lenz's law does not apply here because Faraday's law does not. The right hand rule you use to see where the electrons will go is nothing more than that defined in the vector(cross) product.

• There are no two different mechanisms that cause an induced voltage. In order to measure an induced voltage on a wire segment one needs non-moving segments to pick it up. Those segments will form a flux loop, whether you like it, or not, so you can as well calculate with the integral. – CuriousOne Jan 11 '16 at 23:02
• @CuriousOne "one needs non-moving segments to pick it up". Can I not attach a voltmeter to the two ends of the bar, such that the voltmeter moves with the wire? – Declan Jan 12 '16 at 3:43
• @Declan: Yes, you can, but that brings you back to the case of the closed moving loop and you won't see a potential. Whatever you gain on the wires segment you lose on the voltmeter leads. There are three ways of looking at this: local differential approach with Lorentz force, local integral approach with the integral over the loop area or relativistic approach with a moving observer who sees an electric field where the resting observer only sees a magnetic field. It all leads to the same results. – CuriousOne Jan 12 '16 at 6:50