# EMF induced in a wire loop rotating in a magnetic field

LCSphysicist
Homework Statement:
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Relevant Equations:
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To solve this problem, we need to evaluate the following integral: $$\epsilon = \int_{P}^{C} (\vec v \times \vec B) \vec dl$$

The main problem is, in fact, how do we arrive at it! I can't see why a Electric field arises at the configuration here. The magnetic field of the rotating sphere is time independent ##(\frac{ d \omega }{dt}) = 0##. The magnetic dipole at the center is also time independent.

So why do a electric field arise? Worst: Why do it arise and is equal to ##\vec v \times \vec B##?

Homework Helper
Gold Member
Have you studied motional emf?

LCSphysicist
Have you studied motional emf?
It is more about the difficult to see that the area to calculate the flux is constant in avarage, but in infinitesimally time it is varying?

Homework Helper
Gold Member
So why do a electric field arise? Worst: Why do it arise and is equal to ##\vec v \times \vec B##?
The Lorenz force ##F_L=(v\times B)q## makes the electrons move, and this creates charge separation or simply charge density ##\rho\neq 0##. This charge density creates electric field according to Gauss's law $$\nabla\cdot \mathbf{E}=\rho$$.
How do we know that the line integral of this electric field equals to $$\int_C^P \mathbf{E}\cdot d\mathbf{l}=\int_C^P(\mathbf{v}\times\mathbf{B})\cdot d\mathbf{l}$$.
Well we also impose the additional equilibrium condition $$F_E=F_B\Rightarrow \mathbf{E}q=-(\mathbf{v}\times\mathbf{B})q$$