The factors determining the induced EMF in a wire

[Asmaa Mohammad]

The induced emf in a straight wire is determined by the equation (emf=Blv sinθ) where θ is the angle between the direction of the motion and the lines of the magnetic field, and frequently, I see diagrams like these explaining the phenomenon:

In those pictures the wire is perpendicular to the lines of the field, so it starts its motion of that state, and I was wondering what if the wire was like this:

In that diagram the angle between the wire and the field is not 90 it is θ1 and the angle between the direction of the motion and the magnetic field lines is θ2.
So, in the equation (emf=Blv sinθ) which angle will be involved here? θ1 or θ2? Or θ1=θ2? And then there will be no difference.
Note: in the last diagram the wire moves from position 1 to position 2.

 

[Orodruin]

The basic tule id that the induced emf is dependent on the change of magnetic flux through the circuit. Then you may compute it and get some special case results in given setups.

 

[Asmaa Mohammad]

The basic tule id that the induced emf is dependent on the change of magnetic flux through the circuit. Then you may compute it and get some special case results in given setups.

Actually I don't have given setups, this question came to my mind while studying the induced emf in a straight wire, and all the figures show that the wire starts its motion from the position where it is perpendicular to the field lines.
I wonder whether the equation still the same if the wire starts its motion from the position where there is an angle <90 between it and the magnetic field lines.

 

[Orodruin]

Actually I don't have given setups, this question came to my mind while studying the induced emf in a straight wire, and all the figures show that the wire starts its motion from the position where it is perpendicular to the field lines.
I wonder whether the equation still the same if the wire starts its motion from the position where there is an angle <90 between it and the magnetic field lines.

But you do have closed circuits in your examples. Regardless, you can compute the emf in the wire by considering the magnetic flux through the area swept by the wire per time unit.

 

[cnh1995]

The induced emf in a straight wire is determined by the equation (emf=Blv sinθ) where θ is the angle between the direction of the motion and the lines of the magnetic field, and frequently, I see diagrams like these explaining the phenomenon:

In those pictures the wire is perpendicular to the lines of the field, so it starts its motion of that state, and I was wondering what if the wire was like this:

In that diagram the angle between the wire and the field is not 90 it is θ1 and the angle between the direction of the motion and the magnetic field lines is θ2.
So, in the equation (emf=Blv sinθ) which angle will be involved here? θ1 or θ2? Or θ1=θ2? And then there will be no difference.
Note: in the last diagram the wire moves from position 1 to position 2.

In motional emf equation, θ is the angle between the velocity vector and magnetic field vector.

 

[Asmaa Mohammad]

In motional emf equation, θ is the angle between the velocity vector and magnetic field vector.

So, we would ignore the angle between the wire and the magnetic field lines and only consider the angle between the direction of motion and the magnetic field lines?

 

[Orodruin]

In motional emf equation, θ is the angle between the velocity vector and magnetic field vector.

This is not precisely true. It is true only when the wire is orthogonal to both velocity and field. The actual induced emf would be proportional to the triple product $\vec B\cdot (d\vec \ell\times \vec v)$. The cross product would be the area element swept per unit time and taking its scalar product with $\vec B$ gives the flux.