Calculating Motional EMF when v not perpendicular to B

[vaizard]

Hi,

I'm trying to figure out how you can calculate motional EMF when the velocity of the object in question is not perpendicular to the magnetic field. There are two equations in my textbook, but the text describing them is not very helpful.

The first is [tex]\varepsilon = vBL[/tex], which can be used when [tex]B\perp v[/tex] and [tex]B\perp L[/tex]. The second is [tex]\varepsilon = \oint (\vec v \times \vec B) \cdot d\vec l[/tex] which is the general form. The first one won't work if they're not perpendicular, and I don't understand what the [tex]d\vec l[/tex] is for in the second one. Could someone explain that to me?

Thanks!

 

[tiny-tim]

… I don't understand what the [tex]d\vec l[/tex] is for in the second one. Could someone explain that to me?

Hi vaizard!

v x B is the magnetic force vector

dl is a short length of the wire

the force won't necessarily be along the wire, so (v x B).dl is the component of force along the wire

 

[astrokreng]

Hello,

Calculating motional EMF when the velocity is not perpendicular to the magnetic field can be a bit confusing at first, but it's actually quite simple. The key is to understand the concept of the magnetic flux, which is the measure of the magnetic field passing through a given area. In this case, the area is represented by the path of the object, which is described by the vector d\vec l in the second equation you mentioned.

To explain further, the second equation is known as the general form of the motional EMF equation. This is because it takes into account all possible orientations of the velocity and magnetic field. The vector d\vec l represents an infinitesimal length along the path of the object, and when multiplied by the cross product of the velocity and magnetic field, it gives the magnetic flux through that small segment of the path. By integrating this equation over the entire path, we get the total motional EMF.

Now, you may be wondering why the first equation only works when the magnetic field and velocity are perpendicular. This is because in this case, the magnetic flux through the entire path is simply the product of the magnetic field, velocity, and length of the path (vBL). This is a simplified version of the general equation, which assumes that the magnetic field and velocity are always perpendicular to each other.

I hope this helps clarify the concept for you. If you have any other questions, please don't hesitate to ask. Best of luck in your studies!