EMF induced in moving rod in B-field, why is "L" length of wire frame?

[phantomvommand]

Homework Statement

See picture below

Relevant Equations

E = BLV

Why is the EMF induced, per the formula E = BLV, calculated with L as the length of the wire frame, instead of the length of the rod?
Don't charges throughout the rod (including in the parts beyond the wire frame) move due to a Lorentz Force qvB, so EMF = work done in moving a unit charge through the rod = 1/q(qvBL), where L is the length of the entire rod, not just the part along the wire frame?

Thanks

 

[BvU]

Hi,

You want to distinguish the EMF over he length of the rod on the one hand
and the EMF that causes a current in the resistor on the other hand.

What is the verbatim text of the exercise as given to you ?

$\$

 

[phantomvommand]

Hi,

You want to distinguish the EMF over he length of the rod on the one hand
and the EMF that causes a current in the resistor on the other hand.

What is the verbatim text of the exercise as given to you ?

$\$

Taken from https://pressbooks.online.ucf.edu/osuniversityphysics2/chapter/motional-emf/

I had assumed that the answer would require the length of the rod, not just the width of the wire frame.

 

[BvU]

I understand your problem. Can't call it confusion because IMO the problem statement is just too vague.

$\$

 

[kuruman]

I understand your problem. Can't call it confusion because IMO the problem statement is just too vague.

I agree with this assessment. We are not told the two points across which the induced emf is to be found. We have$$\text{emf}=\int_a^b( \mathbf v\times\mathbf B)~\cdot d\mathbf l.$$If the rails were not there, the only sensible assignment to points $a$ and $b$ is the ends of the rod. There is no current in the rod.

With the rails in place, we have a rod moving at constant velocity that is part of a closed circuit. There will be an induced current, $I_{\text{ind}}$. We further have to assume that the rod has mass $m$ and the closed circuit some resistance. All this implies a constant force pulling the rod which has equal magnitude to the opposing Lorentz force $\mathbf {F}_{\!L}=I_{\text{ind}}\mathbf L\times \mathbf B$. In this case, it is sensible to identify points $a$ and $b$ as the points of intersection of the rod with the rails because it is only part of the overall induced emf across the ends of the rod that drives the induced current.

This ambiguity is analogous to asking for the potential energy of a system where there could be more than one sensible points to take as the zero of potential energy.