Why does BIlvt equal Iet in the V=Blv derivation?

[jsmith613]

when proving the equation V=Blv (V = emf induced) my book says

"Consider the arrangement shown in the attachment. The rod moves as a result of the applied force and an emf is induced, causing electrons to flow round in a circuit.
However, the the rod is now a 'current carrying rod' in a magnetic field so feels a magnetic force opposing the applied force.

To move at a constant speed, the applied Force must be the same as the magnetic force.
Hence F(applied force)=BIl
As the rod is moving at a constand speed the work done by the applied force is equal to: W=F*d
Work done = BIl*vt"

So for this all makes sense. The next bit is confusing.
Electrical energy is produced from the work done (this I understand). When an emf produces a current for time t, the electrical energy produced = Iet (E = emf)"
This also makes sense!

Hence Iet (e = emf) = BIl*vt
Why is this true...apparently it has something to do with energy conservation but surely the work done by the force is not the same as the energy disappated by the current?

Thanks

 

[Muphrid]

My question is: why would you think they're not equal? The rod is not accelerating anymore. The magnetic field is trying to accelerate it, but all the work it does is being dissipated. It's this equilibrium that keeps the rod moving at a constant speed (instead of speeding up or slowing down).

 

[jsmith613]

My question is: why would you think they're not equal? The rod is not accelerating anymore. The magnetic field is trying to accelerate it, but all the work it does is being dissipated. It's this equilibrium that keeps the rod moving at a constant speed (instead of speeding up or slowing down).

but the eVt is NOT doing work on the rod (there is an external force doing work on the rod)?

 

[Muphrid]

Ah, I see, I misunderstood the situation. It's an applied force moving the rod and a magnetic force opposing the motion. Fair enough.

The point is that the magnetic field is responsible for the EMF: you're calculating the same quantity in two different ways--one in terms of the EMF induced by the magnetic field, and the other by just using the magnetic field directly. They're two different routes to getting the same result.

 

[jsmith613]

Ah, I see, I misunderstood the situation. It's an applied force moving the rod and a magnetic force opposing the motion. Fair enough.

The point is that the magnetic field is responsible for the EMF: you're calculating the same quantity in two different ways--one in terms of the EMF induced by the magnetic field, and the other by just using the magnetic field directly. They're two different routes to getting the same result.

oh i see..thanks