Why P = F*v = ε^2/R = (vBl)^2/R when there's no friction?

[Karagoz]

Homework Statement

This is taken from a problem with its solution. But there's one thing I didn't understand with the solution:

A closed conductor loop is located perpendicular to the field lines in a homogeneous magnetic field B. The conductor slice CD is first in rest. Then we pull a constant force F to the right. See figure above. The absolute value of the force is 1.8 N. The conductor slice slides without friction.

After a while the speed v is constant and is equal to 4.0 m/s. The length of the conductor slice CD is 12 cm.

The electrical power in the circuit is given by:

The effect of the pull force F is given by: P = F*v

Since we do not have friction, these powers must be the same. And this gives:

How is it that the effect on the pull force P = F*v is equal to the electrical power in the circuit given by:

?

What's the proof?

Homework Equations

P = F*v

P = ε^2/R = (vBl)^2/R

The Attempt at a Solution

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[kuruman]

You can calculate the current $I$ from Faraday's Law. You can then do two separate calculations, (a) find the force from $F=IlB$ and (b) find the power dissipated in the resistor from $P=I^2R$. If you put the two expressions side by side, you will see that $I^2R = Fv$.

On edit: When the rod is moving at constant speed (terminal velocity), the net force on it is zero, and that's when the magnetic force $IlB$ matches the externally applied constant force $F$.

 

[Karagoz]

You can calculate the current $I$ from Faraday's Law. You can then do two separate calculations, (a) find the force from $F=IlB$ and (b) find the power dissipated in the resistor from $P=I^2R$. If you put the two expressions side by side, you will see that $I^2R = Fv$.

On edit: When the rod is moving at constant speed (terminal velocity), the net force on it is zero, and that's when the magnetic force $IlB$ matches the externally applied constant force $F$.

But how can we find current using Faraday's law? We can only find ε using Faraday's induction formula.

How is P = F*v derived form P = I^2*R in this case?

 

[cnh1995]

But how can we find current using Faraday's law?

Using Ohm's law.

How is P = F*v derived form P = I^2*R in this case?

What did you not understand in #2?

 

[Karagoz]

(I use capital L instead of l to make it easier to read).

Power formula: P = U^2/R

Faraday's law: U = vBL (I use U instead of ε)

Into the power formula:
P = (vBL)^2 / R

Formula for speed from Faraday's formula:
U = vBL
v = U/BL

Formula for the force on a conductor slice:
F = ILB

Current:
I = U/R

Into the formula for the force:
F = U/R * LB = ULB/R

Both the speed and power formula into the speed formula:
F*v = U/BL * ULB/R = U^2/R = P.

P = F*v = U^2/R.

 

[cnh1995]

(I use capital L instead of l to make it easier to read).

Power formula: P = U^2/R

Faraday's law: U = vBL (I use U instead of ε)

Into the power formula:
P = (vBL)^2 / R

Formula for speed from Faraday's formula:
U = vBL
v = U/BL

Formula for the force on a conductor slice:
F = ILB

Current:
I = U/R

Into the formula for the force:
F = U/R * LB = ULB/R

Both the speed and power formula into the speed formula:
F*v = U/BL * ULB/R = U^2/R = P.

P = F*v = U^2/R.

Looks good!
Is there any question?

 

[Karagoz]

Looks good!
Is there any question?

No, I got it. thanks