Induced EMF in a Triangle: Determining EMF and External Force

[SataSata]

Homework Statement

Magnetic Field B is going into the plane.
Bar is moving to the right with velocity v.
Neglect resistance of the conducting bar and all contacts.
a.) Determine the induced emf in the circuit.
b.) To maintain the uniform motion of the conductive bar, there must be an external force F_app to pull the conductive bar. How much should F_app be?

Homework Equations

Let the length of conducting bar in the triangle be $l$
Hall Effect: $emf = Blv$
Faraday: $emf = -d\Phi/dt$
$\Phi = BA$
$A=xl/2$
$l=xtan\theta$
$F_B=IlB$

The Attempt at a Solution

Using Faraday's Law:
$emf = -d\Phi/dt = -BdA/dt$
$dA/dt = dA/dx \times dx/dt$
$dA/dt = vxtan\theta$
$\therefore emf = -Bvxtan\theta$
Or, Using Hall Effect:
$emf = Blv = Bvxtan\theta$
Next, $F_app = -F_B$
$Current I = emf/R = -Bvxtan\theta/R$
$F_app= IlB = B^2vx^2tan^2\theta/R$

Can anybody check if my attempt is correct? Are all the negative signs correct?
Faraday's Law and Hall Effect give different signs. Which one is correct?
F_app sign is supposed to be positive or negative?

 

[mfb]

Faraday's Law and Hall Effect give different signs. Which one is correct?

Depends on the unspecified direction of your current flow. You don't need that direction, however.
I would choose a positive sign for the force needed to pull the bar, which means the induction leads to a negative force. But that is just an aesthetic choice (have force and velocity with the same sign convention if v is positive), the physical direction of the force is given in the problem statement already.

 

[SataSata]

Thank you for your reply mfb.
But is it right to express my answer in terms of x? Since x is a function of time, I can simply write it as x(t) and so emf and Fapp will both be a function of time too. However, the question simply asked for emf induced and Fapp, which caused me to doubt my answers.
In a non-triangle shape like a rectangle that's extending in length, the breadth is a constant and hence I can write my answer with it. But in this case, all sides of the triangle are varying and so I'm not sure how to express my answers.

 

[mfb]

Well, both depend on time here. Expressing them as function of x or t should be fine.