Faraday's Law vs. V = vBl for determing potential difference of a conducting rod

[sdcmma]

Homework Statement

A rod rotates clockwise about a point as pivot with the constant frequency 5 rev/s. Find the potential difference between its two ends, which are 80 cm apart, due to the magnetic field B = 0.3T directed into the page.

Homework Equations

There are two methods to find the induced emf (V) in a conducting rod.

One is uses the equation V = vlB, which is derived from the fact that at equilibrium the magnetic force separating the charges within the bar must equal the opposing electric force of attraction between them (v is the tangential velocity of the bar, l is the length, and B is the strength of the magnetic field acting perpendicular to the bar).

The other uses Faraday's law, emf = -d(flux)/dt (flux = BA in this case since the magnetic field is always perpendicular to the area A the bar traces out as it moves)

The Attempt at a Solution

The two solution methods yield different answers, and I am trying to figure out which one is the correct answer and why.

When solving using the equation V = vlB, the tangential velocity is computed as 5rev/s * (2Pi * 0.80 m)/1rev = 25.1 m/s. Substituting the appropriate values from the problem into the equation yields V = (25.1 m/s)(0.80 m)(0.3T) = 6.02 V

Using Faraday's Law, we can determine the change in flux by assigning the flux when the bar is straight up and down above its pivot point to 0 Wb. After it rotates 30 degrees clockwise, it has traced out an area of 0.17 m^2 [30 degrees/360 degrees = x m^2/(Pi * (0.80m)^2)]; the flux at this point is (0.3T)(0.17m^2) = 0.051 Wb. The change in flux, therefore, is 0.051 Wb.
To determine the corresponding change in time required for this change in flux, we calculate the angular velocity of the bar from the frequency and divide the angular distance traveled by the angular velocity:
w = 5rev/1s * 360 degrees/1rev = 1800 degrees/s
delta t = 30 degrees * 1s/1800 degrees = 0.017s
Plugging these values for delta flux and delta t into the Faraday's Law equation, we arrive at a value for the induced emf in the bar:
abs(emf) = abs( - 0.051Wb/0.017s) = 3.06 V

Obviously, these two solution methods yield different answers. Since Faraday's Law is a fundamental law, I am inclined to believe that 3.06 V is the correct answer; however, can someone explain why the first method (using the equation V = vlB) is wrong?

 

[haruspex]

V = vlB, which is derived from the fact that at equilibrium the magnetic force separating the charges within the bar must equal the opposing electric force of attraction between them

Yes, but the voltage gradient along the bar is nonuniform. On average it is only half what it is at the tip.