A conducting bar of length L = 0.40 m rotates in a counterclockwise direction with a constant angular speed of +2.0 rad/s about a pivot P at one end, as shown. A uniform magnetic field (magnitude = 2.8 T) is directed into the paper. What is the potential difference, VP-VA?
24 mV
To find the potential difference VP-VA, we need to calculate the induced electromotive force (emf) in the rotating bar.
The induced emf can be calculated using the equation: emf = B * L * v, where B is the magnetic field strength, L is the length of the bar, and v is the velocity of the bar.
Given:
B = 2.8 T (magnetic field strength)
L = 0.40 m (length of the bar)
To find the velocity of the bar, we need to use the formula for the linear velocity of an object moving along a circular path:
v = ω * r, where ω is the angular speed and r is the distance from the pivot to the bar.
In this case, r is equal to the length of the bar, so r = L = 0.40 m.
Given:
ω = +2.0 rad/s (angular speed)
Using the formula for the velocity, we can find:
v = ω * r = 2.0 rad/s * 0.40 m = 0.80 m/s
Now, we can calculate the induced emf:
emf = B * L * v = 2.8 T * 0.40 m * 0.80 m/s = 0.896 V
Therefore, the potential difference VP-VA is equal to the induced emf, which is 0.896 V.
To find the potential difference between points P and A, we need to use Faraday's law of electromagnetic induction. According to Faraday's law, the induced electromotive force (emf) in a closed loop is equal to the rate of change of magnetic flux through the loop.
Here's how we can approach the problem:
Step 1: Determine the magnetic flux through the bar.
The magnetic flux (Φ) is given by the equation Φ = B*A*cosθ, where B is the magnetic field strength, A is the area, and θ is the angle between the magnetic field direction and the area vector. In this case, the magnetic field is directed into the paper, so the angle θ between the magnetic field and the area vector is 90 degrees.
Since the conducting bar is rotating about pivot P, the area vector of the bar will change in magnitude but not in direction. Therefore, the area A remains constant. We can express the magnetic flux as Φ = B * A * cos(90) = 0, because the cosine of 90 degrees is 0.
Step 2: Calculate the induced emf.
Since the magnetic flux through the bar is zero, there is no induced emf in the bar.
Step 3: Calculate the potential difference (VP-VA).
Since there is no induced emf, the potential difference between points P and A is also zero. Therefore, VP-VA = 0.
In conclusion, the potential difference between points P and A is zero.