Final answer:
The expression for the induced electromotive force (emf) in a conductor moving in a uniform magnetic field with uniform velocity along the x-axis can be derived using Faraday's law and the formula for motional emf. The total emf around the loop is 2Blv sin(theta), where theta is the angle between the velocity and the magnetic field. To find the time dependence of the emf, assume the conductor rotates at a constant angular velocity w.
Step-by-step explanation:
The expression for the induced electromotive force (emf) in a conductor moving in a uniform magnetic field with uniform velocity along the x-axis can be derived using Faraday's law and the formula for motional emf. The formula for motional emf in a straight wire moving at velocity v through a magnetic field B is E = Blv. However, in this case, as the conductor is moving at an angle to the magnetic field, the formula becomes E = Blv sin(theta), where theta is the angle between the velocity and the magnetic field.
To find the total emf around the loop, we need to consider both sides of the conductor. The emf induced on each segment of the conductor is given by E = Blv sin(theta). As the sides work in the same direction, the total emf is then 2Blv sin(theta).
It's important to note that this expression gives the emf as a function of position. To find the time dependence of the emf, we can assume the conductor rotates at a constant angular velocity w. In this case, the angle can be related to the angular velocity as theta = wt. Substituting this in the formula, we get E = 2Blv sin(wt).