A rectangular loop with sides l and w are inside of a region with constant magnetic field B is pulled to exit the region with velocity v.
a) find the magnitude of the net induced emf on the loop
b) find the direction of the current induced on the loop
e=-d/dt(magnetic flux)
= -Bd/dt(area in field)
dt/dt area in field = v*length perpendicular to motion (I do not have your figure to know if that is w or L)
To find the magnitude of the net induced emf on the loop, we need to use Faraday's Law of electromagnetic induction, which states that the induced emf is equal to the rate of change of magnetic flux through the loop.
a) The magnetic flux through the loop is given by the product of the magnetic field strength (B) and the area of the loop (A). In this case, since the loop is rectangular, the area of the loop is equal to the product of its length (l) and width (w): A = l * w.
The rate of change of flux can be calculated by taking the derivative of the flux with respect to time: dΦ/dt.
Therefore, the magnitude of the induced emf (ε) is given by ε = -dΦ/dt.
Since the loop is being pulled to exit the region with a velocity v, the change in the area of the loop with respect to time is given by dA/dt = vw, where v represents the velocity of the loop.
Substituting this back into the formula for the magnitude of the induced emf: ε = -dΦ/dt = -d/dt (B * A) = -B * (dA/dt) = -B * vw.
Therefore, the magnitude of the net induced emf on the loop is ε = B * v * w.
b) To determine the direction of the induced current on the loop, we can use Lenz's Law, which states that the induced current will flow in a direction that opposes the change in magnetic flux that causes it.
In this case, as the loop is being pulled to exit the region, the magnetic flux through the loop is decreasing. Therefore, the induced current will flow in a direction to oppose this decrease in flux.
By using the right-hand rule for electromagnetic induction, we can determine the direction of the induced current. When the thumb of the right hand points in the direction opposite to the change in the magnetic flux (in this case, pointing inward, since the flux is decreasing as the loop is pulled out), the curled fingers will indicate the direction of the induced current.
So, the direction of the induced current on the loop will be such that it creates a magnetic field that opposes the external magnetic field, thus opposing the motion of the loop.