A 2.40 m length of wire is held in an east-west direction and moves horizontally to the north with a speed of 14.9 m/s. The vertical component of Earth's magnetic field in this region is 40.0 µT directed downward. Calculate the induced emf between the ends of the wire and determine which end is positive.
E=B*l*v*cos(theta)
=(40)(0.024)(14.9)cos(0)<---east-west
solve, the answer is in micro teslas
west end in positive direction
To calculate the induced emf between the ends of the wire, we can use Faraday's law of electromagnetic induction:
emf = -N dΦ/dt
Where:
emf is the induced electromotive force,
N is the number of turns of the wire,
dΦ/dt is the rate of change of magnetic flux.
In this case, the wire is moving horizontally to the north, so we need to calculate the rate of change of magnetic flux due to this motion. The magnetic flux through a loop of wire is given by:
Φ = B * A
Where:
Φ is the magnetic flux,
B is the magnetic field strength,
A is the area of the loop.
In this case, the wire is moving horizontally to the north, so the area of the loop is given by:
A = length * width
Given that the length of the wire is 2.40 m, we can assume that the width is negligible compared to the length, so we can use the full length as the width.
Now, let's calculate the area of the loop:
A = 2.40 m * 2.40 m
Next, we need to calculate the rate of change of magnetic flux. Since the wire is moving horizontally to the north, the magnetic field lines are being cut by the wire, resulting in a change of magnetic flux. The rate of change of magnetic flux, in this case, is given by:
dΦ/dt = B * v * sin(θ)
Where:
dΦ/dt is the rate of change of magnetic flux,
B is the magnetic field strength,
v is the velocity of the wire,
θ is the angle between the velocity and the magnetic field direction.
Given that the magnetic field strength is 40.0 µT (or 40.0 × 10^-6 T), the velocity of the wire is 14.9 m/s, and the angle between the velocity and the magnetic field direction is 90 degrees (since the wire is moving horizontally while the magnetic field is vertical), we can calculate the rate of change of magnetic flux:
dΦ/dt = (40.0 × 10^-6 T) * (14.9 m/s) * sin(90 degrees)
Finally, we can substitute the values into Faraday's law to calculate the induced emf:
emf = -N * dΦ/dt
Since we do not have the number of turns of the wire, we cannot calculate the exact value of the induced emf. However, we can determine which end of the wire is positive based on the direction of the induced current. According to Lenz's law, the induced current in the wire will flow in the direction that opposes the change that produced it. In this case, since the wire is moving horizontally to the north, the induced current will flow in such a way that it generates a magnetic field pointing downward to oppose the external magnetic field. Therefore, the end of the wire facing north (the direction of the wire's motion) will be positive.