n automobile has a vertical radio antenna 1.40 m long. The automobile travels at 30.0 km/h on a horizontal road where Earth's magnetic field is 50.0 µT, directed toward the north and downward at an angle of 65.0° below the horizontal.
(a) Specify the direction the automobile should move so as to generate the maximum motional emf in the antenna, with the top of the antenna positive relative to the bottom.
(b) Calculate the magnitude of this induced emf.
____ mV
so i know that a) would be east, but i am having a hard time doing part B.
Go east or west.
emf= change in B A /change in time
but the area swept out is 1.4*distance/time
distance time= 30km/hr*1hr/3600sec*1000m/km=30'/3.6 m/s
emf=30e-6* (30/3.6)cos65deg
To calculate the magnitude of the induced emf, we can use Faraday's law of electromagnetic induction. This law states that the induced emf is equal to the rate of change of magnetic flux through a loop of wire. In this case, the loop of wire is formed by the vertical radio antenna on the automobile.
Let's break down the steps to find the induced emf:
Step 1: Determine the velocity of the automobile in meters per second (m/s).
Convert the velocity from km/h to m/s. Since 1 km/h = 1000 m/3600 s:
30.0 km/h * (1000 m/3600 s) = 8.33 m/s
Step 2: Calculate the magnetic flux through the loop.
The magnetic flux (Φ) through a loop is given by the equation:
Φ = B * A * cos(θ)
where B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop's surface.
In this case, the magnetic field strength is given as 50.0 µT (microtesla) = 50.0 x 10^(-6) T.
The area of the loop is equal to the length of the antenna multiplied by the thickness of the loop, which can be approximated as the area of a rectangle with side length equal to the thickness of the antenna.
Let's assume the thickness of the antenna is negligible, so we can simply use the length of the antenna as the height of the rectangle. Therefore, A = 1.40 m * 1.40 m.
The angle between the magnetic field and the normal to the loop's surface is 65.0° below the horizontal. In terms of the angle between the magnetic field and the horizontal, it is 180° - 65.0° = 115.0°.
Now, we can plug these values into the formula to calculate the magnetic flux:
Φ = (50.0 x 10^(-6) T) * (1.40 m * 1.40 m) * cos(115.0°)
Step 3: Calculate the rate of change of magnetic flux.
Since the automobile is moving horizontally, the rate of change of magnetic flux can be represented by the change in the area of the loop per unit time.
The antenna is perpendicular to the direction of motion, so only the length contributes to the rate of change. Thus, the rate of change of magnetic flux (dΦ/dt) is equal to the rate of change of the length of the antenna.
Since the velocity of the automobile is constant, the rate of change of the length of the antenna is simply its horizontal velocity, which we computed in Step 1.
Step 4: Calculate the induced emf using Faraday's law of electromagnetic induction.
The induced emf (ε) is given by the equation:
ε = -dΦ/dt
Plugging in the values from Steps 2 and 3:
ε = -[(50.0 x 10^(-6) T) * (1.40 m * 1.40 m) * cos(115.0°)] / 8.33 m/s
Calculate this expression to find the induced emf in volts.