A pickup truck has a width of 78.9 in. If it is traveling north at 36 m/s through a magnetic field with vertical component of 33 µT, what magnitude emf is induced between the driver and passenger sides of the truck?
To determine the magnitude of the induced electromotive force (emf) between the driver and passenger sides of the truck, we can use Faraday's law of electromagnetic induction.
Faraday's law states that the emf induced in a conductor is proportional to the rate of change of magnetic flux through the conductor.
The magnetic flux (Φ) can be calculated as the product of the magnetic field (B), the area (A), and the cosine of the angle (θ) between the magnetic field and the normal vector of the area.
Φ = B * A * cos(θ)
In this case, the magnetic field B is given as 33 µT (microtesla). The area (A) is the width of the truck, which is given as 78.9 inches. However, we need to convert it to meters to be consistent with the other units. Since 1 inch is equal to 0.0254 meters, the width of the truck in meters is:
Width = 78.9 inches * 0.0254 meters/inch
Next, we need to find the angle θ between the magnetic field and the normal vector of the area. Since the truck is traveling due north, and the magnetic field is vertical, the angle between them is 90 degrees or π/2 radians.
Therefore, θ = π/2 radians.
Substituting the values into the formula:
Φ = (33 µT) * (Width) * cos(π/2)
Now, we need to convert the magnetic field from microtesla to tesla, and the width from meters to square meters:
1 µT = 10^-6 T
1 m = 10^3 mm
Substituting the values:
Φ = (33 * 10^-6 T) * (78.9 inches * 0.0254 meters/inch) * cos(π/2)
Next, we need to find the rate of change of magnetic flux (dΦ/dt), which represents the rate at which the magnetic field is changing.
Since the truck is traveling north at 36 m/s, the rate of change of magnetic flux is given by:
dΦ/dt = -B * V * sin(θ)
Substituting the values into the formula:
dΦ/dt = -(33 * 10^-6 T) * (36 m/s) * sin(π/2)
Finally, we can calculate the induced electromotive force (emf) using Faraday's law:
emf = -dΦ/dt
Substituting the value we calculated for dΦ/dt:
emf = -(-(33 * 10^-6 T) * (36 m/s) * sin(π/2))
Calculating this expression will give you the magnitude of the induced emf between the driver and passenger sides of the truck.
To find the magnitude of the electromotive force (emf) induced between the driver and passenger sides of the truck, you can use Faraday's law of electromagnetic induction. This law states that the emf induced in a loop is given by the rate of change of magnetic flux through the loop.
The magnetic flux is defined as the product of the magnetic field strength (B) and the area (A) through which the magnetic field passes:
ϕ = B * A
In this case, the magnetic field has a vertical component of 33 µT and is passing through the width of the truck. To find the area, we need to convert the width from inches to meters:
Width = 78.9 in = 78.9 * 0.0254 m = 2.00306 m (approx.)
Now we can calculate the area:
Area (A) = width * length (assuming length is much larger than the width)
Assuming the length is sufficiently large compared to the width, we can ignore it for this calculation. Therefore, the area is equal to the width:
A = 2.00306 m
Now, we can calculate the magnetic flux:
ϕ = B * A = (33 x 10^-6 T) * (2.00306 m)
Next, we need to differentiate the magnetic flux with respect to time. However, we are not given any information about the rate of change of the magnetic field or the area. So, for the purpose of this question, let’s assume that the magnetic field and area are constant.
Therefore, the rate of change of magnetic flux is zero:
dϕ/dt = 0
Since the rate of change of the magnetic flux is zero, it means that no emf is induced between the driver and passenger sides of the truck.
So, the magnitude of the induced emf is zero.