The hood ornament of Abe's car is shaped like a ring 0.06m in diameter. Abe is driving toward the west so that the earth's 5.0x10^-5 T field provides no flux through the hood ornament. What is the induced voltage in the metal ring as Abe turns from this street onto one where he is traveling north, if he takes 5.0 seconds to make the turn?
To find the induced voltage in the metal ring, we can use Faraday's Law of electromagnetic induction. According to this law, the induced voltage is equal to the rate of change of magnetic flux through the area enclosed by the metal ring.
The equation for the induced voltage (emf) is given by:
emf = - N * d(flux)/dt
Where:
- emf is the induced voltage (in volts)
- N is the number of turns in the metal ring
- d(flux)/dt is the rate of change of magnetic flux
First, let's find the magnetic flux through the metal ring.
The magnetic flux through a loop is given by the equation:
flux = B * A * cos(theta)
Where:
- B is the magnetic field strength (in teslas)
- A is the area of the loop (in square meters)
- theta is the angle between the magnetic field and the normal to the loop
In this case, since the magnetic field provides no flux through the hood ornament, we can say that the angle between the magnetic field and the normal to the loop is 90 degrees (theta = 90 degrees). Therefore, the magnetic flux through the metal ring is zero.
Now, let's work on finding the rate of change of magnetic flux (d(flux)/dt).
Since the magnetic flux is zero initially and will remain zero throughout the turn, the rate of change of magnetic flux (d(flux)/dt) is also zero.
Therefore, the induced voltage (emf) in the metal ring is zero.
In conclusion, when Abe turns from the street where he is driving west to the street where he is traveling north, the induced voltage in the metal ring is zero.
To find the induced voltage in the metal ring, we need to use Faraday's law of electromagnetic induction, which states that the induced voltage (emf) is equal to the rate of change of magnetic flux.
Here's how you can calculate the induced voltage in the metal ring:
1. Calculate the initial magnetic flux:
The problem statement states that the Earth's magnetic field provides no flux through the hood ornament when Abe is driving west. Since the Earth's magnetic field is perpendicular to the surface of the hood ornament, the initial flux through the ring is zero.
Initial Magnetic Flux (Φi) = 0
2. Calculate the final magnetic flux:
When Abe turns onto the street traveling north, the magnetic field lines coming from the Earth will intersect the ring. The magnetic field passing through the ring will change, resulting in an induced voltage.
Final Magnetic Flux (Φf) = Magnetic Field (B) * Area (A)
The area of the ring can be calculated using the formula for the area of a circle: A = π * r^2, where r is the radius of the ring.
Given that the diameter of the ring is 0.06m, the radius (r) is equal to half the diameter: r = 0.06m / 2 = 0.03m
Substitute the values into the formula to find the area of the ring:
A = π * (0.03m)^2 ≈ 0.00283 m^2 (rounded to 5 decimal places)
Now, substitute the values for the Earth's magnetic field and the calculated area into the formula to find the final magnetic flux:
Φf = (5.0x10^-5 T) * (0.00283 m^2) = 1.415x10^-7 T·m^2 (rounded to 4 decimal places)
3. Calculate the rate of change of flux:
Since Abe takes 5.0 seconds to make the turn, we need to find the rate at which the magnetic flux through the ring changes.
Rate of Change of Flux (dΦ/dt) = (Φf - Φi) / (time)
Substituting the values into the formula:
dΦ/dt = (1.415x10^-7 T·m^2 - 0) / (5.0 s) = 2.83x10^-8 T·m^2/s (rounded to 3 decimal places)
4. Calculate the induced voltage (emf):
Finally, we can calculate the induced voltage using Faraday's law of electromagnetic induction, which states that the induced voltage (emf) is equal to the rate of change of magnetic flux:
Induced Voltage (emf) = -dΦ/dt
Substituting the value for the rate of change of flux:
emf = -2.83x10^-8 V (rounded to 3 decimal places)
Therefore, the induced voltage in the metal ring as Abe turns from the street traveling west to north is approximately -2.83x10^-8 volts. The negative sign indicates that the induced voltage will produce a current flowing in the opposite direction.