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 calculate the induced voltage in the metal ring, we need to apply Faraday's law of electromagnetic induction. According to Faraday's law, the induced voltage (emf) in a loop is equal to the rate of change of flux through the loop.
Let's break down the problem and solve it step by step:
1. Determine the change in flux:
The change in flux through the metal ring occurs as Abe turns from the west street to the north street. Initially, when driving west, the Earth's magnetic field provides no flux through the ring. However, as Abe turns to the north, the magnetic field lines start passing through the ring, resulting in a change in magnetic flux.
2. Calculate the change in flux:
The change in flux (∆Φ) can be calculated by multiplying the area of the ring by the change in magnetic field (∆B). Since the ring is shaped like a circle, the area (A) can be calculated using the formula A = πr^2, where r is the radius of the ring. In this case, the diameter of the ring is given as 0.06m, so the radius (r) is 0.03m.
A = π(0.03)^2 = 0.00283 m^2
The change in magnetic field (∆B) is equal to the Earth's magnetic field at the north street since it was previously zero.
∆B = 5.0x10^-5 T
∆Φ = A * ∆B
= 0.00283 m^2 * 5.0x10^-5 T
3. Determine the time interval (∆t) for the change to occur:
In the problem, it is mentioned that Abe takes 5.0 seconds to make the turn. This represents the time interval (∆t) over which the change in flux occurs.
∆t = 5.0 seconds
4. Calculate induced voltage (emf):
The induced voltage (emf) can be calculated by dividing the change in flux (∆Φ) by the time interval (∆t).
emf = ∆Φ / ∆t
Now, substituting the values we calculated,
emf = (0.00283 m^2 * 5.0x10^-5 T) / 5.0 seconds
5. Solve for emf:
Simply multiplying the values and dividing by seconds gives us the induced voltage or emf.
emf = 0.0000283 V
Therefore, the induced voltage in the metal ring as Abe turns from the west street to the north street is 0.0000283 V (or 28.3 μV).