Question

A supersonic aircraft with a wingspan of \(10.0 \mathrm{~m}\) is flying over the north magnetic pole (in a magnetic field of magnitude 0.500 G oriented perpendicular to the ground) at a speed of three times the speed of sound (Mach 3). What is the potential difference between the tips of the wings? Assume that the wings are made of aluminum.

Short answer

Answer: The potential difference between the tips of the wings of the supersonic aircraft is approximately \(5.145 \times 10^{-3} \mathrm{~V}\).

Step-by-step solution

Calculate the Area per Unit Time Covered by the Moving Aircraft (rate of change of area)

To calculate the electromotive force (EMF) induced by the aircraft moving through the magnetic field, we first need to determine the rate at which the aircraft's wings cover area, or sweep through the magnetic field. The aircraft is flying horizontally, so its speed can be used to determine the area covered per unit time. Area per unit time, \(A_t\) can be calculated as: \(A_t = l \times v\) where \(l\) is the wingspan (length of the wings) and \(v\) is the speed of the aircraft. We are given that the speed of the aircraft is three times the speed of sound (Mach 3). The speed of sound at the aircraft's altitude is approximately \(343 \mathrm{~m/s}\), so the speed of the aircraft, v, is: \(v = 3 \times 343 \mathrm{~m/s} = 1029 \mathrm{~m/s}\) Now, let's calculate the area per unit time using the wingspan \(l = 10.0 \mathrm{~m}\) and the velocity of the aircraft. \(A_t = 10.0 \mathrm{~m} \times 1029 \mathrm{~m/s} = 10290 \mathrm{~m^2/s}\)

Calculate the EMF Produced by the Moving Aircraft in the Magnetic Field

Now that we have the area per unit time, we can use Faraday's law to calculate the electromotive force (EMF) induced in the wings. Faraday's law states: \(EMF = - \frac{d \Phi}{dt}\), where \(\Phi\) is the magnetic flux through the wings, and \(t\) is the time. The negative sign indicates that the EMF opposes the change in magnetic flux. In this case, we can calculate the change in magnetic flux, \(d \Phi\), as: \(d \Phi = B \times dA\), where \(B\) is the magnitude of the magnetic field and \(dA\) is the change in area. Since the magnetic field is perpendicular to the ground and the wings are parallel to the ground, the change in magnetic flux is the product of the rate of change of area and the magnetic field strength. \(d \Phi/dt = B \times A_t\) Given the magnetic field strength, \(B = 0.500 \mathrm{~G}\) or \(B = 0.500 \times 10^{-4} \mathrm{~T}\), we can calculate the EMF as: \(EMF = (0.500 \times 10^{-4} \mathrm{~T}) \times (10290 \mathrm{~m^2/s})\) \(EMF = 5.145 \times 10^{-3} \mathrm{~V}\)

Find the Potential Difference Between the Tips of the Wings

In this case, the EMF induced in the wings is equal to the potential difference between the tips of the wings. Thus, the potential difference is: \(\Delta V = 5.145 \times 10^{-3} \mathrm{~V}\) The potential difference between the tips of the wings of the supersonic aircraft is approximately \(5.145 \times 10^{-3} \mathrm{~V}\).