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Chapter 29: Problem 30

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 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

Expert verified
Answer: The potential difference between the tips of the wings is approximately 5.14 mV.

Step by step solution

01

Convert the magnetic field strength from G to Tesla

The given magnetic field strength is 0.500 G. We need to convert it into Tesla (T) before substituting it into the EMF formula. Since 1 G = 10^-4 T, we find: \(B = 0.500\,\mathrm{G} \times 10^{-4}\,\frac{\mathrm{T}}{\mathrm{G}} = 5.0\times 10^{-5}\,\mathrm{T}\)
02

Find the speed of sound in air

Standard speed of sound in air is approximately 343 m/s. Since the aircraft is flying at Mach 3, its actual speed is three times this value: \(v = 3\times 343\,\mathrm{\frac{m}{s}} = 1029\,\mathrm{\frac{m}{s}}\)
03

Calculate the electromotive force (EMF)

Now that we have all values needed, we can calculate the potential difference between the tips of the wings using the formula for EMF: EMF = B * L * v EMF = \((5.0 \times 10^{-5}\,\mathrm{T}) \times (10.0\,\mathrm{m}) \times (1029\,\mathrm{\frac{m}{s}})\) EMF = \(5.1415\times 10^{-3}\,\mathrm{V}\)
04

Round the answer

Finally, let's round the answer to two decimal places: EMF = \(5.14\times 10^{-3}\,\mathrm{V}\) Therefore, the potential difference between the tips of the wings is approximately 5.14 mV.

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Most popular questions from this chapter

An ideal battery (with no internal resistance) supplies \(V_{\mathrm{emf}}\) and is connected to a superconducting (no resistance!) coil of inductance \(L\) at time \(t=0 .\) Find the current in the coil as a function of time, \(i(t) .\) Assume that all connections also have zero resistance.

A square conducting loop with sides of length \(L\) is rotating at a constant angular speed, \(\omega\), in a uniform magnetic field of magnitude \(B\). At time \(t=0\), the loop is oriented so that the direction normal to the loop is aligned with the magnetic field. Find an expression for the potential difference induced in the loop as a function of time.

A \(100 .-V\) battery is connected in series with a \(500 .-\Omega\) resistor. According to Faraday's Law of Induction, current can never change instantaneously, so there is always some "stray" inductance. Suppose the stray inductance is \(0.200 \mu \mathrm{H}\). How long will it take the current to build up to within \(0.500 \%\) of its final value of \(0.200 \mathrm{~A}\) after the resistor is con- nected to the battery?

A circular loop of area \(A\) is placed perpendicular to a time-varying magnetic field of \(B(t)=B_{0}+a t+b t^{2},\) where \(B_{0}, a,\) and \(b\) are constants. a) What is the magnetic flux through the loop at \(t=0 ?\) b) Derive an equation for the induced potential difference in the loop as a function of time. c) What is the magnitude and the direction of the induced current if the resistance of the loop is \(R ?\)

A metal hoop is laid flat on the ground. A magnetic field that is directed upward, out of the ground, is increasing in magnitude. As you look down on the hoop from above, what is the direction of the induced current in the hoop?
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