You and your roommate are headed to Cancún for spring break. Your roommate, who has had only high school physics, has read that an emf can be induced in the wings of an airplane and wonders whether this would give enough voltage to power a portable music player. What's your answer? (Assume that the wingspan of your 747 is \(60 \mathrm{m},\) the plane is flying at 600 mph, and Earth's magnetic field is 0.3 G.)

Michael Faraday's pioneering research in electromagnetism gave us one of the key principles in the field, known as Faraday's Law of Electromagnetic Induction. It states that a changing magnetic field within a closed loop induces an electromotive force (emf) within the loop. Mathematically, it's often simplified to the expression \( emf = -N \frac{\Delta\Phi_B}{\Delta t} \), where \( \Delta\Phi_B \) represents the change in magnetic flux through the loop over a given time interval \( \Delta t \) and \( N \) is the number of loops.

In the context of our exercise, the moving airplane wing acts like a loop slicing through Earth's magnetic field, thus experiencing a change in magnetic flux and producing an emf. This emf is what we evaluate to see if it could potentially power an electronic device, such as a portable music player. Imagine the wing cutting through the magnetic lines at a perpendicular angle—the faster the speed or the stronger the magnetic field, the greater the induced emf will be. It's essential when applying Faraday's Law to ensure that all units are consistent, hence the need for unit conversions in our exercise.

Working with scientific equations often requires consistent units to get accurate and meaningful results. This is where SI units conversion comes into play. In physics, the International System of Units (SI) is the standard and ensures that all involved variables are expressed in compatible units. For instance, speeds should be in meters per second (m/s), distances in meters (m), and magnetic fields in tesla (T).

In our exercise, the initial challenge is converting miles per hour (mph) to m/s and gauss (G) to tesla (T). Conversions from mph to m/s can be done using the fact that 1 mile equals 1609.34 meters and 1 hour equals 3600 seconds. For magnetic fields, knowing that 1 G equals \(10^{-4}\) T, allows us to easily convert values given in gauss to the SI unit of tesla. Keeping track of these conversions is imperative as using incorrect units would result in an incorrect emf value.

The process of calculating electromotive force (emf) involves applying the appropriate version of Faraday's Law. In scenarios where a conductor moves through a magnetic field, such as an airplane wing, we can often use a simplified form of Faraday's Law, \( emf = B \cdot v \cdot d \). In this equation, \( B \) represents the magnetic field strength (in tesla), \( v \) is the velocity of the conductor (in m/s), and \( d \) is the length of the conductor—in this case, the wingspan of the airplane.

This formula produces the emf in volts, which is the pressure that would drive an electric current if a complete circuit were available. By plugging in the converted SI units for the speed and magnetic field strength, along with the appropriate dimensions (or wingspan), we calculate the induced emf. After calculation, one would compare the resulting emf with the operating voltage of a portable music player to evaluate if the airplane's induced emf could potentially be harnessed to power the device.