A helicopter hovers above the north magnetic pole in a magnetic field of magnitude 0.426 G perpendicular to the ground. The helicopter rotors are \(10.0 \mathrm{~m}\) long, are made of aluminum, and rotate about the hub with a rotational speed of \(10.0 \cdot 10^{4} \mathrm{rpm} .\) What is the potential difference from the hub of the rotor to the end?

A magnetic field is an invisible force that exerts magnetic forces on substances which are sensitive to magnetism. A magnetic field is created by moving electric charges and inherent magnetic moments of elementary particles associated with a fundamental quantum property known as the spin. In the context of the given exercise, the helicopter hovers above the north magnetic pole where the magnetic field is quite strong and is oriented perpendicular to the surface of the Earth.

When a conductive material moves within a magnetic field, an electromotive force (EMF) is induced across the material—a phenomenon known as electromagnetic induction. In this scenario, the aluminum rotors of the helicopter, which conduct electricity, are moving due to the rotational motion within the Earth's magnetic field, causing a potential difference, or voltage, to be induced across the length of the rotor blades. This process is governed by Faraday’s law of electromagnetic induction.

Potential difference, also known as voltage, is the difference in electric potential between two points in an electric field. It provides the push that drives charge through a circuit. In our helicopter example, potential difference is generated due to the rotors cutting through the Earth's magnetic field. This movement creates a flow of electric charges within the rotors, which is manifested as voltage.

To calculate the potential difference, one would typically use the formula \( \Delta V = B \times L^2 \times \omega / 2 \) where \( \Delta V \) is the potential difference, \( B \) is the magnetic field strength in Tesla, \( L \) is the length of the rotor, and \( \omega \) is the angular velocity in radians per second. The factor of \( 1/2 \) comes from a more sophisticated understanding of the electromagnetic effects at play—essentially, the average of the potential across the rotor from one point to another.

Rotational motion is the circular movement of an object around a center or axis of rotation. A three-dimensional object always rotates around an imaginary line called a rotation axis. In our example, the helicopter’s rotors are experiencing rotational motion as they spin at high speeds around their hub or axis. This type of motion is often measured in revolutions per minute (rpm) but can also be expressed in radians per second (rad/s) which is needed when applying formulas in physics.

The helicopter’s rotor converting rotational speed from rpm to rad/s involves understanding that a full revolution is equal to \(2\pi\) radians and 1 minute is equal to 60 seconds. This conversion is crucial for calculating the potential difference accurately, which, as highlighted in your exercise improvement advice, ensures students understand the importance of units in physics equations and the need to convert them appropriately for calculations to be correct.