Question

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?

Short answer

Answer: The potential difference induced from the hub of the rotor to the end is approximately 22.3 Volts.

Step-by-step solution

Convert the magnetic field to Tesla

The given magnetic field is in Gauss (G), but we need Tesla (T) to use the formula for the potential difference. To convert Gauss to Tesla, we can use the following relation: 1 Tesla = 10000 Gauss. B = 0.426 G * (1 T / 10000 G) = 0.0000426 T

Convert the rotational speed to radians per second

The given rotational speed is in revolutions per minute (rpm). We need to convert it to radians per second (rad/s) to use the formula for the potential difference. To do this, we will first convert rpm to revolutions per second (rps), and then to radians per second: 1 rpm = 1/60 rps, 1 revolution = 2π radians Rotational speed = 10.0 × 10^4 rpm * (1/60 rps/rpm) * (2π rad/revolution) = 1.047 × 10^5 rad/s

Calculate the potential difference

Now we can use the formula for the potential difference induced in a rotating conducting rod in a magnetic field: ΔV = B * L²ω/2 ΔV = 0.0000426 T * (10 m)² * (1.047 × 10^5 rad/s) / 2 ΔV ≈ 22.3 Volts Therefore, the potential difference from the hub of the rotor to the end is approximately 22.3 Volts.

Key concepts

Magnetic Field

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

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

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.