A Boeing- 747 jet with a wing span of 60.0 m is flying...

A Boeing- 747 jet with a wing span of $60.0 \mathrm{~m}$ is flying horizontally at a speed of $300 \mathrm{~m} / \mathrm{s}$ over Phoenix, Arizona, at a location where the Earth's magnetic field is $50.0 \mu \mathrm{T}$ at $58.0^{\circ}$ below the horizontal. What voltage is generated between the wingtips?

Key Concepts

Motional Electromotive Force (EMF)

Motional EMF is the voltage generated when a conductor moves through a magnetic field. It arises from the separation of charges due to the magnetic Lorentz force acting on these charges within the moving conductor. The induced voltage can be calculated using the relationship ? = B · L · v · sin(?), where B is the magnetic field strength, L is the effective length of the conductor perpendicular to the field, v is the speed of the conductor, and ? is the angle between the direction of motion and the magnetic field direction.

Lorentz Force

The Lorentz force is the fundamental force experienced by charged particles moving in a magnetic field, and it is given by F = q(v × B). In a moving conductor, this force acts on free charges, pushing them to accumulate at opposite ends of the conductor. This separation of charges results in an electric field and ultimately an electromotive force, allowing for the conversion of kinetic energy into electrical energy.

Vector Decomposition of the Magnetic Field

When a conductor moves through a magnetic field that is not perpendicular to its direction of motion, only the component of the magnetic field that is perpendicular to the velocity contributes to the induced EMF. By decomposing the magnetic field into components, the effective field strength for motional induction can be expressed as B · sin(?), where ? is the angle between the magnetic field and the direction of motion. This decomposition is essential for accurately calculating the induced voltage in cases where the field is tilted with respect to the motion.

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