An automobile has a vertical radio antenna 1.20 m long. The...

An automobile has a vertical radio antenna $1.20 \mathrm{~m}$ long. The automobile travels at $65.0 \mathrm{~km} / \mathrm{h}$ on a horizontal road where the Earth's magnetic field is $50.0 \mu \mathrm{T}$ directed toward the north and downward at an angle of $65.0^{\circ}$ below the horizontal. (a) Specify the direction that the automobile should move to generate the maximum motional emf in the antenna, with the top of the antenna positive relative to the bottom. (b) Calculate the magnitude of this induced emf.

Key Concepts

Motional Electromotive Force (EMF)

Motional EMF is the voltage generated across a conductor that moves through a magnetic field. It arises because the magnetic field exerts a force on the free electrons in the conductor, causing them to separate, which creates a potential difference. The induced EMF depends on the magnitude of the magnetic field, the speed of the conductor, and the length of the conductor moving perpendicularly to the field.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field, given by the vector cross product F = q(v × B), where q is the charge, v is its velocity, and B is the magnetic field. This force is responsible for the separation of charges in a moving conductor, and it is fundamental to understanding why and how a motional EMF is generated.

Vector Cross Product

The vector cross product is used to determine both the magnitude and the direction of the force acting on a moving charge in a magnetic field. In the context of motional EMF, it explains how the orientation of the velocity of the conductor relative to the magnetic field affects the induced voltage. The maximum EMF is achieved when the velocity and the magnetic field are perpendicular, as the sine of the angle between them is maximized.

Magnetic Field Orientation

Understanding the orientation of the magnetic field is essential when analyzing induced EMF, as only the component of the field that is perpendicular to both the direction of motion and the length of the conductor contributes to the induced voltage. Decomposing the magnetic field into its directional components allows for determining the most effective alignment of motion to maximize the induced EMF.

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