A 25.0-cm -long metal rod lies in the x y -plane and makes...

A $25.0-\mathrm{cm}$ -long metal rod lies in the $x y$ -plane and makes an angle of $36.9^{\circ}$ with the positive $x$ -axis and an angle of $53.1^{\circ}$ with the positive $y$ -axis. The rod is moving in the $+x$ -direction with a speed of $6.80 \mathrm{~m} / \mathrm{s}$. The rod is in a uniform magnetic field $\overrightarrow{\boldsymbol{B}}=(0.120 \mathrm{~T}) \hat{\imath}-(0.220 \mathrm{~T}) \hat{\jmath}-(0.0900 \mathrm{~T}) \hat{\boldsymbol{k}} .$ (a) What is the magni- tude of the emf induced in the rod? (b) Indicate in a sketch which end of the rod is at higher potential.

A $25.0-\mathrm{cm}$ -long metal rod lies in the $x y$ -plane and makes an angle of $36.9^{\circ}$ with the positive $x$ -axis and an angle of $53.1^{\circ}$ with the positive $y$ -axis. The rod is moving in the $+x$ -direction with a speed of $6.80 \mathrm{~m} / \mathrm{s}$. The rod is in a uniform magnetic field $\overrightarrow{\boldsymbol{B}}=(0.120 \mathrm{~T}) \hat{\imath}-(0.220 \mathrm{~T}) \hat{\jmath}-(0.0900 \mathrm{~T}) \hat{\boldsymbol{k}} .$ (a) What is the magni-
tude of the emf induced in the rod? (b) Indicate in a sketch which end of the rod is at higher potential.

Key Concepts

Potential Difference Distribution

Understanding the distribution of potential along a conductor involves knowing which end of the conductor is at a higher or lower potential due to the separation of charges caused by the Lorentz force. The relative orientation of the conductor, its velocity, and the magnetic field establishment the direction of the induced electric field, which in turn indicates the potential difference between the ends.

Vector Cross Product

The calculation of the induced emf in a moving conductor relies on the vector cross product between the velocity vector and the magnetic field vector. This mathematical operation determines both the magnitude and the direction of the force (and thus the induced electric field) acting on the moving charges.

Motional EMF

This concept refers to the electromotive force induced in a conductor as it moves through a magnetic field. In such a scenario, the moving charges in the conductor experience a force due to the magnetic field, which in turn sets up a potential difference (voltage) along its length.

Lorentz Force

The Lorentz force is the fundamental force acting on a charged particle moving in a magnetic field. For a conductor moving in a magnetic field, the free charges within it experience this force, which is given by F = q(v × B), thereby causing charge separation and an induced emf.

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