Figure 23.27 shows a zero-resistance rod sliding toward the...

Figure $23.27$ shows a zero-resistance rod sliding toward the right on two zero-resistance rails that are separated by the distance $L=0.450 \mathrm{~m}$. The rails are connected by a $12.5-\Omega$ resistor, and the entire system is in a uniform magnetic field with a magnitude of $0.750 \mathrm{~T}$. Find the speed at which the bar must move to produce a current of $0.125 \mathrm{~A}$ in the resistor.

Key Concepts

Motional Electromotive Force (emf)

This concept refers to the voltage created when a conductor moves through a magnetic field. As the conductor cuts across magnetic field lines, an emf is induced across its ends due to the motion, and the magnitude of this emf depends on the speed of the conductor, the magnetic field strength, and the length of the conductor within the magnetic field.

Faraday’s Law of Electromagnetic Induction

Faraday’s Law states that an electromotive force is induced in a closed circuit when the magnetic flux through the circuit changes. In the context of a moving conductor in a uniform magnetic field, the change in flux is directly related to the speed of the conductor, and this induced emf is fundamental to understanding how mechanical motion can generate electrical energy.

Ohm’s Law

Ohm’s Law is a basic principle in circuit theory that relates voltage, current, and resistance through the equation V = IR. In a circuit where an emf is induced, this law is used to determine the current that flows through a resistor, linking the induced voltage to the measurable electrical current in the system.

Uniform Magnetic Field

A uniform magnetic field has the same magnitude and direction at all points in a region, which simplifies the analysis of electromagnetic induction. In problems involving moving conductors, the uniformity of the field ensures that the induced emf can be calculated directly using the product of the magnetic field strength, the velocity of the conductor, and the effective length within the field.

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