A metal rod of length l and mass m is suspended in a...

A metal rod of length $l$ and mass $m$ is suspended in a magnetic field from two light wires that are connected to a battery with emf $\varepsilon$ . When a vertical magnetic field $\vec{B}$ is turned on, the supporting wires make an angle $\theta$ with the vertical. What can you learn about the circuit using this information? Make a list of physical quantities and explain how to determine three of them.

Key Concepts

Electromechanical Equilibrium

This concept involves the balance of forces acting on the rod. In the presence of gravity pulling downward and a horizontal magnetic force due to the current in the rod, the supporting wires adjust their angle (?) from the vertical to maintain equilibrium. Analyzing this balance allows us to relate mechanical quantities (such as the weight mg and the tension in the wires) to electrical quantities in the circuit.

Magnetic Force on a Current?Carrying Conductor

A conductor carrying an electrical current in a magnetic field experiences a force given by F = I·l·B, where I is the current, l is the length of the rod, and B is the magnetic field strength. This magnetic force is responsible for displacing the rod and can be linked to the observed angle ? when balanced against the gravitational force.

Determining the Electrical Current

By examining the equilibrium of forces, the horizontal magnetic force (I·l·B) can be equated to the horizontal component of the tension in the supporting wires, which is related to the weight (mg) through the angle (tan ?). Rearranging this relationship yields an expression for the current, typically I = (mg·tan ?)/(l·B), provided that m, g, l, B, and ? are known.

Determining the Circuit Resistance

Once the current I in the circuit is known, and if the battery’s electromotive force (?) is given, the overall resistance of the circuit can be determined using Ohm’s law: R = ?/I. This allows one to infer resistive properties from the mechanical deflection caused by the magnetic interaction.

Tension in the Supporting Wires

The tension in the wires replaces the role of the support force for the rod. Resolving this tension into vertical and horizontal components, the vertical component must balance the gravitational force (mg) while the horizontal component is balanced by the magnetic force (I·l·B). The measured angle ? directly relates to these components (via tan ?), which couples the mechanical behavior to the electrical parameters of the circuit.

Transcript

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