A small object of mass m carries a charge q and is suspended...

A small object of mass $m$ carries a charge $q$ and is suspended by a thread between the vertical plates of a parallel-plate capacitor. The plate separation is $d$. If the thread makes an angle $\theta$ with the vertical, what is the potential difference between the plates?

Key Concepts

Static Equilibrium

This concept involves analyzing an object that remains at rest under the action of multiple forces. In the context of the problem, the gravitational force, tension in the thread, and the electric force acting on the charged object are resolved into their components and balanced. Understanding static equilibrium is essential to determine how forces interact and counteract each other, ultimately allowing one to set up the proper equations to solve for unknown quantities.

Electric Force on a Charge

The electric force experienced by a charged particle in an electric field is given by F = qE, where q is the charge and E is the electric field magnitude. In a uniform electric field, such as that between the plates of a parallel-plate capacitor, this force acts in a consistent direction and its magnitude forms one of the primary components in the force balance equation. Recognizing and quantifying the electric force is key to determining the conditions of equilibrium for charged objects.

Electric Field in a Parallel-Plate Capacitor

A uniform electric field exists between the plates of a parallel-plate capacitor, and its magnitude is directly related to the potential difference across the plates and inversely related to the plate separation (E = V/d). This relationship connects the mechanical aspects of the problem (via the force on the charge) with the electrical property (voltage), allowing one to solve for the potential difference once the electric field is determined through balance of forces.

Potential Difference

The potential difference, or voltage, reflects the work done per unit charge to move a charge between two points in an electric field. In a capacitor setting, the electric field established between the plates leads to a specific voltage, which is calculated using the relationship V = Ed. Understanding how to link the mechanical equilibrium conditions with the electric field concept enables the determination of the potential difference based on the forces acting on the charged object.

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