A parallel-plate capacitor, at rest in 𝒮0 and tilted at a 45 ^∘ angle to the x0 axis, carries ch

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A parallel-plate capacitor, at rest in 𝒮0 and tilted at a 45...

A parallel-plate capacitor, at rest in $\mathcal{S}_{0}$ and tilted at a 45 $^{\circ}$ angle to the $x_{0}$ axis, carries charge densities $\pm \sigma_{0}$ on the two plates (Fig. 12.41 ). System $\mathcal{S}$ is moving to the right at speed $v$ relative to $\mathcal{S}_{0}$ (a) Find $\mathbf{E}_{0},$ the field in $\mathcal{S}_{0}$ (b) Find $\mathbf{E},$ the field in $\mathcal{S}$ (c) What angle do the plates make with the $x$ axis? (d) Is the field perpendicular to the plates in $\mathcal{S}$ ?

A parallel-plate capacitor, at rest in $\mathcal{S}_{0}$ and tilted at a 45 $^{\circ}$ angle to the $x_{0}$ axis, carries charge densities $\pm \sigma_{0}$ on the two plates (Fig. 12.41 ). System $\mathcal{S}$ is moving to the right at speed $v$ relative to $\mathcal{S}_{0}$
(a) Find $\mathbf{E}_{0},$ the field in $\mathcal{S}_{0}$
(b) Find $\mathbf{E},$ the field in $\mathcal{S}$
(c) What angle do the plates make with the $x$ axis?
(d) Is the field perpendicular to the plates in $\mathcal{S}$ ?

Key Concepts

Relativistic Length Contraction and Geometric Orientation

Length contraction occurs along the direction of motion in special relativity, affecting how spatial dimensions and angles are measured in a moving frame. For objects like a tilted capacitor, this contraction alters the perceived geometry, including the tilt angle of its plates relative to the coordinate axes. Understanding how geometry transforms is essential for determining the orientation of the capacitor’s plates in different inertial frames.

Field Perpendicularity in Different Reference Frames

In electromagnetic problems, determining whether the electric field remains perpendicular to the capacitor plates in a moving frame requires analyzing how both the field components and the orientation of the plates transform under Lorentz boosts. Although the field is perpendicular to the plates in the rest frame, relativistic effects can cause the measured angle between the field and the plates to differ in another inertial frame, highlighting the subtle interplay between field transformations and geometric contractions.

Electric Field in a Capacitor

A parallel-plate capacitor creates a uniform electric field between its plates determined by the charge density and the permittivity of free space. In the rest frame of the capacitor, the field is normally calculated using Gauss’s law, which yields a magnitude independent of the separation of the plates, assuming edge effects can be ignored. This concept forms the basis for understanding the behavior of fields in an idealized capacitor configuration.

Lorentz Transformation of Electromagnetic Fields

In special relativity, the electric and magnetic fields observed in different inertial frames are related through Lorentz transformations. When one switches from the rest frame of a charge distribution to a moving frame, the fields mix such that a purely electric field in one frame can appear as both an electric field and a magnetic field in another. This transformation is crucial for analyzing how the fields of a capacitor appear to observers in different inertial frames.

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