Suppose that a circular parallel-plate capacitor has radius r₀ = 3.0 cm and plate separation d = 5.0 mm. A sinusoidal potential difference V = V₀ sin (2𝝅ft) is applied across the plates, where V₀ = 180 V and f = 60 Hz. In the region between the plates, show that the magnitude of the induced magnetic field is given by B = B₀(r) cos (2𝝅ft), where B₀(r) is a function of the radial distance r from the capacitor’s central axis.

Verified step by step guidance

1. Start by recalling that a time-varying electric field between the plates of a capacitor induces a magnetic field according to Maxwell's equations. Specifically, the displacement current density, which arises from the changing electric field, acts as a source for the magnetic field.
2. The displacement current density, J_d, is given by: $J{d}_{}=\epsilon {₀}_{}\frac{d}{\mathrm{dt}}E$, where $\epsilon {₀}_{}$ is the permittivity of free space and $E$ is the electric field between the plates. The electric field is related to the potential difference by $E=\frac{V}{d}$, where $d$ is the plate separation.
3. Substitute the sinusoidal potential difference $V=V{₀}_{}\sin \left(2\pi ft\right)$ into the expression for $E$. This gives $E=\frac{V}{{₀}_{}}d$. Differentiate $E$ with respect to time to find $\frac{d}{\mathrm{dt}}E$.
4. Using Ampere-Maxwell's law in its integral form, relate the induced magnetic field to the displacement current. The law states: ${\oint }_{C}B·\mathrm{dl}=\mu {₀}_{}I{d}_{}$, where $I{d}_{}$ is the displacement current enclosed by the loop. For a circular loop of radius $r$, the magnetic field is uniform along the loop, so $B=\frac{\mu }{{₀}_{}}$.
5. Finally, substitute the expression for $\frac{d}{\mathrm{dt}}E$ into the equation for $B$. Simplify to show that the magnetic field has the form $B=B{₀}_{}\left(r\right)\cos \left(2\pi ft\right)$, where $B{₀}_{}\left(r\right)$ is a function of the radial distance $r$.

Key Concepts

Capacitance and Electric Fields

A capacitor stores electrical energy in an electric field created between its plates. The capacitance, defined as the ability to store charge per unit voltage, is influenced by the plate area and separation. In this scenario, the sinusoidal voltage creates a time-varying electric field between the plates, which is essential for understanding how the electric field influences the surrounding space.

Maxwell's Equations

Maxwell's Equations describe how electric and magnetic fields interact and propagate. Specifically, Faraday's law of induction states that a changing electric field induces a magnetic field. This principle is crucial for deriving the relationship between the time-varying electric field in the capacitor and the resulting magnetic field in the surrounding region.

Induced Magnetic Field

The induced magnetic field arises from the changing electric field between the capacitor plates. According to electromagnetic theory, the magnitude and direction of this magnetic field depend on the rate of change of the electric field and the distance from the source. In this case, the magnetic field's dependence on the radial distance r from the capacitor's axis is key to understanding the behavior of the field in the specified configuration.