A parallel plate capacitor made of circular plates is being charged such that the surface charge density on its plates is increasing at a constant rate with time. The magnetic field arising due to displacement current is:

The correct option is: Non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates

Let's analyze the magnetic field produced by the displacement current inside and outside a parallel plate capacitor with circular plates of radius $R$ being charged at a rate such that the surface charge density $\sigma$ increases at a constant rate with time $t$.

The surface charge density $\sigma$ is changing at a constant rate:
$\frac{d\sigma }{dt}=\text{constant}$
The electric field $E$ between the plates of a parallel plate capacitor (neglecting fringing fields) is given by:
$E=\frac{\sigma }{{\epsilon }_0}$
Since $\sigma$ increases at a constant rate, the electric field also increases at a constant rate:
$\frac{dE}{dt}=\frac{1}{{\epsilon }_0}$

According to the Maxwell-Ampere law, a changing electric field gives rise to a displacement current density $J_d$:
$J_d={\epsilon }_0\frac{dE}{dt}=\frac{d\sigma }{dt}$
This displacement current density exists in the region between the plates. We can find the magnetic field $B$ at a distance $r$ from the central axis of the circular plates using Ampere's Law:
$\oint B·dl={\mu }_0{I}_{d,\text{enclosed}}$
For a circular loop of radius $r$ coaxial with the plates, the line integral is:
$B·2\pi r={\mu }_0{I}_{d,\text{enclosed}}$

Let's evaluate this in two regions:
Case 1: Inside the plates ($r\le R$)
The enclosed displacement current is:
$I_{d,\text{enclosed}}=J_d·\pi r^2$
Substituting this into Ampere's Law:
$B·2\pi r={\mu }_0{J}_d·\pi r^2$
$B=\frac{{\mu }_0{J}_d}{2}r$
Thus, inside the plates, the magnetic field is non-zero (except at the axis $r=0$) and increases linearly with $r$, reaching its maximum value at the boundary $r=R$.

Case 2: Outside the plates ($r>R$)
The total enclosed displacement current is limited to the region between the plates:
$I_{d,\text{enclosed}}=J_d·\pi R^2$
Substituting this into Ampere's Law:
$B·2\pi r={\mu }_0{J}_d·\pi R^2$
$B=\frac{{\mu }_0{J}_d{R}^2}{2r}$
Thus, outside the plates, the magnetic field decreases as $\frac{1}{r}$, meaning it is non-zero everywhere and decays as we move further away.

Comparing the two regions, the magnetic field is non-zero everywhere (except at the center line $r=0$ and infinity) and reaches its maximum value precisely at the boundary $r=R$, which corresponds to the imaginary cylindrical surface connecting the peripheries of the plates.