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 answer is: non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates.

Let us analyze the magnetic field inside and outside a charging parallel plate capacitor with circular plates of radius $R$.
Let the surface charge density on the plates at any time $t$ be $\sigma \left(t\right)$. We are given that $\sigma$ is increasing at a constant rate, so:
$\frac{d\sigma }{dt}=k$ (where $k$ is a constant).

The electric field $E$ between the plates of a parallel plate capacitor is uniform (neglecting fringing effects near the edges) and is given by:
$E=\frac{\sigma }{{\epsilon }_0}$
Since $\sigma$ varies with time, the electric field between the plates also varies with time:
$\frac{dE}{dt}=\frac{1}{{\epsilon }_0}\frac{d\sigma }{dt}=\frac{k}{{\epsilon }_0}$

According to Maxwell's Ampere Law, a changing electric field creates a displacement current. The displacement current density $J_d$ is given by:
$J_d={\epsilon }_0\frac{dE}{dt}={\epsilon }_0\left(\frac{k}{{\epsilon }_0}\right)=k$
This displacement current acts as a source of magnetic field. Let us find the magnetic field $B$ at a distance $r$ from the central axis of the circular plates using Ampere's Law:
$\oint B·dl={\mu }_0{I}_{d,\mathrm{enclosed}}$

Case 1: Inside the plates ($r\le R$)
The displacement current enclosed by a loop of radius $r$ is:
$I_{d,\mathrm{enclosed}}=J_d·\left(\pi r^2\right)=k\pi r^2$
Applying Ampere's Law:
$B·\left(2\pi r\right)={\mu }_0\left(k\pi r^2\right)$
$B=\frac{{\mu }_{0}k}{2}r$
Thus, inside the plates, the magnetic field $B$ increases linearly with $r$ ($B\propto r$).

Case 2: Outside the plates ($r>R$)
The total displacement current is confined within the plates of radius $R$. Therefore, for any loop of radius $r>R$, the enclosed displacement current is constant and equal to:
$I_{d,\mathrm{enclosed}}=J_d·\left(\pi R^2\right)=k\pi R^2$
Applying Ampere's Law:
$B·\left(2\pi r\right)={\mu }_0\left(k\pi R^2\right)$
$B=\frac{{\mu }_{0}k{R}^2}{2r}$
Thus, outside the plates, the magnetic field decreases inversely with $r$ ($B\propto \frac{1}{r}$).

Conclusion:
The magnetic field is non-zero everywhere (except at the exact center axis $r=0$). It increases linearly inside the capacitor, reaches its maximum value at the periphery of the plates where $r=R$ (which forms an imaginary cylindrical surface connecting the peripheries of the plates), and then decreases as $1/r$ outside the plates.