Question

Plates capacitor made of circular utes 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 :

• A. non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates
• B. zero between the plates and non-zero outside
• C. zero at all places
• D. constant between the plates and zero outside the plates

Answer 1

Answer: (A)

non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates

Answer 2

The charging parallel plate capacitor has a uniform displacement current density $\vec{J}_d$ between the plates, directed from the positive plate to the negative plate. For circular plates of radius $R$, the displacement current density is $J_d = \epsilon_0 \frac{\partial E}{\partial t}$. Since the surface charge density is increasing at a constant rate, $\frac{d\sigma}{dt}$ is constant. The electric field between the plates is $E = \sigma/\epsilon_0$, so $\frac{\partial E}{\partial t} = \frac{1}{\epsilon_0}\frac{d\sigma}{dt}$, which is constant and uniform between the plates. Thus, $\vec{J}_d$ is constant and uniform between the plates.

The magnetic field arising due to this displacement current can be found using Ampere-Maxwell's law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{d,enc}$, where $I_{d,enc}$ is the displacement current enclosed by the Amperian loop. Due to the symmetry of the uniform displacement current flowing axially between the circular plates, the magnetic field lines are concentric circles in planes parallel to the plates, centered on the axis of the capacitor. The magnitude of the magnetic field $B$ depends on the distance $r$ from the axis.

Consider a circular Amperian loop of radius $r$ centered on the axis, in a plane between the plates.

1. For $r < R$ (inside the plates): The enclosed displacement current is $I_{d,enc} = J_d \cdot (\pi r^2)$. Applying Ampere-Maxwell's law: $B(r) \cdot (2\pi r) = \mu_0 J_d (\pi r^2)$. $B(r) = \frac{\mu_0 J_d r}{2}$. The magnetic field increases linearly with $r$ from $B=0$ at $r=0$ to $B = \frac{\mu_0 J_d R}{2}$ at $r=R$.

2. For $r > R$ (outside the plates): The enclosed displacement current is the total displacement current flowing between the plates, $I_d = J_d \cdot (\pi R^2)$. Applying Ampere-Maxwell's law: $B(r) \cdot (2\pi r) = \mu_0 I_d = \mu_0 J_d (\pi R^2)$. $B(r) = \frac{\mu_0 J_d R^2}{2r}$. The magnetic field decreases with $r$ as $1/r$ for $r > R$.

Comparing the expressions for $B(r)$, we see that for $r \le R$, the maximum value of $B(r)$ is at $r=R$, with $B(R) = \frac{\mu_0 J_d R}{2}$. For $r > R$, $B(r)$ decreases from this value. Thus, the magnitude of the magnetic field is maximum at $r=R$, which corresponds to the cylindrical surface connecting the peripheries of the plates.

The magnetic field is non-zero everywhere except on the axis ($r=0$). Option (1) states "non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates". While strictly speaking the field is zero on the axis, compared to the other options, this is the most accurate description. Option (2) is incorrect because the field is non-zero between the plates (for $r>0$). Option (3) is incorrect because displacement current creates a magnetic field. Option (4) is incorrect because the field between the plates is not constant, it varies with $r$.

Therefore, the magnetic field arising due to displacement current is non-zero everywhere (except the axis) and has its maximum magnitude at the cylindrical surface connecting the peripheries of the plates.