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Electromagnetic Waves
Displacement Current and Induced Magnetic Field
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- Displacement Current Distribution
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2025 1 question
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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:
Displacement current induces a magnetic field that grows linearly with radius inside ($B \propto r$), peaks at plate boundary $r = R$, and drops as $1/r$ outside.
Since $\frac{d\sigma}{dt} = \text{constant}$, the rate of change of electric flux inside the capacitor plates is constant, producing a uniform displacement current density $J_d = \frac{d\sigma}{dt}$. Applying Ampere-Maxwell's law $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{d,enc}$: For $r \le R$: $B (2\pi r) = \mu_0 I_d \left(\frac{r^2}{R^2}\right) \implies B = \frac{\mu_0 I_d r}{2\pi R^2} \propto r$. The field is zero at $r=0$ and grows linearly to reach its maximum at the edge $r=R$. For $r > R$: $B (2\pi r) = \mu_0 I_d \implies B = \frac{\mu_0 I_d}{2\pi r} \propto \frac{1}{r}$. Hence, the field is non-zero everywhere with a maximum at the cylindrical boundary connecting the plate peripheries.