A parallel-plate capacitor with circular plates is charging so that surface charge density increases at a constant rate. How does the magnetic field caused by the displacement current vary spatially?
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A parallel-plate capacitor with circular plates is charging so that surface charge density increases at a constant rate. How does the magnetic field caused by the displacement current vary spatially?

G1 Physics · Accepted Answer

Step 1: The displacement current density is j𝒅 = ∂D/∂t = ε₀∂E/∂t = dσ/dt, uniform over the circular plate area. Step 2: For a circular Amperian loop of radius r less than plate radius a, enclosed displacement current is Iₑₙ𝒄 = j𝒅 πr². Step 3: Ampère–Maxwell law gives B(2πr) = μ₀ Iₑₙ𝒄, hence B = μ₀j𝒅r/2 for r ≤ a: B rises linearly with r. Step 4: For r ≥ a, enclosed current is j𝒅πa², giving B = μ₀j𝒅a²/(2r), which decreases as 1/r. Step 5: Both expressions peak at r = a, the cylindrical su…