A circular parallel-plate capacitor has plate radius R = 0.20 m. The charge on its positive plate varies as Q(t) = 4t² C and is uniformly distributed. At t = 0.5 s, find the magnetic field between the plates at r = 0.10 m. If B = N×10⁻⁶ T, find N. Take μ₀ = 4π×10⁻⁷ T·m/A.
Physics · Electromagnetic wave · displacement current and properties of electromagnetic wave
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Question

A circular parallel-plate capacitor has plate radius R = 0.20 m. The charge on its positive plate varies as Q(t) = 4t² C and is uniformly distributed. At t = 0.5 s, find the magnetic field between the plates at r = 0.10 m. If B = N×10⁻⁶ T, find N. Take μ₀ = 4π×10⁻⁷ T·m/A.

Physics · Electromagnetic wave · displacement current and properties of electromagnetic wave

G1 Physics · Accepted Answer

First find the charging current: I = dQ/dt. Given, Q = 4t², so I = 8t. At t = 0.5 s, I = 4 A. Since r<R, the enclosed displacement current is I(enclosed) = I(r²/R²). Thus, I(enclosed) = 4(0.10²/0.20²) = 1 A. Ampere-Maxwell law gives B(2πr) = μ₀I(enclosed). Therefore, B = μ₀I(enclosed)/(2πr) = (4π×10⁻⁷)(1)/[2π(0.10)] = 2×10⁻⁶ T. Hence N = 2.