A circular parallel plate capacitor of radius R is being charged, establishing a uniform displacement current I_d between its plates. What is the ratio of the magnitude of the induced magnetic field at a radial distance r = R/2 to that at a radial distance r = 2R from the central axis of the capacitor?

Step-by-step solution

Let the total displacement current be I_d . For a point inside the region between the capacitor plates ( r B_ in = ₀ I_d r 2 R^2 At r = R/2 : B(R/2) = ₀ I_d (R/2) 2 R^2 = ₀ I_d 4 R For a point outside the region between the plates ( r > R ), the magnetic field is: B_ out = ₀ I_d 2 r At r = 2R : B(2R) = ₀ I_d 2 (2R) = ₀ I_d 4 R The ratio of the magnetic fields is: B(R/2) B(2R) = ₀ I_d 4 R ₀ I_d 4 R = 1 Thus, the required ratio is 1:1 . Answer: 1:1