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Chapter 17: Problem 33

A parallel plate capacitor consists of two circular plates each of radius \(12 \mathrm{~cm}\) and separated by \(5.0 \mathrm{~mm}\). The capacitor is being charged by an external source. The charging is being charged and is equal to \(0.15 \mathrm{~A}\). The rate of change of potential difference between the plates will be (A) \(8.173 \times 10^{7} \mathrm{~V} / \mathrm{s}\) (B) \(7.817 \times 10^{8} \mathrm{~V} / \mathrm{s}\) (C) \(1.873 \times 10^{9} \mathrm{~V} / \mathrm{s}\) (D) \(3.781 \times 10^{10} \mathrm{~V} / \mathrm{s}\)

Short Answer

Expert verified
The rate of change of potential difference between the plates is approximately \(2.494 * 10^{9} \mathrm{~V}/\mathrm{s}\). However, the closest answer in the options is (C) \(1.873 \times 10^{9} \mathrm{~V}/\mathrm{s}\).

Step by step solution

01

Calculate the capacitance of the capacitor

To find the capacitance, use the formula C = 蔚鈧� * A/d, where A = 蟺r虏, r is the radius of the plate, 蔚鈧� is the permittivity of free space (8.85 * 10^{-12} F/m), and d is the separation between the plates. First, calculate the area of the plates: A = 蟺r虏 = 蟺(0.12 m)虏 Now, find the capacitance: C = 蔚鈧� * A/d = (8.85 * 10^{-12} F/m) * (蟺(0.12 m)虏) / (0.005 m)
02

Calculate the charge as a function of time

The charging current (I) is equal to 0.15 A. The charge on the capacitor (Q) is the integral of the current (I) with respect to time (t): Q(t) = 鈭� Idt = 鈭� (0.15 A) dt = 0.15t [C]
03

Calculate the rate of change of the potential difference between the plates

We can use the formula Q = C*V (Charge = Capacitance * Voltage) to find the potential difference (V) between the plates: V(t) = Q(t)/C = 0.15t [C] / C Now, we need to calculate the rate of change of potential difference with respect to time (dV/dt): dV/dt = d(0.15t/C)/dt = 0.15/C [V/s]
04

Plug in the values and find the answer

First, calculate the capacitance from Step 1: C = (8.85 * 10^{-12} F/m) * (蟺(0.12 m)虏) / (0.005 m) 鈮� 6.015 * 10^{-11} F Now, calculate the rate of change of potential difference between the plates: dV/dt = 0.15/C = 0.15 / (6.015 * 10^{-11} F) 鈮� 2.494 * 10^{9} V/s The closest answer is (C) \(1.873 \times 10^{9} \mathrm{~V}/\mathrm{s}\). However, the calculated value doesn't match any of the given options.

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