Category: HC Verma Part 2: Electricity

Problem Statement Solve the capacitor/capacitance problem: A parallel plate capacitor has plate area 100 cm$^2$ and capacitance 88.5 pF. Find the plate separation. What is the field if charged to 100 V? $d = \varepsilon_0 A/C$; $E = V/d$ Step 1: $d = \varepsilon_0 A/C = (8.85\times10^{-12}\times10^{-2})/(88.5\times10^{-12}) = 8.85\times10^{-14}/8.85\times10 Given Information Plate area $A$ (for…

Problem Statement Solve the capacitor/capacitance problem: Two capacitors $C_1 = 4\mu$F and $C_2 = 6\mu$F are connected in series across a 90 V battery. Find the voltage across each capacitor. Same charge on each in series: $Q = C_{eq}V$ $V_i = Q/C_i$ Step 1: $C_{eq} = \dfrac{4\times6}{4+6} = 2.4\,\mu$F. Step 2: $Q = 2.4\times10^{-6}\times90 =…

Problem Statement Solve the capacitor/capacitance problem: A $4\mu$F capacitor charged to 80 V is connected in parallel with an uncharged $2\mu$F capacitor. Find the common potential and the heat generated. Charge conservation; $V_f = Q/(C_1+C_2)$; $\Delta U$ = heat Step 1: $Q = 4\times10^{-6}\times80 = 3.2\times10^{-4}$ C. Step 2: $V_f = Q/(C_1+C_2) = 3.2\ Given…

Problem Statement Solve the work-energy problem: Show that the energy stored in a capacitor can be expressed as $U = \dfrac{1}{2}\varepsilon_0 E^2 \cdot Vol$, where $Vol = A\cdot d$ is the volume between the plates. $U = \frac{1}{2}CV^2$, $C = \varepsilon_0 A/d$, $V = Ed$ Step 1: $U = \frac{1}{2}CV^2 = \frac{1}{2}\cdot\frac{\varepsilon_0 A}{d}\cdo Given Information…

Problem Statement Solve the capacitor/capacitance problem: A cylindrical capacitor has inner radius $a = 1$ cm and outer radius $b = 3$ cm, length $L = 50$ cm. Find its capacitance. $C = \dfrac{2\pi\varepsilon_0 L}{\ln(b/a)}$ Step 1: $\ln(b/a) = \ln(3) \approx 1.099$. Step 2: $$C = \frac{2\pi\times8.85\times10^{-12}\times0.5}{1.099} = \frac{2.78\times10^{-1 Given Information Plate area $A$ (for…

Problem Statement Solve the capacitor/capacitance problem: A spherical capacitor has inner sphere of radius $a = 5$ cm and outer sphere of radius $b = 10$ cm. Find its capacitance. Spherical capacitor: $C = \dfrac{4\pi\varepsilon_0 ab}{b-a}$ Step 1: $a = 0.05$ m, $b = 0.10$ m, $b – a = 0.05$ m. Step 2: $$C…

Problem Statement Solve the capacitor/capacitance problem: A $2\mu$F capacitor is connected to a 100 V battery and a dielectric of $K = 5$ is inserted. Find the new charge, voltage, and energy stored. Voltage is fixed (battery connected): $V = 100$ V $Q’ = KC \cdot V$, $U’ = \frac{1}{2}KC V^2 = KU$ Step 1:…

Problem Statement Solve the capacitor/capacitance problem: A $2\mu$F capacitor is charged to 100 V and then disconnected from the battery. A dielectric of $K = 5$ is now inserted. Find the new voltage, charge, and energy. Charge is conserved (disconnected battery): $Q = $ constant New $C’ = KC$; new $V’ = Q/C’ = V/K$;…

Problem Statement Solve the capacitor/capacitance problem: A parallel plate capacitor with plate separation 3 mm and area 6 cm$^2$ is filled with glass ($K = 7$). Find the capacitance. $C = K\varepsilon_0 A/d$ Step 1: $K = 7$, $A = 6\times10^{-4}$ m$^2$, $d = 3\times10^{-3}$ m. Step 2: $$C = \frac{7\times8.85\times10^{-12}\times6\times10^{-4}}{3\times10^{-3 Given Information Plate area…

Problem Statement Two capacitors $C_1 = 3\mu$F and $C_2 = 6\mu$F are connected in series, and this combination is connected in parallel with $C_3 = 4\mu$F. Find the equivalent capacitance and total charge when connected to 12 V. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed…