Category: HC Verma Part 2: Electricity

Problem Statement Solve the capacitor/capacitance problem: Derive the expression for the time-varying voltage across a capacitor $C$ being charged through resistance $R$ from battery $V_0$: $V_C(t) = V_0(1-e^{-t/RC})$. Kirchhoff’s loop: $V_0 = V_R + V_C = IR + Q/C$ $I = dQ/dt$ Step 1: Loop equation: $V_0 = R\dfrac{dQ}{dt} + \dfrac{Q}{C}$. Step 2: Separate v…

Problem Statement Solve the capacitor/capacitance problem: $C_1 = 2\mu$F (charged to 100 V) and $C_2 = 3\mu$F (uncharged) are connected in series via a switch. When switch closes, find the final charge on each. When connected in series with a discharged cap, redistribution occurs; conservation of charge Step 1: Initial charge on $C_1$: $Q_0 =…

Problem Statement Solve the capacitor/capacitance problem: A capacitor of $8\mu$F is charged using a 6 V battery. Calculate (a) charge stored, (b) energy stored in capacitor, (c) total energy supplied by battery, and (d) energy lost as heat. Standard capacitor charging energy analysis (a) $Q = CV = 8\times10^{-6}\times6 = 48\,\mu$C. (b) $U_C = \frac{1}{2}QV…

Problem Statement Solve the capacitor/capacitance problem: Two coaxial metallic cylinders of length $L$, inner radius $a$, outer radius $b$ are used as a capacitor. Derive the capacitance formula. Gauss law gives $E = \lambda/(2\pi\varepsilon_0 r)$; integrate to find $V$ Step 1: Field between cylinders: $E = \lambda/(2\pi\varepsilon_0 r)$. Step 2: Potential Given Information Plate area…

Problem Statement Solve the Newton’s Laws / mechanics problem: A parallel plate capacitor (width $b$, separation $d$, length $L$) is connected to battery $V$. A dielectric of constant $K$ is inserted a length $x$ inside. Find the force on the dielectric slab. $C(x) = (\varepsilon_0 b/d)[x(K-1) + L]$; $U = \frac{1}{2}C(x)V^2$; $F = dU/dx$ (at…

Problem Statement Solve the work-energy problem: Find the energy density in the electric field of a parallel plate capacitor with dielectric constant $K$, plate voltage $V$, and separation $d$. $u = \frac{1}{2}K\varepsilon_0 E^2 = \frac{1}{2}\varepsilon_0 KE^2$ Step 1: $E = V/d$. Step 2: $$u = \frac{1}{2}K\varepsilon_0 E^2 = \frac{1}{2}K\varepsilo Given Information Mass $m$, velocity $v$,…

Problem Statement Solve the kinematics problem: Define the electric displacement vector $\vec{D}$ and write Gauss’s law in terms of $\vec{D}$. For a parallel plate capacitor with dielectric $K$, find $D$, $E$, and $P$. $\vec{D} = \varepsilon_0\vec{E} + \vec{P} = K\varepsilon_0\vec{E}$ Gauss: $\oint\vec{D}\cdot d\vec{A} = Q_{free}$ Step 1: $D = \s Given Information Initial velocity $u$…

Problem Statement A capacitor charged to 90 V is isolated. When a dielectric is inserted, the voltage drops to 30 V. Find the dielectric constant. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This problem draws on fundamental physical…

Problem Statement Explain how a dielectric reduces the electric field inside a capacitor and define polarization $P$ and electric susceptibility $\chi_e$. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This problem draws on fundamental physical principles. The key is…

Problem Statement Solve the capacitor/capacitance problem: In the circuit: 10 V battery, $R_1 = 2\,\Omega$, $R_2 = 3\,\Omega$, and capacitor $C = 5\mu$F are in a circuit where $R_1$ and $(R_2, C)$ are in parallel across the battery. In steady state, find the charge on $C$. In steady state, no current through capacitor; all current…