Category: HC Verma Part 2: Electricity

Problem Statement Solve the capacitor/capacitance problem: A parallel plate capacitor has plates of radius $R$ and separation $d$. Derive an expression for capacitance and discuss the effect of fringing fields (guard ring). $C = \varepsilon_0 A/d = \varepsilon_0 \pi R^2/d$ (ideal, no fringing) Guard rings eliminate fringing effects at plate edges Step 1: Fo Given…

Problem Statement Solve the capacitor/capacitance problem: A capacitor with $K = 5$ dielectric has capacitance $200$ pF. The dielectric is removed while battery (100 V) remains connected. Find the change in charge. With battery: $V$ constant; $C’ = C/K$; $Q’ = C’V$ Step 1: Initial: $C = 200$ pF, $Q_i = CV = 200\times10^{-12}\times100 =…

Problem Statement Solve the capacitor/capacitance problem: A $1\mu$F capacitor is needed with plate separation 0.1 mm. Find the minimum plate area required (no dielectric). $A = Cd/\varepsilon_0$ Step 1: $C = 10^{-6}$ F, $d = 10^{-4}$ m. Step 2: $$A = \frac{Cd}{\varepsilon_0} = \frac{10^{-6}\times10^{-4}}{8.85\times10^{-12}} = \frac{10^{-10}}{8.85\times10^{ Given Information Plate area $A$ (for parallel plate) or…

Problem Statement Solve the capacitor/capacitance problem: Show that when a charged capacitor $C_1$ shares charge with an uncharged $C_2$, the energy loss is $\Delta U = \dfrac{C_1 C_2}{2(C_1+C_2)}V_0^2$ where $V_0$ is the initial voltage. Energy before – Energy after = heat generated Step 1: $Q_0 = C_1 V_0$. Final voltage: $V_f = Q_0/(C_1+C_2) = C_1…

Problem Statement Solve the capacitor/capacitance problem: Four capacitors $C_1 = C_2 = 4\mu$F and $C_3 = C_4 = 2\mu$F form a bridge network connected to 12 V. $C_5 = 1\mu$F connects the midpoints. If the bridge is balanced, find the charge on $C_5$. Balanced bridge: $C_1/C_2 = C_3/C_4 \Rightarrow$ no charge on $C_5$ Step 1:…

Problem Statement Solve the capacitor/capacitance problem: A parallel plate capacitor (area $A$, separation $d$) has a dielectric ($K = 3$) inserted to cover half the area. Find the effective capacitance. Two capacitors in parallel: half with dielectric, half without Step 1: Without dielectric (area $A/2$): $C_1 = \varepsilon_0(A/2)/d$. Step 2: With dielect Given Information Plate…

Problem Statement Solve the capacitor/capacitance problem: A $100\mu$F capacitor charged to 12 V is discharged through a $1000\,\Omega$ resistor. Find (a) the initial current, (b) the time constant, and (c) the energy dissipated. $I_0 = V_0/R$; $\tau = RC$; energy = $\frac{1}{2}CV_0^2$ (a) $I_0 = V_0/R = 12/1000 = 0.012$ A $= 12$ mA. (b)…

Problem Statement Solve the capacitor/capacitance problem: A parallel plate capacitor of plate area $A$ and separation $d$ has a metal slab of thickness $t$ inserted (not touching either plate). Find the new capacitance. Metal slab reduces effective separation to $d – t$ (field inside metal = 0) Step 1: Inside the metal slab, $E =…

Problem Statement Solve the capacitor/capacitance problem: Three conducting plates A, B, C are parallel, each of area $A$. A–B separation = $d_1$, B–C separation = $d_2$. Plate B has charge $Q_B$ on it; A and C are connected to a battery $V$. Find the capacitance of the system. The three-plate system forms two capacitors A–B…

Problem Statement Solve the capacitor/capacitance problem: A capacitor of plate area $A$ and separation $d$ has two dielectric slabs of thickness $d/2$ each with constants $K_1 = 2$ and $K_2 = 4$. Find the capacitance. Two dielectrics stacked: two capacitors in series Step 1: $C_1 = K_1\varepsilon_0 A/(d/2) = 2K_1\varepsilon_0 A/d = 4\varepsilon_0 A/d$. Ste…