Category: HC Verma Part 2: Electricity

Problem Statement Solve the capacitor/capacitance problem: Three capacitors $3\mu$F, $6\mu$F, and $9\mu$F are connected in parallel. Find the equivalent capacitance. $C_{eq} = C_1 + C_2 + C_3$ Step 1: $C_{eq} = 3 + 6 + 9 = 18\,\mu$F. $$\boxed{C_{eq} = 18\,\mu\text{F}}$$ Given Information Plate area $A$ (for parallel plate) or geometry as given Separation $d$…

Problem Statement Solve the capacitor/capacitance problem: Three capacitors $C_1 = 2\mu$F, $C_2 = 4\mu$F, $C_3 = 8\mu$F are connected in series. Find the equivalent capacitance. $1/C_{eq} = 1/C_1 + 1/C_2 + 1/C_3$ Step 1: $$\frac{1}{C_{eq}} = \frac{1}{2}+\frac{1}{4}+\frac{1}{8} = \frac{4+2+1}{8} = \frac{7}{8}$$ $$C_{eq} = \frac{8}{7}\,\mu\text{F} \approx 1.1 Given Information Plate area $A$ (for parallel plate) or…

Problem Statement Solve the capacitor/capacitance problem: A $4\mu$F capacitor is charged to 200 V. Find the energy stored. $U = \frac{1}{2}CV^2$ Step 1: $C = 4\times10^{-6}$ F, $V = 200$ V. Step 2: $U = \frac{1}{2}\times4\times10^{-6}\times(200)^2 = \frac{1}{2}\times4\times10^{-6}\times4\times10^4 = 8\times10^{-2}$ J $= 80$ mJ. $$\boxed{U = 0.08 \text{ J} Given Information Plate area $A$ (for…

Problem Statement Solve the capacitor/capacitance problem: A $10\mu$F capacitor is charged to a potential difference of 50 V. Find the charge on the plates. $Q = CV$ Step 1: $C = 10\times10^{-6}$ F, $V = 50$ V. Step 2: $Q = CV = 10\times10^{-6}\times50 = 5\times10^{-4}$ C $= 500\,\mu$C. $$\boxed{Q = 500\,\mu\text{C}}$$ Given Information Plate…

Problem Statement Solve the capacitor/capacitance problem: A parallel plate capacitor has plates of area $25$ cm$^2$ separated by 1.0 mm. Find its capacitance. $C = \varepsilon_0 A/d$ Step 1: $A = 25\times10^{-4}$ m$^2$, $d = 1.0\times10^{-3}$ m. Step 2: $$C = \frac{8.85\times10^{-12}\times25\times10^{-4}}{10^{-3}} = \frac{2.2125\times10^{-14}}{10^{-3}} = 2 Given Information Plate area $A$ (for parallel plate) or…

Problem Statement The electric field at the Earth’s surface is 150 N/C (directed inward). Using Gauss’s law, find the total charge on Earth and whether it is positive or negative. 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 Statement Determine the electric field for the configuration described: Determine the electric field for the configuration described: State Gauss’s law in both integral and differential forms, and explain the physical meaning of each. Integral: $\oint_S \vec{E}\cdot d\vec{A} = Q_{enc}/\varepsilon_0$ Differential: $\nabla\cdot\vec{E} = \rho/\varepsilon_0$ Integral form: Given Information Geometry and charge distribution as given in…

Problem Statement Three concentric shells of radii $r_1 < r_2 < r_3$ carry charges $Q_1$, $Q_2$, $Q_3$. Find the electric field in each of the four regions. 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…

Problem Statement Determine the electric field for the configuration described: Determine the electric field for the configuration described: Prove using Gauss’s law that a uniformly charged spherical shell exerts no force on a charge placed inside it. Gauss’s law; spherical Gaussian surface inside the shell Step 1: Draw a Gaussian sphere of radius $r Step…

Problem Statement Compare the electric field just outside and just inside the surface of: (a) a uniformly charged non-conducting sphere, and (b) a conducting sphere, both carrying total charge $Q$ and radius $R$. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts…