Category: Part 3: Electricity

Problem Statement Analyze the circuit: Irodov Problem 3.108. $F = ILB\sin\theta = 10\times0.5\times0.3\times\sin30° = \boxed{0.75\,\text{N}}$. Given Information Resistance values $R_1, R_2, \ldots$ as specified EMF $\mathcal{E}$ and internal resistance $r$ of battery Any additional circuit elements given Physical Concepts & Formulas Ohm’s Law $V = IR$ and Kirchhoff’s two laws are the complete toolkit for…

Problem Statement Determine the electric field for the configuration described: Irodov Problem 3.27. $$\boxed{\nabla\cdot\mathbf{E} = \rho/\varepsilon_0}$$ (1st Maxwell equation). Positive $\rho$: source of field; negative: sink. Charge-free region: $\nabla^2 V = 0$. Given Information Geometry and charge distribution as given in the problem Permittivity of free space $\varepsilon_0 = 8.85\times10^{-12}\,\text{C}^2\text{N}^{-1}\text{m}^{-2}$ Coulomb constant $k = 1/(4\pi\varepsilon_0)…

Problem Statement Solve the capacitor/capacitance problem: Irodov Problem 3.61. $V(t) = V_0 e^{-t/RC}$, $\tau = RC = 1.0\,\text{s}$. $V(2s) = 6.77\,\text{V}$. Half-life: $t_{1/2} = \tau\ln2 = 0.693\,\text{s}$. Given Information Plate area $A$ (for parallel plate) or geometry as given Separation $d$ or radii as given Dielectric constant $\kappa$ (if applicable, else $\kappa=1$ for vacuum) Permittivity…

Problem Statement Solve the magnetic field/force problem: Irodov Problem 3.105. $$B = \frac{\mu_0 NI}{2\pi R} = \frac{4\pi\times10^{-7}\times500\times2}{2\pi\times0.10} = \boxed{2.0\,\text{mT}}$$ $B=0$ outside. Given Information Current $I$ or charge $q$ and velocity $v$ as given Geometry (straight wire, loop, solenoid) as specified Permeability of free space $\mu_0 = 4\pi\times10^{-7}\,\text{T m A}^{-1}$ Physical Concepts & Formulas Magnetic fields…

Problem Statement Solve the Newton’s Laws / mechanics problem: Irodov Problem 3.28. 4 properties: $E=0$ inside, charge on surface only, surface is equipotential, field just outside $= \sigma/\varepsilon_0$. Field strongest at sharp points. Given Information Mass(es), forces, angles, and coefficients of friction as given $g = 9.8\,\text{m/s}^2$ (acceleration due to gravity) Physical Concepts & Formulas…

Problem Statement Analyze the circuit: Irodov Problem 3.62. Self-similar: $R_{eq}^2 – R_1 R_{eq} – R_1 R_2 = 0$. For $R_1=1$, $R_2=2\,\Omega$: $R_{eq} = \boxed{2\,\Omega}$. Given Information Resistance values $R_1, R_2, \ldots$ as specified EMF $\mathcal{E}$ and internal resistance $r$ of battery Any additional circuit elements given Physical Concepts & Formulas Ohm’s Law $V = IR$…

Problem Statement Solve the magnetic field/force problem: Irodov Problem 3.106. $\mathbf{F} = q\mathbf{v}\times\mathbf{B}$. Proton at $3\times10^6\hat{x}\,\text{m/s}$ in $0.5\hat{z}\,\text{T}$: $\mathbf{F} = -2.4\times10^{-13}\hat{y}\,\text{N}$. Given Information Current $I$ or charge $q$ and velocity $v$ as given Geometry (straight wire, loop, solenoid) as specified Permeability of free space $\mu_0 = 4\pi\times10^{-7}\,\text{T m A}^{-1}$ Physical Concepts & Formulas Magnetic fields…

Problem Statement Solve the rotational mechanics problem: Irodov Problem 3.25. $$\boldsymbol{\tau} = \mathbf{p}\times\mathbf{E},\quad U = -\mathbf{p}\cdot\mathbf{E} = -pE\cos\theta$$ Stable equilibrium at $\theta=0$. Given Information Mass $m$, geometry (radius $R$, length $L$, etc.) Angular velocity $\omega$ or torque $\tau$ Axis of rotation Physical Concepts & Formulas Rotational mechanics is the angular analogue of linear mechanics. The…

Problem Statement Solve the work-energy problem: Irodov Problem 3.59. Max power to load $R$ when $R = r$ (impedance matching). $P_{\max} = \mathcal{E}^2/(4r)$, efficiency $= 50\%$. Given Information Mass $m$, velocity $v$, height $h$, or other given quantities Any forces doing work (conservative or non-conservative) as specified Physical Concepts & Formulas The Work-Energy Theorem states…

Problem Statement Solve the magnetic field/force problem: Irodov Problem 3.103. $$\boxed{B = \frac{\mu_0 IR^2}{2(R^2+x^2)^{3/2}}}$$ At centre: $\mu_0 I/(2R)$. Far: dipole field. Given Information Current $I$ or charge $q$ and velocity $v$ as given Geometry (straight wire, loop, solenoid) as specified Permeability of free space $\mu_0 = 4\pi\times10^{-7}\,\text{T m A}^{-1}$ Physical Concepts & Formulas Magnetic fields…