Quantum Mechanics - H.K. Malik

Cheatsheet Content

1. Introduction to Electromagnetism Electromagnetism: The branch of physics that studies the interactions between electric charges and currents. It unifies electricity and magnetism, showing they are two aspects of the same fundamental force. Key Concepts: Electric fields, magnetic fields, electromagnetic waves, forces on charges and currents. Fundamental Principles: Coulomb's Law, Ampere's Law, Faraday's Law, Gauss's Laws (for electricity and magnetism). 2. Electrostatics 2.1. Electric Charge Properties: Quantized ($q = ne$, where $e = 1.602 \times 10^{-19}$ C), conserved, two types (positive and negative). Units: Coulomb (C). 2.2. Coulomb's Law Describes the electrostatic force between two point charges. $$F = k \frac{|q_1 q_2|}{r^2} \hat{r}$$ where $k = \frac{1}{4\pi\epsilon_0} \approx 9 \times 10^9 \text{ N m}^2/\text{C}^2$ is Coulomb's constant, $\epsilon_0 = 8.854 \times 10^{-12} \text{ C}^2/\text{N m}^2$ is the permittivity of free space. Vector Form: $\vec{F}_{12} = k \frac{q_1 q_2}{r_{12}^2} \hat{r}_{12}$ (force on $q_2$ due to $q_1$). Superposition Principle: The total force on a charge is the vector sum of forces due to individual charges. 2.3. Electric Field ($\vec{E}$) Definition: Force per unit positive test charge. $$\vec{E} = \frac{\vec{F}}{q_0}$$ Electric Field due to a Point Charge $q$: $$\vec{E} = k \frac{q}{r^2} \hat{r}$$ Electric Field due to Continuous Charge Distribution: $$\vec{E} = k \int \frac{dq}{r^2} \hat{r}$$ Line charge density: $\lambda = dq/dl$ Surface charge density: $\sigma = dq/dA$ Volume charge density: $\rho = dq/dV$ Electric Field Lines: Originate from positive charges, terminate on negative charges, never cross, density indicates field strength. 2.4. Gauss's Law for Electrostatics Relates the electric flux through a closed surface to the net charge enclosed within that surface. $$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$$ Applications: Calculating electric fields for symmetric charge distributions (sphere, cylinder, infinite plane). Differential Form: $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$ (Maxwell's 1st equation). 2.5. Electric Potential ($V$) Definition: Work done per unit charge by an external agent to move a test charge from a reference point (usually infinity) to a specific point without acceleration. $$V = \frac{U}{q_0}$$ Relation to Electric Field: $$\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{l}$$ $$\vec{E} = -\nabla V$$ Electric Potential due to a Point Charge $q$: $$V = k \frac{q}{r}$$ Electric Potential due to Continuous Charge Distribution: $$V = k \int \frac{dq}{r}$$ Equipotential Surfaces: Surfaces where the electric potential is constant. Electric field lines are always perpendicular to equipotential surfaces. 2.6. Electric Potential Energy ($U$) Definition: Energy stored in a system of charges due to their configuration. $$U = qV$$ For two point charges $q_1, q_2$: $$U = k \frac{q_1 q_2}{r}$$ For a system of multiple charges: Sum of potential energies for all unique pairs. 2.7. Capacitance ($C$) Definition: Ability of a conductor to store electric charge. $C = Q/V$. Units: Farad (F). Parallel Plate Capacitor: $$C = \frac{\epsilon_0 A}{d}$$ where $A$ is plate area, $d$ is separation. Energy Stored in a Capacitor: $$U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$$ Capacitors in Series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ Capacitors in Parallel: $C_{eq} = \sum C_i$ Dielectrics: Insulating materials placed between capacitor plates. Increase capacitance by a factor $\kappa$ (dielectric constant). $$C = \kappa C_0 = \kappa \frac{\epsilon_0 A}{d}$$ Permittivity of dielectric: $\epsilon = \kappa \epsilon_0$. 3. Magnetostatics 3.1. Magnetic Field ($\vec{B}$) Definition: Generated by moving charges (currents). Exerts force on moving charges and current-carrying wires. Units: Tesla (T) or Gauss (G); $1 \text{ T} = 10^4 \text{ G}$. Magnetic Field Lines: Form closed loops, never cross, density indicates field strength, direction given by tangent. 3.2. Magnetic Force On a Moving Charge: Lorentz Force Law. $$\vec{F}_B = q(\vec{v} \times \vec{B})$$ Direction given by right-hand rule. No force if $\vec{v} \parallel \vec{B}$. No work done by magnetic force ($W = 0$). On a Current-Carrying Wire: $$\vec{F}_B = I (\vec{L} \times \vec{B})$$ For non-uniform field or wire, $\vec{F}_B = I \int (d\vec{l} \times \vec{B})$. Torque on a Current Loop: $$\vec{\tau} = \vec{\mu} \times \vec{B}$$ where $\vec{\mu} = IA\hat{n}$ is the magnetic dipole moment. Potential Energy of a Magnetic Dipole: $$U = -\vec{\mu} \cdot \vec{B}$$ 3.3. Biot-Savart Law Calculates the magnetic field generated by a current element $d\vec{l}$. $$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$$ where $\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$ is the permeability of free space. Applications: Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Center of a Circular Loop: $B = \frac{\mu_0 I}{2R}$ Center of a Solenoid: $B = \mu_0 n I$ (n = turns per unit length) 3.4. Ampere's Law Relates the line integral of the magnetic field around a closed loop to the total current passing through the loop. $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$ Applications: Calculating magnetic fields for symmetric current distributions (long wire, solenoid, toroid). Differential Form (for magnetostatics): $\nabla \times \vec{B} = \mu_0 \vec{J}$ (Maxwell's 4th equation, without displacement current). 3.5. Gauss's Law for Magnetism States that the net magnetic flux through any closed surface is always zero. This implies that magnetic monopoles do not exist (magnetic field lines always form closed loops). $$\oint \vec{B} \cdot d\vec{A} = 0$$ Differential Form: $\nabla \cdot \vec{B} = 0$ (Maxwell's 2nd equation). 4. Electrodynamics 4.1. Faraday's Law of Induction States that a changing magnetic flux through a loop of wire will induce an electromotive force (EMF) in the loop. $$\mathcal{E} = -\frac{d\Phi_B}{dt}$$ where $\Phi_B = \int \vec{B} \cdot d\vec{A}$ is the magnetic flux. Lenz's Law: The direction of the induced current (and hence induced EMF) is such that it opposes the change in magnetic flux that produced it (negative sign in Faraday's Law). Motional EMF: EMF induced in a conductor moving through a magnetic field. $$\mathcal{E} = \int (\vec{v} \times \vec{B}) \cdot d\vec{l}$$ For a straight conductor of length $L$ moving perpendicular to a uniform $\vec{B}$: $\mathcal{E} = BLv$. Differential Form: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ (Maxwell's 3rd equation). 4.2. Inductance ($L$) Definition: Property of an electrical conductor to oppose a change in the electric current flowing through it. $$\mathcal{E} = -L \frac{dI}{dt}$$ Units: Henry (H). Self-Inductance of a Solenoid: $L = \mu_0 n^2 A l$. Energy Stored in an Inductor: $$U = \frac{1}{2}LI^2$$ Inductors in Series: $L_{eq} = \sum L_i$ Inductors in Parallel: $\frac{1}{L_{eq}} = \sum \frac{1}{L_i}$ Mutual Inductance ($M$): When a changing current in one coil induces an EMF in a nearby coil. $$\mathcal{E}_2 = -M \frac{dI_1}{dt}$$ 4.3. Displacement Current (Maxwell's Correction to Ampere's Law) Maxwell realized Ampere's Law was incomplete for time-varying fields (e.g., charging capacitor). He added the displacement current term. $$I_D = \epsilon_0 \frac{d\Phi_E}{dt}$$ where $\Phi_E = \int \vec{E} \cdot d\vec{A}$ is the electric flux. Modified Ampere's Law (Ampere-Maxwell Law): $$\oint \vec{B} \cdot d\vec{l} = \mu_0 (I_{cond} + I_D) = \mu_0 I_{cond} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$$ 5. Maxwell's Equations The four fundamental equations that describe the behavior of electric and magnetic fields and their interactions. They unify electricity, magnetism, and optics. Integral Form: Gauss's Law for Electricity: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ (Electric charges create electric fields; field lines originate/terminate on charges). Gauss's Law for Magnetism: $\oint \vec{B} \cdot d\vec{A} = 0$ (Magnetic monopoles do not exist; magnetic field lines are closed loops). Faraday's Law of Induction: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$ (Changing magnetic fields create electric fields). Ampere-Maxwell Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ (Electric currents and changing electric fields create magnetic fields). Differential Form: $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$ $\nabla \cdot \vec{B} = 0$ $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$ where $\vec{J}$ is the current density. 6. Electromagnetic Waves Maxwell's equations predict the existence of electromagnetic waves, which are self-propagating oscillations of electric and magnetic fields. Properties: Propagate at the speed of light in vacuum: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$. Are transverse waves: $\vec{E}$ and $\vec{B}$ fields are perpendicular to each other and to the direction of propagation. $\vec{E}$ and $\vec{B}$ fields are in phase and their magnitudes are related by $E = cB$. Do not require a medium to propagate. Carry energy and momentum. Electromagnetic Spectrum: The full range of electromagnetic waves, ordered by frequency (or wavelength). Includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Poynting Vector ($\vec{S}$): Represents the direction and rate of energy flow per unit area (power per unit area) of an electromagnetic wave. $$\vec{S} = \frac{1}{\mu_0} (\vec{E} \times \vec{B})$$ The average intensity is $I = \langle S \rangle = \frac{E_{max}B_{max}}{2\mu_0} = \frac{E_{max}^2}{2\mu_0 c} = \frac{c B_{max}^2}{2\mu_0}$. Radiation Pressure: EM waves exert pressure on surfaces they strike. $$P_{rad} = \frac{I}{c} \quad (\text{for total absorption})$$ $$P_{rad} = \frac{2I}{c} \quad (\text{for total reflection})$$ 7. Properties of Materials in EM Fields Conductors: Free charges move easily. $\vec{E}=0$ inside a conductor in electrostatic equilibrium. Excess charge resides on the surface. Potential is constant throughout the conductor. Dielectrics (Insulators): Charges are bound, but can be polarized by an external E-field. Reduce the internal E-field by a factor of $\kappa$. Permittivity $\epsilon = \kappa \epsilon_0$. Magnetic Materials: Diamagnetism: Weak repulsion from magnetic fields (e.g., water, copper). Induced magnetic moment opposes the applied field. Paramagnetism: Weak attraction to magnetic fields (e.g., aluminum, oxygen). Randomly oriented atomic magnetic moments align with the field. Ferromagnetism: Strong attraction to magnetic fields (e.g., iron, nickel, cobalt). Domains of aligned atomic moments. Exhibits hysteresis. Permeability of Material: $\mu = \kappa_m \mu_0$, where $\kappa_m$ is relative permeability. 8. Circuits (DC and AC Fundamentals) 8.1. DC Circuits Ohm's Law: $V = IR$. Resistance: $R = \rho L/A$, where $\rho$ is resistivity. Resistors in Series: $R_{eq} = \sum R_i$. Resistors in Parallel: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$. Power Dissipated: $P = IV = I^2R = V^2/R$. Kirchhoff's Rules: Junction Rule: Sum of currents entering a junction equals sum of currents leaving ($ \sum I_{in} = \sum I_{out}$). (Conservation of charge) Loop Rule: Sum of potential changes around any closed loop is zero ($ \sum \Delta V = 0$). (Conservation of energy) RC Circuits: Charging: $Q(t) = Q_{max}(1 - e^{-t/RC})$, $I(t) = I_{max}e^{-t/RC}$. Discharging: $Q(t) = Q_0 e^{-t/RC}$, $I(t) = I_0 e^{-t/RC}$. Time Constant: $\tau = RC$. RL Circuits: Current Growth: $I(t) = I_{max}(1 - e^{-t/\tau})$, where $\tau = L/R$. Current Decay: $I(t) = I_0 e^{-t/\tau}$. 8.2. AC Circuits (RLC Series Circuit) Impedance ($Z$): Total opposition to current flow in AC circuits. $V_{rms} = I_{rms}Z$. Reactance: Inductive Reactance: $X_L = \omega L = 2\pi f L$. Capacitive Reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$. Impedance of RLC Series Circuit: $$Z = \sqrt{R^2 + (X_L - X_C)^2}$$ Phase Angle ($\phi$): Phase difference between voltage and current. $$\tan\phi = \frac{X_L - X_C}{R}$$ Resonance: Occurs when $X_L = X_C$, leading to minimum impedance $Z=R$. $$f_0 = \frac{1}{2\pi\sqrt{LC}}$$ Power in AC Circuits: Average Power: $P_{avg} = I_{rms}V_{rms}\cos\phi = I_{rms}^2 R$. Power Factor: $\cos\phi = R/Z$. 9. Electromagnetic Waves in Materials Refractive Index: $n = c/v$, where $v$ is the speed of light in the medium. Related to permittivity and permeability: $v = \frac{1}{\sqrt{\mu\epsilon}} = \frac{1}{\sqrt{\kappa_m \mu_0 \kappa \epsilon_0}} = \frac{c}{\sqrt{\kappa_m \kappa}}$. Dispersion: Refractive index depends on wavelength/frequency. Absorption: Energy of EM waves can be absorbed by the material.