Electrostatics Essentials

Cheatsheet Content

1. Introduction to Electrostatics Charges: Positive (shortage of electrons) Negative (excess of electrons) Neutrons are not transferred. Forces: Same charges repel, opposite charges attract. Electrostatics: Study of electric charges at rest. Charging Objects: Friction: Electrons transferred by rubbing. Conduction (Touch): Electrons transferred by contact. Polarization (Induction): Charge rearrangement in a neutral object due to a nearby charged object. Conservation of Charge: Charge cannot be created or destroyed, only transferred. 2. Electric Field Source Charge: Produces an electric field ($q_1, q_2, \dots$). Can be point or continuous. Test Charge: Small, electropositive charge ($Q$) that does not disturb the field. Principle of Superposition: The interaction between any two charges is unaffected by others. Total force $\vec{F} = \vec{F}_1 + \vec{F}_2 + \dots$. Coulomb's Law: Force between two point charges $q$ and $Q$ separated by distance $\mathcal{r}$. $$ \vec{F} = \frac{1}{4\pi\epsilon_0} \frac{qQ}{\mathcal{r}^2} \hat{\mathcal{r}} $$ where $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2\text{ N}^{-1}\text{ m}^{-2}$ is the permittivity of free space. Electric Field ($\vec{E}$): Force per unit charge experienced by a test charge $Q$. $$ \vec{E}(\vec{r}) = \frac{\vec{F}}{Q} = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{q_i}{\mathcal{r}_i^2} \hat{\mathcal{r}}_i $$ where $\mathcal{r}_i$ is the separation vector from $q_i$ to $\vec{r}$. 3. Continuous Charge Distributions For continuous charge distributions, sums become integrals: Line Charge: $\lambda$ (charge/unit length), $dq = \lambda dl'$ $$ \vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{\mathcal{L}} \frac{\lambda(\vec{r}')}{\mathcal{r}^2} \hat{\mathcal{r}} dl' $$ Surface Charge: $\sigma$ (charge/unit area), $dq = \sigma da'$ $$ \vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{\mathcal{S}} \frac{\sigma(\vec{r}')}{\mathcal{r}^2} \hat{\mathcal{r}} da' $$ Volume Charge: $\rho$ (charge/unit volume), $dq = \rho d\tau'$ $$ \vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{\mathcal{V}} \frac{\rho(\vec{r}')}{\mathcal{r}^2} \hat{\mathcal{r}} d\tau' $$ 4. Electric Flux and Gauss's Law Electric Flux ($\Phi_E$): Measure of the "number of electric field lines" passing through a surface. Uniform $\vec{E}$: $\Phi_E = \vec{E} \cdot \vec{S} = ES \cos\theta$ Non-uniform $\vec{E}$: $\Phi_E = \int \vec{E} \cdot d\vec{s}$ Gauss's Law (Integral Form): Total electric flux through any closed surface is proportional to the total enclosed charge. $$ \oint \vec{E} \cdot d\vec{s} = \frac{Q_{enc}}{\epsilon_0} $$ Flux is positive if field lines leave the surface. Flux is negative if field lines enter the surface. Gauss's Law (Differential Form): $$ \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} $$ Applications of Gauss's Law: Used to find electric fields for symmetric charge distributions (e.g., sphere, infinite plane, parallel plates). 5. Electric Potential Curl of $\vec{E}$: For electrostatic fields, $\nabla \times \vec{E} = 0$. This implies the line integral of $\vec{E}$ is path-independent. Electric Potential ($V$): Work/energy needed to move a unit charge from a reference point (often infinity) to a specific point in an electric field. $$ V(\vec{r}) = -\int_O^{\vec{r}} \vec{E} \cdot d\vec{l} $$ where $O$ is the reference point. Potential Difference: $V_b - V_a = -\int_a^b \vec{E} \cdot d\vec{l}$. Relationship between $\vec{E}$ and $V$: $\vec{E} = -\nabla V$. Superposition Principle for Potential: $V = V_1 + V_2 + \dots$ Units: Volts (V), where $1 \text{ V} = 1 \text{ J/C}$. Potential of a Point Charge $q$ at origin: $V(r) = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$ Potential of a Collection of Point Charges: $V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{q_i}{\mathcal{r}_i}$ Potential of Continuous Charge Distributions: Line: $V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{\mathcal{L}} \frac{\lambda(\vec{r}')}{\mathcal{r}} dl'$ Surface: $V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{\mathcal{S}} \frac{\sigma(\vec{r}')}{\mathcal{r}} da'$ Volume: $V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{\mathcal{V}} \frac{\rho(\vec{r}')}{\mathcal{r}} d\tau'$ Equipotential Surface: A surface over which the potential is constant. $\vec{E}$ is always perpendicular to equipotential surfaces. 6. Boundary Conditions Normal Component of $\vec{E}$: Discontinuous across a surface charge $\sigma$. $$ E_{\perp, \text{above}} - E_{\perp, \text{below}} = \frac{\sigma}{\epsilon_0} $$ If no surface charge, $E_\perp$ is continuous. Tangential Component of $\vec{E}$: Always continuous. $$ E_{\parallel, \text{above}} = E_{\parallel, \text{below}} $$ Potential $V$: Always continuous across any boundary. $$ V_{\text{above}} = V_{\text{below}} $$ Normal Derivative of $V$: Related to the normal component of $\vec{E}$. $$ \frac{\partial V_{\text{above}}}{\partial n} - \frac{\partial V_{\text{below}}}{\partial n} = -\frac{\sigma}{\epsilon_0} $$ 7. Energy of a Charge Distribution Work Done to Move a Test Charge $Q$: $$ W_{a \to b} = Q [V(b) - V(a)] $$ If reference is infinity, $W = QV(r)$. Energy of a Point Charge Distribution: Work to assemble a collection of point charges. $$ W = \frac{1}{2} \sum_{i=1}^{n} q_i V(\vec{r}_i) = \frac{1}{2} \sum_{i=1}^{n} \sum_{j \ne i} \frac{1}{4\pi\epsilon_0} \frac{q_i q_j}{\mathcal{r}_{ij}} $$ Energy of a Continuous Charge Distribution: $$ W = \frac{1}{2} \int \rho V d\tau $$ Alternatively, in terms of $\vec{E}$: $$ W = \frac{\epsilon_0}{2} \int |\vec{E}|^2 d\tau $$ (integrated over all space) 8. Conductors Properties of Ideal Conductors in Electrostatics: $\vec{E} = 0$ inside a conductor. $\rho = 0$ inside a conductor (any net charge resides on the surface). A conductor is an equipotential ($V$ is constant throughout). $\vec{E}$ is perpendicular to the surface just outside a conductor. Induced Charges: External charges induce charges on conductor surfaces. If a charge $+q$ is near an uncharged conductor, negative charges are pulled to the near side, positive are repelled to the far side. In a cavity within a conductor, the total induced charge on the inner surface is equal and opposite to the charge inside the cavity. Surface Charge and Force: Electric field just outside a conductor: $\vec{E} = \frac{\sigma}{\epsilon_0} \hat{n}$ Force per unit area (electrostatic pressure) on the surface: $f = \frac{1}{2} \sigma \vec{E} = \frac{\sigma^2}{2\epsilon_0} \hat{n}$ 9. Capacitors Definition: Two conductors carrying equal and opposite charges, $+Q$ and $-Q$. Capacitance ($C$): Ratio of charge to the potential difference between conductors. $$ C = \frac{Q}{V} $$ (where $V$ is the potential of the positive conductor relative to the negative one). Units: Farads (F), $1 \text{ F} = 1 \text{ C/V}$. Often expressed in microfarads ($\mu\text{F}$) or picofarads ($\text{pF}$). Energy Stored in a Capacitor: Work required to charge a capacitor. $$ W = \frac{1}{2}CV^2 = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2}QV $$ Examples: Parallel-Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ (A = area, d = separation) Spherical Capacitor: (concentric spheres, radii $a, b$) $C = 4\pi\epsilon_0 \frac{ab}{b-a}$ 10. Poisson's and Laplace's Equations Poisson's Equation: Derived from $\nabla \cdot \vec{E} = \rho/\epsilon_0$ and $\vec{E} = -\nabla V$. $$ \nabla^2 V = -\frac{\rho}{\epsilon_0} $$ Laplace's Equation: Poisson's equation in regions where there is no charge ($\rho = 0$). $$ \nabla^2 V = 0 $$ Laplacian Operator ($\nabla^2$): Rectangular Coordinates: $\nabla^2 V = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2}$ Cylindrical Coordinates: $\nabla^2 V = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial V}{\partial \rho}\right) + \frac{1}{\rho^2} \frac{\partial^2 V}{\partial \phi^2} + \frac{\partial^2 V}{\partial z^2}$ Spherical Coordinates: $\nabla^2 V = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial V}{\partial r}\right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left(\sin\theta \frac{\partial V}{\partial \theta}\right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 V}{\partial \phi^2}$ Uniqueness Theorem: If a solution to Laplace's equation satisfies the boundary conditions, it is the unique solution. 11. Image Problem Method of Images: Technique to solve electrostatic problems involving conductors by replacing the conductors with "image charges." For a point charge $q$ above an infinite grounded conducting plane, the field in the region above the plane is equivalent to the field produced by the original charge $q$ and an image charge $-q$ located symmetrically below the plane. The conducting plane is replaced by the image charge, and the potential on the plane ($V=0$) is satisfied. Total charge induced on the surface of the conductor is $-q$. 12. Dielectric Polarization Dielectrics: Insulating materials that become polarized in an electric field. Polarization ($\vec{P}$): Dipole moment per unit volume. $$ \vec{P} = \chi_e \epsilon_0 \vec{E} $$ where $\chi_e$ is the electric susceptibility. Bound Charges: Polarization in a dielectric leads to bound volume charge density ($\rho_b$) and bound surface charge density ($\sigma_b$). $$ \rho_b = -\nabla \cdot \vec{P} $$ $$ \sigma_b = \vec{P} \cdot \hat{n} $$ Electric Displacement Field ($\vec{D}$): $$ \vec{D} = \epsilon_0 \vec{E} + \vec{P} $$ In linear dielectrics: $\vec{D} = \epsilon_0 \vec{E} + \chi_e \epsilon_0 \vec{E} = \epsilon_0 (1 + \chi_e) \vec{E} = \epsilon \vec{E}$ Permittivity ($\epsilon$): $\epsilon = \epsilon_0 (1 + \chi_e)$. Relative Permittivity (Dielectric Constant, $\epsilon_r$): $\epsilon_r = 1 + \chi_e = \epsilon/\epsilon_0$. Gauss's Law in Dielectrics: $$ \oint \vec{D} \cdot d\vec{s} = Q_{f,enc} $$ where $Q_{f,enc}$ is the free charge enclosed. 13. Boundary Conditions for $\vec{D}$ and $\vec{P}$ Normal Component of $\vec{D}$: Discontinuous across a free surface charge $\sigma_f$. $$ D_{\perp, \text{above}} - D_{\perp, \text{below}} = \sigma_f $$ If no free surface charge, $D_\perp$ is continuous. Tangential Component of $\vec{E}$: Still continuous. $$ E_{\parallel, \text{above}} = E_{\parallel, \text{below}} $$ This implies: $\frac{D_{\parallel, \text{above}}}{\epsilon_{\text{above}}} = \frac{D_{\parallel, \text{below}}}{\epsilon_{\text{below}}}$ 14. Energy in Dielectric Systems The energy stored in an electrostatic system with linear dielectrics: $$ W = \frac{1}{2} \int (\vec{D} \cdot \vec{E}) d\tau $$ (integrated over all space) For a capacitor filled with a linear dielectric, its capacitance increases by a factor of $\epsilon_r$: $C = \epsilon_r C_{vac}$.