Chapter 12: Electrostatics
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Introduction to Electrostatics
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- Electrostatics: Deals with charges at rest.
- Electrodynamics: Deals with moving charges.
- Electric force: Holds positive and negative charges in atoms, molecules, bodies.
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Charge & Its Relationship with Friction
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- Bodies get charged by friction (rubbing).
- Rubbing: electrons gain energy, detach, causing positive charge.
- Similar charges repel; opposite charges attract.
- Excess electrons: negative charge. Deficiency of electrons: positive charge.
- SI unit of charge: Coulomb (C).
- $1~C$: force between two equal and opposite charges 1m apart is $9 \times 10^9~N$.
- $1~C$: charge when 1A current flows for 1s.
- $1~C = 6.25 \times 10^{18}$ electrons.
- Charge is conserved and quantized.
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Coulomb's Law
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- Force between two point charges.
- Directly proportional to product of magnitudes of charges.
- Inversely proportional to square of distance between them.
- Acts along the line joining the charges.
- $F = k \frac{q_1 q_2}{r^2}$.
- $k = \frac{1}{4\pi\epsilon_o}$ (for vacuum/air).
- $k \approx 9 \times 10^9~N m^2 C^{-2}$.
- $\epsilon_o$: permittivity of free space ($8.85 \times 10^{-12}~C^2 N^{-1} m^{-2}$).
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Coulomb's Law in Vector Form
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- $\vec{F}_{12} = k \frac{q_1 q_2}{r^2} \hat{r}_{12}$ (force on $q_1$ due to $q_2$).
- $\hat{r}_{12}$: unit vector from $q_2$ to $q_1$.
- $\vec{F}_{12} = -\vec{F}_{21}$ (Newton's 3rd Law).
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Effect of Medium (Dielectric)
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- Force in medium: $F_m = \frac{1}{4\pi\epsilon} \frac{q_1 q_2}{r^2}$.
- $\epsilon = \epsilon_r \epsilon_o = K \epsilon_o$.
- $\epsilon_r$ or $K$: relative permittivity or dielectric constant.
- $F_m = \frac{F_{vacuum}}{K}$.
- $K$ for vacuum/air = 1. For water $\approx 80$. For conductors, $K=\infty$.
- Force decreases in a dielectric medium.
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Electric Field (E)
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- Region around a charge where it exerts force on other charges.
- Electric field intensity: Force per unit positive test charge.
- $\vec{E} = \frac{\vec{F}}{q_o}$.
- Unit: N/C or V/m.
- Dimensions: $[MLT^{-3}A^{-1}]$.
- Vector quantity, direction same as force on positive test charge.
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Electric Field due to a Point Charge
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- $E = k \frac{q}{r^2}$.
- $\vec{E} = k \frac{q}{r^2} \hat{r}$.
- For positive charge, $\vec{E}$ is radially outward.
- For negative charge, $\vec{E}$ is radially inward.
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Electric Field Lines (Lines of Force)
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- Imaginary lines representing electric field.
- Originate from positive charges, terminate on negative charges.
- Tangent at any point gives direction of $\vec{E}$.
- Density of lines indicates strength of $\vec{E}$.
- Never cross each other.
- Never form closed loops.
- Perpendicular to conductor surfaces.
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Electric Field of a Parallel Plate Capacitor
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- Uniform field between plates (neglecting edge effects).
- $E = \frac{\sigma}{\epsilon_o}$ (in vacuum).
- $\sigma$: surface charge density ($q/A$).
- $E = \frac{\sigma}{\epsilon_r \epsilon_o} = \frac{\sigma}{K \epsilon_o}$ (with dielectric).
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Electric Flux ($\Phi_E$)
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- Measure of the number of electric field lines passing through a given area.
- $\Phi_E = \vec{E} \cdot \vec{A} = EA \cos\theta$.
- Unit: $N m^2 C^{-1}$ or V m.
- Scalar quantity.
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Gauss's Law
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- Total electric flux through any closed surface (Gaussian surface) is equal to the total charge enclosed divided by the permittivity of free space.
- $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_o}$.
- Used to calculate electric field for symmetric charge distributions.
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Applications of Gauss's Law
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- Electric Field due to an Infinite Sheet of Charge: $E = \frac{\sigma}{2\epsilon_o}$.
- Electric Field due to an Infinite Line of Charge: $E = \frac{\lambda}{2\pi\epsilon_o r}$. ($\lambda$: linear charge density).
- Electric Field due to a Hollow Charged Sphere (outside): $E = k \frac{Q}{r^2}$.
- Electric Field due to a Hollow Charged Sphere (inside): $E = 0$.
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Electric Potential (V)
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- Work done per unit positive test charge to bring it from infinity to a point in the electric field without acceleration.
- $V = \frac{W}{q_o}$.
- Unit: Volt (V) = J/C.
- Dimensions: $[ML^2T^{-3}A^{-1}]$.
- Scalar quantity.
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Electric Potential due to a Point Charge
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- $V = k \frac{q}{r}$.
- For positive charge, V is positive. For negative charge, V is negative.
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Electric Potential Difference ($\Delta V$)
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- Work done per unit positive test charge to move it from one point to another in an electric field.
- $\Delta V = V_B - V_A = \frac{W_{AB}}{q_o}$.
- Work done by electric field: $W_{AB} = -q_o(V_B - V_A) = -q_o\Delta V$.
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Equipotential Surfaces
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- Surfaces where electric potential is constant.
- No work is done moving a charge along an equipotential surface.
- Electric field lines are always perpendicular to equipotential surfaces.
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Relation between Electric Field and Potential
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- $E = -\frac{\Delta V}{\Delta r}$ (for uniform field).
- Electric field points in direction of decreasing potential.
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Capacitance (C)
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- Ability of a conductor to store electric charge.
- $C = \frac{Q}{V}$.
- Unit: Farad (F) = C/V.
- Dimensions: $[M^{-1}L^{-2}T^4A^2]$.
- Scalar quantity.
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Capacitance of a Parallel Plate Capacitor
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- $C = \frac{\epsilon_o A}{d}$ (in vacuum/air).
- $C = \frac{\epsilon A}{d} = \frac{K\epsilon_o A}{d}$ (with dielectric).
- $A$: area of plates.
- $d$: distance between plates.
- Capacitance increases with dielectric.
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Combinations of Capacitors
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Series Combination:
- Charge (Q) is same across each capacitor.
- Total voltage: $V = V_1 + V_2 + ...$.
- Equivalent capacitance: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$.
- For two capacitors: $C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$.
- $C_{eq}$ is always less than the smallest individual capacitance.
Parallel Combination:
- Voltage (V) is same across each capacitor.
- Total charge: $Q = Q_1 + Q_2 + ...$.
- Equivalent capacitance: $C_{eq} = C_1 + C_2 + ...$.
- $C_{eq}$ is always greater than the largest individual capacitance.
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Energy Stored in a Capacitor
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- $U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$.
- Unit: Joule (J).
- Energy density: Energy stored per unit volume.
- $u = \frac{1}{2}\epsilon_o E^2$ (in vacuum/air).
- $u = \frac{1}{2}K\epsilon_o E^2$ (with dielectric).
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Millikan's Oil Drop Experiment
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- Determined the elementary charge (charge of an electron).
- Balanced gravitational force on oil drop with electric force.
- At terminal velocity (no electric field): $mg = 6\pi\eta r v_t$.
- With electric field, balanced: $qE = mg$.
- Charge on oil drop: $q = \frac{mg}{E}$.
- Found charge $q$ is always an integer multiple of elementary charge $e$.
- $e = 1.602 \times 10^{-19}~C$.
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Charging and Discharging of a Capacitor
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- RC circuit: Resistor and capacitor in series.
- Time constant ($\tau$): Time for charge to reach 63% of max during charging, or discharge to 37% during discharging.
- $\tau = RC$.
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Applications of Electrostatics
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- Photocopiers (Xerography).
- Inkjet printers.
- Electrostatic paint spraying.
- Dust precipitators.
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