Electrostatic Potential & Capacitance Full Notes.
Chapter 2 exhaustive notes, theory derivations, equipotential geometries, dielectric shifts, and board numericals by Jatin Sandilya.
build your potential! ⚡
Physics Meets Daily Life.
Hover or tap to flip for technological uses and profound life parallels.
Defibrillators & Energy Release
Batteries deliver energy steadily and slowly. A capacitor accumulates charge over seconds and discharges thousands of volts in a millisecond to shock a stopped heart back to rhythm.
Preparation is like charging a capacitor. The hours of unseen, disciplined study quietly build internal potential. When exam day arrives, that stored energy is released effortlessly.
Touchscreens & Equipotential Planes
Projected capacitive touchscreens maintain an equipotential electrostatic grid. Touching the glass alters the field gradient and pinpoints coordinate touches.
When your mind is balanced and free of internal conflict, navigating daily challenges requires minimum emotional friction. Move without wasting mental energy on doubt.
Dielectric Slabs & Capacity Boost
Air breaks down under high electric fields ($3 \times 10^6\text{ V/m}$). Inserting an insulating dielectric slab dampens the internal field, allowing the capacitor to hold $K$-times more charge safely.
Life will test your breakdown threshold with pressure and stress. Developing emotional resilience acts like a dielectric insulator—it dampens external negativity and expands your capacity to handle challenges.
Lightning Protection & Aircraft Cages
Airplanes are struck by lightning regularly during storms, yet passengers are untouched because charge stays on the outer metallic skin while the interior field is strictly zero ($E = 0$).
External storms of distractions, criticism, and chaos will strike daily. Maintain a solid mental shield around your goals so external turbulence never penetrates your core focus.
📌 Core NCERT Definitions
1 Mark EssentialsA force field where total work done in transporting a charge between two points is completely independent of the path taken.
The electrostatic force experienced per unit infinitesimal positive test charge placed at that spatial location.
External mechanical work required to bring a unit positive test charge from infinity up to a designated point without acceleration.
Energy transferred to or from a system via vector force acting across displacement ($W = \vec{F} \cdot \vec{d}$).
Energy stored within a collective system of charges governed by their relative spatial geometric configuration.
An infinitesimal charge magnitude chosen explicitly so it does not distort the source electric field being mapped.
The localized scalar potential generated at distance $r$ from an isolated point charge ($V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$).
Pair of equal and opposite charges ($\pm q$) separated by $2a$. Vector $\vec{p} = q(2\vec{a})$ points conventionally from $-q$ to $+q$.
Scalar superposition property: total potential equals direct algebraic sum of potentials produced by all individual charges.
💡 High-Yield Conceptual Pearls
Board HighlightsWork done moving a charge in an electrostatic field is independent of path, proving the conservative nature of electrostatic forces.
By standard universal reference convention, both electrostatic potential and potential energy are set to zero at infinity ($\infty$).
Point charge potential varies inversely with distance: $V \propto 1/r$, while electric field varies as $E \propto 1/r^2$.
Electrostatic potential along the entire perpendicular bisector (equatorial line) of a dipole is always exactly zero ($V = 0$), though electric field $\vec{E} \neq 0$.
In static equilibrium, electric potential inside any hollow or solid conductor is completely uniform and equal to its surface potential.
Potential is positive for positive source charges ($+q$) and negative for negative charges ($-q$). It is strictly a scalar quantity.
📐 Essential Formula Reference Bank
Formula SummaryWork required to move charge $q$ against field $\vec{E}$:
Scalar potential at distance $r$ from charge $Q$:
Axial ($V = \pm\frac{kp}{r^2}$) | Equatorial ($V = 0$):
Superposition sum across $n$ discrete charges:
📊 Comparison: Point Charge vs Dipole Potential
Symmetry Matrix| Characteristic | Isolated Point Charge | Electric Dipole ($\vec{p}$) |
|---|---|---|
| Distance Scaling | Inversely proportional to distance: $V \propto \frac{1}{r}$ | Falls off faster: $V \propto \frac{1}{r^2}$ |
| Angular Dependence | Spherically symmetric (independent of angle $\theta$) | Depends explicitly on polar angle ($\propto \cos\theta$) |
| Equatorial Potential | Non-zero at all finite points | Strictly zero at all equatorial bisector points ($V=0$) |
| Spatial Symmetry | Spherical symmetry | Axial / Cylindrical symmetry |
⚔️ Contrast: Electric Field vs Electric Potential
2–3 Mark Questions| Parameter | Electric Field ($\vec{E}$) | Electrostatic Potential ($V$) |
|---|---|---|
| Physical Nature | Vector quantity (magnitude + spatial direction) | Scalar quantity (magnitude + algebraic sign) |
| Physical Role | Electrostatic force experienced per unit test charge | External work required per unit test charge from $\infty$ |
| SI Unit | $\text{N/C}$ or $\text{V/m}$ | $\text{Volt (V)}$ or $\text{J/C}$ |
| Gradient Formula | $\vec{E} = -\frac{dV}{dr}\hat{r}$ (negative spatial gradient) | $V = -\int_\infty^r \vec{E} \cdot d\vec{r}$ |
| Field Direction | Points toward decreasing potential | Has no spatial direction vector attribute |
⚡ Derivation: Electric Potential Due to a Point Charge
Core ProofFig 2.2: Coordinate layout for deriving electrostatic potential at point $P$ from charge $+q$.
📈 Graphical Decay: Potential ($V$) vs Field ($E$)
Asymptotic CurvesFig 2.3: Comparison showing steeper inverse-square drop ($E \propto 1/r^2$) vs smoother decay ($V \propto 1/r$).
🌐 Equipotential Geometries & Energy Assemblies
Visual TopologyEquipotential Contours (Point Charge)
Concentric spherical surfaces perpendicular to radial field lines ($V_1 > V_2 > V_3$).
Three-Charge Energy Assembly
Total Stored Energy: $U = \frac{1}{4\pi\varepsilon_0}\left[\frac{q_1q_2}{r_{12}} + \frac{q_2q_3}{r_{23}} + \frac{q_3q_1}{r_{31}}\right]$.
⚡ Mathematical Framework & Quantitative Relations
Analytical OverviewSpatial gradient relationships linking fields and scalar potentials, multi-charge interaction energies within external fields, and dipole rotational dynamics.
Spatial Field-Potential Link
The magnitude of the local electric field intensity equals the maximum spatial rate of potential decrease.
Charge Energy in External Potential
Electrostatic potential energy acquired by positioning a charge within an existing spatial potential distribution.
Dipole Alignment Energy
Potential energy stored in a dipole oriented at angle $\theta$ relative to a uniform electric field.
Rotational Work on Dipole
Net external mechanical work required to pivot an electric dipole between two angular orientations.
Potential Energy of Two Charges in External Field
Includes external field interactions of each individual charge along with their mutual Coulombic interaction energy:
Spatial position vectors $\vec{r}_1$, $\vec{r}_2$, and mutual separation $r_{12}$.
🗺️ Spatial Field Mapping & Equipotential Topologies
Geometry SpecsIsolated Point Charge
Concentric spherical shells radiating outward symmetrically from charge center.
Uniform Electric Field
Equidistant parallel planar sheets perpendicular to electric field vectors.
Electric Dipole Pairing
Non-uniform distorted envelopes closing around opposing charges; flat plane at center ($V=0$).
Two Positive Charges (+q, +q)
Repelling surfaces close to charges, merging into a single dumbbell envelope further away.
Infinite Line Charge
Coaxial cylindrical surfaces concentric with the charged wire axis.
Proof: Electric Field as Negative Potential Gradient ($E = -dV/dr$)
Core DerivationDisplacement $d\vec{r}$ connecting adjacent equipotential levels within field $\vec{E}$.
1. External work done to move unit test charge $q_0$ from surface $A$ to $B$ against electric field $\vec{E}$:
2. By potential difference definition: $\frac{dW}{q_0} = V_B - V_A = V - (V + dV) = -dV$.
3. Equating both expressions:
Physical Takeaway: The negative sign dictates that electric field lines point in the direction of steepest potential decline.
Potential Energy Assembly: Two Point Charges ($U = \frac{kq_1q_2}{r_{12}}$)
Core Derivation⚡ Mathematical Framework & Quantitative Relations
Analytical OverviewSpatial gradient relationships linking fields and potentials, multi-charge interaction energies within external fields, and dipole rotational dynamics.
Spatial Field-Potential Link
The magnitude of local electric field intensity equals the maximum spatial rate of potential decrease.
Charge Energy in External Potential
Electrostatic potential energy acquired by positioning a charge within an existing spatial potential distribution.
Dipole Alignment Energy
Potential energy stored in a dipole oriented at angle $\theta$ relative to a uniform electric field.
Rotational Work on Dipole
Net external mechanical work required to pivot an electric dipole between two angular orientations.
Potential Energy of Two Charges in External Field
Includes external field interactions of each charge along with their mutual Coulombic interaction energy:
Spatial position vectors $\vec{r}_1$, $\vec{r}_2$, and mutual separation $r_{12}$.
🗺️ Spatial Field Mapping & Equipotential Topologies
Geometry SpecsIsolated Point Charge
Concentric spherical shells radiating outward symmetrically from charge center.
Uniform Electric Field
Equidistant parallel planar sheets perpendicular to electric field vectors.
Electric Dipole Pairing
Distorted envelopes closing around opposing charges; flat plane at center ($V=0$).
Two Positive Charges (+q, +q)
Repelling surfaces close to charges, merging into a dumbbell envelope further away.
Infinite Line Charge
Coaxial cylindrical surfaces concentric with the charged wire axis.
Analytical Coupling: Field as Potential Gradient ($E = -dV/dr$)
Core DerivationDisplacement $d\vec{r}$ between adjacent equipotential levels within field $\vec{E}$.
1. External work done to move unit test charge $q_0$ from surface $A$ to $B$ against field $\vec{E}$:
2. By potential difference definition: $\frac{dW}{q_0} = V_B - V_A = V - (V + dV) = -dV$.
3. Equating both expressions:
Physical Takeaway: Electric field vectors point in the direction of steepest potential decline.
4. Potential from known field via integration ($\vec{r}_1 \to \infty, V_1 = 0$):
Potential Energy Assembly: Two Point Charges ($U = \frac{kq_1q_2}{r_{12}}$)
Core DerivationPositioning charge $q_1$ followed by the arrival of charge $q_2$ from infinity.
Electrostatic Potential of a Charged Spherical Shell (Radius $R$, Charge $Q$)
Radial ZonesSpherical shell of radius $R$ and observation point $P$.
Radial potential decay: constant inside, $1/r$ outside.
Acts as if entire charge $Q$ is concentrated at center:
Potential at the outer conductive boundary:
Because internal field $E = 0$, potential remains constant:
Conductors in Equilibrium & Electrostatic Shielding
Physical LawsFree electrons redistribute instantly until induced internal fields cancel external fields completely ($\vec{E} = 0$).
Electric field lines are strictly perpendicular to the outer surface. Any tangential component would trigger continuous current.
Applying Gauss's Law inside the metal reveals zero net volume charge; all excess charge resides strictly on the outer boundary.
Because $\vec{E} = -\frac{dV}{dr} = 0$, the potential gradient vanishes, maintaining identical potential everywhere inside and on the surface.
Surface Field Derivation: Gaussian Pillbox ($E = \sigma / \varepsilon_0$)
Constructing a tiny cylindrical pillbox across the surface of a charged conductor with cross-sectional area $\Delta S$:
Faraday Cage Principle: Field inside a cavity within a conductor is zero regardless of external fields. Protects passengers in cars/airplanes during lightning strikes.
Electric Dipole: Torque, Potential Energy & Equilibrium States
Torque & EnergyRotational couple $\vec{\tau} = \vec{p} \times \vec{E}$ acting on dipole in uniform field.
1. Restoring torque magnitude: $\tau = qE(2a\sin\theta) = pE\sin\theta \implies \vec{\tau} = \vec{p} \times \vec{E}$.
2. Integrating work done during rotation from $\theta_1$ to $\theta_2$:
3. Potential energy definition ($U = -\vec{p}\cdot\vec{E} = -pE\cos\theta$):
Capacitance Standards, Metric Units & Isolated Spherical Conductor
The Farad UnitFundamental Definition: $Q \propto V \implies Q = CV \implies C = \frac{Q}{V}$. The SI unit is the Farad ($\text{F}$), where $1\text{ F} = 1\text{ C/V}$. Dimensional formula: $[M^{-1}L^{-2}T^4A^2]$.
Surface potential due to uniform charge $Q$: $V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{R}$.
Planetary Context: For an isolated sphere to have $C = 1\text{ F}$, radius $R = 9 \times 10^9\text{ m} = 9 \times 10^6\text{ km}$, roughly 1,500 times larger than Earth's radius ($R_E \approx 6400\text{ km}$). This proves a single isolated farad is impossibly huge; practical units use $\mu\text{F}, \text{nF}, \text{pF}$.
Solved Exemplar: Coalescence of 8 Identical Charged Droplets
Numerical AnalysisProblem: Eight identical spherical droplets, each containing charge $q = 1\text{ nC}$ and at potential $V = 900\text{ V}$, merge without loss to form a single large drop. Compute the resultant potential $V'$.
1. Volume Conservation: $\frac{4}{3}\pi R^3 = 8\left(\frac{4}{3}\pi r^3\right) \implies R = 2r$.
2. Charge Conservation: Total charge $Q' = 8q$.
3. Capacitance Scaling: $C' = 4\pi\varepsilon_0 R = 4\pi\varepsilon_0(2r) = 2C$.
4. Resultant Potential:
Redistribution of Charges & Proof of Energy Loss ($\Delta U$)
Sharing TheoremConductors linked by a wire equalize at common potential $V$.
1. Common Potential Formula:
2. Charge Sharing Ratio: After sharing, $Q_1' = C_1 V$ and $Q_2' = C_2 V$:
3. Mathematical Proof of Energy Dissipation ($\Delta U = U_i - U_f$):
Because $(V_1 - V_2)^2 > 0$, energy loss is strictly positive ($\Delta U > 0$). This energy dissipates as heat in the connecting wire and spark radiation.
Multi-Dielectric Composite Slabs: Series vs Parallel Formulations
Composite GeometrySlabs stacked sequentially across thickness $d$. System behaves as capacitors in series:
Slabs placed side-by-side spanning separation $d$. System behaves as capacitors in parallel:
Concentric Spheres: Potential Difference & Unidirectional Charge Transfer
Core PrincipleConcentric sphere ($r, q$) inside hollow shell ($R, Q$).
1. Potential of inner sphere: $V_r = \frac{1}{4\pi\varepsilon_0}\left[\frac{q}{r} + \frac{Q}{R}\right]$.
2. Potential of outer shell: $V_R = \frac{1}{4\pi\varepsilon_0}\left[\frac{Q + q}{R}\right]$.
3. Potential Difference:
Because $r < R$, the term $\left(\frac{1}{r} - \frac{1}{R}\right) > 0$. Hence, if $q > 0$, $V_r$ is always strictly greater than $V_R$. When connected by a wire, charge $q$ flows entirely to the outer shell regardless of how large charge $Q$ is.
🛡️️ Conductors in Electrostatic Equilibrium & Shielding
Fundamental LawsFig 2.6: Free electrons migrate until induced internal field completely cancels applied external field ($\vec{E} = 0$).
Induced boundary charges create counter-fields neutralizing any applied electric field inside the metallic bulk.
Field lines meet conductor surfaces strictly at $90^\circ$. Any tangential component would induce continuous surface current.
Gauss's law on any closed loop inside where $\vec{E}=0$ yields $q_{\text{enclosed}} = 0$. All excess charge sits on the outer boundary.
Because $E = -dV/dr = 0$, potential gradient vanishes. Potential throughout the volume and surface is completely uniform.
Surface Electric Field Proof: Gaussian Pillbox ($E = \sigma / \varepsilon_0$)
Gauss ApplicationFig 2.7: Cylindrical Gaussian pillbox spanning the conductor boundary.
1. Inside the conductor, field is zero; bottom cap contributes zero flux.
2. Field is normal to the surface; curved cylindrical walls are parallel to $\vec{E}$, contributing zero flux.
3. Flux emerges solely through the upper cap of area $\Delta S$:
4. Total charge enclosed with surface density $\sigma$ is $q = \sigma \Delta S$. Applying Gauss's law:
Rotational Dynamics & Potential Energy of an Electric Dipole
Torque & WorkFig 2.8: Equal and opposite electrostatic forces creating couple torque $\vec{\tau} = \vec{p} \times \vec{E}$.
1. Net Translatory Force: $\vec{F}_{\text{net}} = (+q\vec{E}) + (-q\vec{E}) = 0$. Dipole experiences no net linear acceleration.
2. Torque Magnitude: Force $\times$ perpendicular arm: $\tau = qE(2a\sin\theta) = pE\sin\theta \implies \vec{\tau} = \vec{p} \times \vec{E}$.
3. Angular Work: Work in rotating dipole from $\theta_1$ to $\theta_2$ against restoring torque:
4. Potential Energy: $U(\theta) = -\vec{p}\cdot\vec{E} = -pE\cos\theta$.
Working Principle of a Capacitor: Electrostatic Induction & Grounding
Capacity BoostFig 2.9: Grounding outer positive charge leaves bound negative charge, dramatically lowering potential and amplifying capacitance ($C = Q/V$).
1. When uncharged plate $B$ approaches charged plate $A$, induced negative charge on $B$'s inner face lowers $A$'s potential, while induced positive charge on $B$'s outer face raises it.
2. Because negative charges are nearer to $A$, the net potential of $A$ decreases slightly.
3. Earthing Effect: When plate $B$ is grounded, free positive charges are neutralized by electrons from earth. Bound negative charges remain, driving potential $V_A$ down substantially.
4. Since $C = Q / V$, lowering potential for the same charge expands capacitance exponentially.
Parallel-Plate Capacitor: Field Superposition & Capacitance Derivation
Core DerivationFig 2.10: Complete regional field distribution showing outer cancellation ($E=0$) and inner reinforcement ($E = \sigma/\varepsilon_0$).
1. Outer Regions: Opposing fields from plates cancel: $E_{\text{outer}} = \frac{\sigma}{2\varepsilon_0} - \frac{\sigma}{2\varepsilon_0} = 0$.
2. Inner Region: Fields add up constructively:
3. Potential Difference: Uniform field across distance $d$ gives $V = E \cdot d = \frac{Q d}{\varepsilon_0 A}$.
4. Capacitance Expression: Substituting into $C = Q / V$:
• Separation ($C \propto 1/d$): Bringing plates closer boosts attraction.
• Medium ($C \propto K$): Dielectrics amplify capacity by factor $K$.
📐 Parallel-Plate Capacitor: Field Superposition & Capacitance Derivation
Core DerivationTwo expansive planar conducting armatures of active area $A$, separated across distance $d$, carrying uniform surface charge densities $\pm\sigma = \pm Q/A$.
Fig 2.10: Regional field distribution showing cancellation in outer regions ($E=0$) and inner constructive reinforcement ($E = \sigma/\varepsilon_0$).
1. Outer Cancellation: Opposing fields from plates cancel: $E_{\text{outer}} = \frac{\sigma}{2\varepsilon_0} - \frac{\sigma}{2\varepsilon_0} = 0$.
2. Inner Reinforcement: Internal fields add constructively: $E = \frac{\sigma}{2\varepsilon_0} + \frac{\sigma}{2\varepsilon_0} = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}$.
3. Terminal Potential Drop: Uniform field across distance $d$ yields: $V = E \cdot d = \frac{\sigma d}{\varepsilon_0} = \frac{Q d}{\varepsilon_0 A}$.
4. Capacitance Expression ($C = Q / V$):
🌐 Concentric Spherical Storage Architectures & Capacitance Derivation
Spherical ShellsDual concentric spherical armatures: inner shell of radius $a$ carrying charge $-Q$, and outer shell of radius $b$ carrying charge $+Q$.
Fig 2.11: Concentric spherical configuration with intermediate Gaussian surface $r$ ($a < r < b$).
1. Field in the Gap: Applying Gauss's law for $a < r < b$: $E \cdot 4\pi r^2 = \frac{Q}{\varepsilon_0} \implies E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}$.
2. Integrating Potential Difference ($V$):
3. Capacitance Formula ($C = Q / V$):
🔢 Solved Exemplars: Voltage-Charge Variations
Board Problem SetsA capacitor holds $360\,\mu\text{C}$ at potential $V$. Dropping potential by $120\text{ V}$ reduces charge to $120\,\mu\text{C}$. Find $V$, $C$, and charge if voltage rises by $120\text{ V}$.
• $\frac{V}{V - 120} = \frac{360}{120} = 3 \implies V = 180\text{ V}$
• $C = \frac{Q}{V} = \frac{360\,\mu\text{C}}{180\text{ V}} = \mathbf{2\,\mu\text{F}}$
• If $V$ rises by $120\text{ V}$: $Q'' = C(V + 120) = 2\,\mu\text{F}(300\text{ V}) = \mathbf{600\,\mu\text{C}}$
A capacitor holds $120\,\mu\text{C}$ at potential $V$. Lowering potential by $40\text{ V}$ drops charge to $40\,\mu\text{C}$. Find base potential $V$, capacity $C$, and charge if voltage increases by $40\text{ V}$.
• $\frac{120}{V} = \frac{40}{V - 40} \implies 3(V - 40) = V \implies V = 60\text{ V}$
• $C = \frac{120\,\mu\text{C}}{60\text{ V}} = \mathbf{2\,\mu\text{F}}$
• If $V$ rises by $40\text{ V}$: $Q_{\text{new}} = C(V + 40) = 2\,\mu\text{F}(100\text{ V}) = \mathbf{200\,\mu\text{C}}$
🔌 Network Architectures: Series vs. Parallel Circuits
Equivalent CapacitanceCharge Constant ($Q$), Voltages Divide
• Total voltage: $V = V_1 + V_2 + V_3$.
• Equivalent capacity is smaller than the smallest component in the chain.
Voltage Constant ($V$), Charges Add
• Total charge: $Q = Q_1 + Q_2 + Q_3$.
• Equivalent capacity is larger than the largest component in the network.
⚡ Stored Electrostatic Energy & Field Energy Density ($u$)
Volumetric DensityFig 2.12: Incremental charge transfer $dQ'$ establishing internal field energy density.
1. Integration of Work Done: $W = \int_{0}^{Q} \frac{Q'}{C} dQ' = \frac{1}{C}\left[\frac{Q'^2}{2}\right]_{0}^{Q} = \frac{1}{2}\frac{Q^2}{C}$.
2. Volumetric Energy Density Derivation ($u$):
Universal Principle: This energy density formula holds true for electric fields of any geometry or origin (measured in $\text{J/m}^3$).
⚡ Mathematical Framework & Quantitative Relations
Analytical OverviewSpatial gradient relationships linking fields and potentials, multi-charge interaction energies within external fields, and dipole rotational dynamics.
Spatial Field-Potential Link
The magnitude of local electric field intensity equals the maximum spatial rate of potential decrease.
Charge Energy in External Potential
Electrostatic potential energy acquired by positioning a charge within an existing spatial potential distribution.
Dipole Alignment Energy
Potential energy stored in a dipole oriented at angle $\theta$ relative to a uniform electric field.
Rotational Work on Dipole
Net external mechanical work required to pivot an electric dipole between two angular orientations.
Potential Energy of Two Charges in External Field
Includes external field interactions of each charge along with their mutual Coulombic interaction energy:
Spatial position vectors $\vec{r}_1$, $\vec{r}_2$, and mutual separation $r_{12}$.
🗺️️ Spatial Field Mapping & Equipotential Topologies
Geometry SpecsIsolated Point Charge
Concentric spherical shells radiating outward symmetrically from charge center.
Uniform Electric Field
Equidistant parallel planar sheets perpendicular to electric field vectors.
Electric Dipole Pairing
Distorted envelopes closing around opposing charges; flat plane at center ($V=0$).
Parallel-Plate Capacitor: Field Superposition & Capacitance Derivation
Core DerivationTwo expansive planar conducting armatures of active area $A$, separated across distance $d$, carrying uniform surface charge densities $\pm\sigma = \pm Q/A$.
Regional field distribution showing cancellation in outer regions ($E=0$) and inner reinforcement ($E = \sigma/\varepsilon_0$).
1. Outer Cancellation: Opposing fields cancel: $E_{\text{outer}} = \frac{\sigma}{2\varepsilon_0} - \frac{\sigma}{2\varepsilon_0} = 0$.
2. Inner Reinforcement: $E = \frac{\sigma}{2\varepsilon_0} + \frac{\sigma}{2\varepsilon_0} = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}$.
3. Potential Difference: $V = E \cdot d = \frac{Q d}{\varepsilon_0 A}$.
⚡ Stored Electrostatic Energy & Field Energy Density ($u$)
Volumetric Density1. Work Done during Charging: $W = \int_{0}^{Q} \frac{Q'}{C} dQ' = \frac{1}{2}\frac{Q^2}{C}$.
2. Volumetric Energy Density ($u$):
1. Parallel-Plate Capacitor: Internal Field & Geometric Capacitance
Fig 2.13: Uniform field formation between plates and cancellation in external regions.
1. Field Formulation: Electric field intensity between plates is given by $E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}$.
2. Terminal Potential Difference: $V = E \cdot d = \frac{Q d}{\varepsilon_0 A}$.
3. Capacitance Equation:
2. Concentric Spherical Capacitor Derivation
Fig 2.14: Concentric spherical configuration with intermediate Gaussian surface $r$.
1. Field in the Gap ($a < r < b$): By Gauss's Law: $E \cdot 4\pi r^2 = \frac{Q}{\varepsilon_0} \implies E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}$.
2. Potential Difference Line Integral:
3. Resultant Capacitance ($C = Q / V$):
3. Solved Board Numericals: Voltage & Charge Variations
A capacitor holds $360\,\mu\text{C}$ at potential $V$. Lowering potential by $120\text{ V}$ reduces charge to $120\,\mu\text{C}$. Calculate $V$, $C$, and charge if voltage rises by $120\text{ V}$.
• Ratio: $\frac{V}{V - 120} = \frac{360}{120} = 3 \implies V = 180\text{ V}$
• Capacity: $C = \frac{360\,\mu\text{C}}{180\text{ V}} = \mathbf{2\,\mu\text{F}}$
• Charge at $(V + 120\text{ V})$: $Q = 2\,\mu\text{F} \times (180 + 120) = \mathbf{600\,\mu\text{C}}$
A capacitor holds $120\,\mu\text{C}$ at potential $V$. Lowering potential by $40\text{ V}$ drops charge to $40\,\mu\text{C}$. Find $V$, $C$, and charge if voltage rises by $40\text{ V}$.
• Ratio: $\frac{120}{V} = \frac{40}{V - 40} \implies 3(V - 40) = V \implies V = 60\text{ V}$
• Capacity: $C = \frac{120\,\mu\text{C}}{60\text{ V}} = \mathbf{2\,\mu\text{F}}$
• Charge at $(V + 40\text{ V})$: $Q = 2\,\mu\text{F} \times (60 + 40) = \mathbf{200\,\mu\text{C}}$
4. Capacitor Combinations: Series & Parallel Topologies
Charge is uniform across each plate; terminal voltages divide: $V = V_1 + V_2 + V_3$.
Voltage is identical across all branches; charges accumulate: $Q = Q_1 + Q_2 + Q_3$.
5. Stored Electrostatic Energy & Field Energy Density ($u$)
Fig 2.15: Incremental charge transfer $dQ'$ against potential $V'$, establishing stored potential energy.
1. Energy Integration: Total work done during charging: $W = \int_{0}^{Q} \frac{Q'}{C} dQ' = \frac{1}{2}\frac{Q^2}{C}$.
2. Volumetric Energy Density ($u$): Dividing total energy by enclosed plate volume ($A \cdot d$):
6. Charge Redistribution & Energy Dissipation ($\Delta U$)
Fig 2.16: Charges flow until potential reaches common value $V$.
1. Common Potential: $V = \frac{Q_1 + Q_2}{C_1 + C_2} = \frac{C_1 V_1 + C_2 V_2}{C_1 + C_2}$.
2. Charge Sharing Ratio: $Q_1' = C_1 V$ and $Q_2' = C_2 V \implies \mathbf{\frac{Q_1'}{Q_2'} = \frac{C_1}{C_2}}$ (Charges divide in direct ratio of capacities).
3. Energy Loss Equation ($\Delta U = U_i - U_f$):
Because $(V_1 - V_2)^2 > 0$, energy loss is strictly positive ($\Delta U > 0$). This energy dissipates as heat in the wire and electromagnetic radiation.
7. Partial Dielectric & Conducting Slab Insertion
Fig 2.17: Field reduction across thickness $t$ inside a dielectric slab.
1. Reduced field inside dielectric slab: $E = \frac{E_0}{K}$.
2. Potential drop summation: $V = E_0(d - t) + \left(\frac{E_0}{K}\right)t = E_0\left(d - t + \frac{t}{K}\right)$.
3. Modified Capacitance Expression:
8. Multi-Dielectric Slabs: Series vs Parallel Configurations
Series layers: division across thickness.
Parallel sections: division across area.
9. Suspended Concentric Spherical Shells: Unidirectional Charge Transfer
Fig 2.18: Inner sphere $r$ connected to outer shell $R$.
1. Inner sphere potential: $V_r = \frac{1}{4\pi\varepsilon_0}\left[\frac{q}{r} + \frac{Q}{R}\right]$.
2. Outer shell potential: $V_R = \frac{1}{4\pi\varepsilon_0}\left[\frac{q + Q}{R}\right]$.
3. Potential difference:
Because $r < R$, term $\left(\frac{1}{r} - \frac{1}{R}\right) > 0$. If $q > 0$, then $V_r > V_R$ always. When joined by a wire, charge $q$ flows entirely to the outer shell, regardless of the magnitude of $Q$.
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