Electrostatics revision notes

Key Concepts & Definitions

Electrostatics:

The study of forces, fields, and potentials arising from static charges.

Electric Charge:

An intrinsic property of matter causing electric interactions. There are two polarities: positive (e.g., glass rubbed with silk) and negative (e.g., plastic rubbed with fur).

Conductors & Insulators:

Conductors (e.g., metals, human body) readily allow the passage of electricity due to free mobile charge carriers (electrons). Insulators (e.g., glass, plastic, nylon) offer high resistance to charge movement.

Gold-Leaf Electroscope:

A device used to detect charge on a body, consisting of a vertical metal rod housed in a box with two thin gold leaves. The degree of divergence indicates the amount of charge.

Point Charge:

An idealization where the size of charged bodies is extremely small compared to the distance between them, assuming all charge is concentrated at one point.

Electric Field (E\mathbf{E}E):

The electrical environment characterized by the force a unit positive test charge would experience if placed at that point, without disturbing the source charges.

Test Charge (q0q_0q0​):

An infinitesimally small positive charge used to map an electric field without altering the source charge configuration.

Electric Field Lines:

Space curves drawn such that the tangent at any point gives the direction of the net electric field. The relative density of lines indicates field strength.

Electric Flux (Φ\PhiΦ):

A measure of the number of electric field lines crossing a given area element.

Solid Angle (Ω\OmegaΩ):

The 3D analog of a plane angle, defined as ΔΩ=ΔS/r2\Delta \Omega = \Delta S / r^2ΔΩ=ΔS/r2, used to define the spread of electric field lines.

Electric Dipole:

A pair of equal and opposite point charges (qqq and −q-q−q) separated by a small distance (2a2a2a). The total charge of a dipole is zero, but its electric field is non-zero.

Electrostatic Potential (VVV):

The work done by an external force in bringing a unit positive test charge from infinity to a specific point without acceleration.

Equipotential Surface:

A surface with a constant value of electrostatic potential at all points.

Conservative Force:

A force (like electrostatic or gravitational) where the work done in moving a particle between two points is independent of the path taken.

Electron Volt (eV):

The energy gained by an electron accelerated through a potential difference of 1 Volt. 1 eV=1.6×10−19 J1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}1 eV=1.6×10−19 J.

Dielectric:

Non-conducting substances lacking free charge carriers but capable of microscopic polarization in an external field.

Polarisation (P\mathbf{P}P):

The induced dipole moment per unit volume in a dielectric.

Capacitor:

A system of two conductors separated by an insulator, used to store electric charge and electrostatic energy.

Electric Charges & Properties

• Additivity of Charges: Total charge is the algebraic sum of individual charges in a system ($q_{net} = q_1 + q_2 + ... + q_n$). Proper signs must be used.
• Conservation of Charge: The total charge of an isolated system is always conserved. Charges can be transferred, and charge-carrying particles can be created/destroyed in equal and opposite pairs (e.g., neutron turning into a proton and electron), but net charge cannot be created or destroyed.
• Quantisation of Charge: All observable free charges are integral multiples of the basic elementary charge $e$ ($q = ne$, where $n = 0, \pm 1, \pm 2...$).
  • Suggested by Faraday’s laws of electrolysis, experimentally proven by Millikan’s oil drop experiment.
  • Macroscopically, quantization is ignored because the scale of charges is enormous compared to $e$.JEE TIPA body emitting $10^9$ electrons per second takes ~198 years to accumulate 1 C of charge. This highlights the macroscopic continuous nature of large charges.

Coulomb's Law & Electric Field

• Coulomb's Law: The mutual electrostatic force between two stationary point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
  • Originally discovered using a torsion balance.
  • Obeys Newton's Third Law: $\mathbf{F}_{12} = -\mathbf{F}_{21}$.
  • Electrical forces are enormously stronger than gravitational forces (e.g., ratio of $F_e/F_g$ for two protons is $\sim 1.3 \times 10^{36}$).
• Superposition Principle: The force on any charge due to a number of other charges is the vector sum of all the forces on that charge taken one at a time. The individual forces are unaffected by the presence of other charges.
• Electric Field: $\mathbf{E} = \lim_{q \to 0} \frac{\mathbf{F}}{q}$. The source charge remains fixed.
• Continuous Charge Distribution:
  • Linear charge density ($\lambda$): Charge per unit length (C/m).
  • Surface charge density ($\sigma$): Charge per unit area (C/m$^2$).
  • Volume charge density ($\rho$): Charge per unit volume (C/m$^3$).
• Field Lines Properties:
  1. Start at positive charges and end at negative charges (or infinity).
  2. Continuous curves without breaks in a charge-free region.
  3. Never cross each other (implies two directions of $\mathbf{E}$ at one point, which is absurd).
  4. Electrostatic field lines do not form closed loops (due to the conservative nature of the field).

Gauss's Law & Electric Flux

• Electric Flux: $\Delta \Phi = \mathbf{E} \cdot \Delta \mathbf{S} = E \Delta S \cos\theta$. For a closed surface, the outward normal is taken as positive.
• Gauss's Law: The net electric flux through any closed surface is equal to $1/\epsilon_0$ times the total net charge enclosed by the surface ($\oint \mathbf{E} \cdot d\mathbf{S} = q_{enc}/\epsilon_0$).
  • True for any closed surface (Gaussian surface) regardless of shape/size.
  • $\mathbf{E}$ on the LHS is the net field due to ALL charges (inside and outside), but $q_{enc}$ on the RHS is ONLY the charge inside.JEE TIPIf a charge is moved outside the Gaussian surface, the flux becomes zero, but the electric field at points on the surface changes!
• Standard Applications of Gauss's Law:
  • Infinitely long straight charged wire: $\mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 r} \hat{\mathbf{n}}$
  • Infinite uniformly charged plane sheet: $\mathbf{E} = \frac{\sigma}{2\epsilon_0} \hat{\mathbf{n}}$ (Independent of distance $r$).
  • Thin spherical shell (Radius $R$, Charge $Q$):
    • Outside ($r \ge R$): $\mathbf{E} = \frac{Q}{4\pi\epsilon_0 r^2} \hat{\mathbf{r}}$ (Acts as a point charge).
    • Inside ($r < R$): $\mathbf{E} = \mathbf{0}$.
  • Solid Non-Conducting Sphere (JEE Advanced Topic):
    • Outside ($r \ge R$): $\mathbf{E} = \frac{Q}{4\pi\epsilon_0 r^2} \hat{\mathbf{r}}$
    • Inside ($r < R$): $\mathbf{E} = \frac{\rho r}{3\epsilon_0} = \frac{Q r}{4\pi\epsilon_0 R^3} \hat{\mathbf{r}}$ (Field increases linearly).

Electric Dipole

• Dipole Moment ($\mathbf{p}$): Magnitude $p = q \times 2a$, directed from the negative charge to the positive charge.
• Field of a Dipole:
  • On Axis: $\mathbf{E}_{axis} = \frac{2\mathbf{p}}{4\pi\epsilon_0 r^3}$ (for $r \gg a$). Direction is parallel to $\mathbf{p}$.
  • On Equatorial Plane: $\mathbf{E}_{eq} = -\frac{\mathbf{p}}{4\pi\epsilon_0 r^3}$ (for $r \gg a$). Direction is anti-parallel to $\mathbf{p}$.
  • JEE TIPThe field of a short dipole falls off as $1/r^3$, much faster than a point charge ($1/r^2$).
• Potential of a Dipole:
  • General Point: $V = \frac{\mathbf{p} \cdot \mathbf{\hat{r}}}{4\pi\epsilon_0 r^2} = \frac{p \cos\theta}{4\pi\epsilon_0 r^2}$ (for $r \gg a$).
  • $V = 0$ everywhere on the equatorial plane ($\theta = 90^\circ$).
  • JEE TIPDipole potential falls off as $1/r^2$, unlike a point charge which falls off as $1/r$.
• Dipole in Uniform External Field:
  • Net force $\mathbf{F}_{net} = \mathbf{0}$.
  • Torque $\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}$ (Magnitude $\tau = pE \sin\theta$). This torque tends to align the dipole with the field.
  • Potential Energy $U = -\mathbf{p} \cdot \mathbf{E} = -pE \cos\theta$. Reference zero energy is chosen at $\theta = \pi/2$.
• Dipole in Non-Uniform External Field:
  • Net force is NON-ZERO. Moves toward increasing field if $\mathbf{p}$ is parallel to $\mathbf{E}$; moves toward decreasing field if anti-parallel.JEE TIPThis explains why a charged comb attracts uncharged paper (it polarizes the paper, then attracts the induced dipole in its non-uniform field).

Electrostatic Potential & Potential Energy

• Electrostatic Potential Energy ($U$): Work done by an external force (without accelerating) in bringing a charge configuration from infinity to its present state. $W_{ext} = \Delta U$.
• Electrostatic Potential ($V$): Work done per unit test charge. $V_P - V_R = \frac{W_{RP}}{q}$.
• Potential due to Point Charge: $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$ (Taking $V_\infty = 0$).
• Potential Energy of Two Charges: $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$.
• Potential Energy of a System of Charges: Derived using the superposition principle. Assemble charges one by one from infinity. $U = \frac{1}{4\pi\epsilon_0} \sum_{i < j} \frac{q_i q_j}{r_{ij}}$.
• Potential Energy in an External Field:
  • For a single charge: $U = qV(\mathbf{r})$.
  • For two charges: $U = q_1 V(\mathbf{r}_1) + q_2 V(\mathbf{r}_2) + \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$.
• Relation between E and V: $\mathbf{E} = -\nabla V$. Or $|E| = -\frac{dV}{dl}$.
  • JEE TIPElectric field points in the direction of the steepest decrease in potential.

Electrostatics of Conductors & Dielectrics

• Electrostatics of Conductors (6 Key Properties):
  1. $\mathbf{E} = \mathbf{0}$ everywhere inside a conductor.
  2. $\mathbf{E}$ just outside a charged conductor is purely normal to the surface: $\mathbf{E} = \frac{\sigma}{\epsilon_0} \hat{\mathbf{n}}$.
  3. No excess charge can reside in the interior.
  4. The entire volume and surface of a conductor are an equipotential region.
  5. Electrostatic Shielding: The field inside a cavity within a conductor is strictly zero, shielding it from all external electrical influences.
  6. Charge Sharing (JEE Advanced Topic): When two conducting spheres are joined, charge flows until potentials equalize. Final surface charge density relates to radius as $\sigma \propto 1/R$ (sharp points have higher charge densities, leading to corona discharge).
• Dielectrics:
  • Non-polar molecules: Centers of + and - charges coincide (e.g., O$_2$, H$_2$). Induce dipoles under external $\mathbf{E}$.
  • Polar molecules: Permanent dipoles exist (e.g., HCl, H$_2$O) but align randomly due to thermal agitation. Align under external $\mathbf{E}$.
• Polarisation ($\mathbf{P}$): $\mathbf{P} = \chi_e \epsilon_0 \mathbf{E}$, where $\chi_e$ is electric susceptibility. It leads to induced surface charge densities ($\sigma_p$ and $-\sigma_p$).
• Dielectric Constant ($K$ or $\epsilon_r$): Reduces net internal field: $\mathbf{E}_{in} = \mathbf{E}_0 / K$. It implies bounded induced surface charges $\sigma_p = \sigma(1 - 1/K)$. $K = 1 + \chi_e$.

Capacitors & Capacitance

• Capacitance ($C$): $C = Q/V$. Unit is Farad (F). Depends purely on the geometry and dielectric medium, not on $Q$ or $V$.
• Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$.
• Dielectric Insertion: If a dielectric slab of constant $K$ completely fills the space, $C = K C_0 = \frac{K \epsilon_0 A}{d}$.
• Combination of Capacitors:
  • Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ (Charge $Q$ is constant, Potential $V$ divides).
  • Parallel: $C_{eq} = C_1 + C_2 + ...$ (Potential $V$ is constant, Charge $Q$ divides).
• Energy Stored: $U = \frac{1}{2} C V^2 = \frac{Q^2}{2C} = \frac{1}{2} Q V$.
• Energy Density: $u = \frac{1}{2} \epsilon_0 E^2$. This holds true for ANY electric field, not just inside a capacitor.
• Dielectric Breakdown: The maximum field a dielectric can withstand without ionizing (for air, $\sim 3 \times 10^6$ V/m).
• Transient Connection (Energy Loss): Connecting a charged capacitor to an uncharged one causes a transient current. Charge flows until potentials equalize. Total energy is strictly lost as heat/electromagnetic radiation. Loss $\Delta H = U_i - U_f$.

Conditions & Limitations

• Coulomb's Law limitations: Valid only for point charges at rest. It is inaccurate for charges moving at high speeds (where magnetic and relativistic effects occur) and macroscopic charged bodies (unless they are perfectly symmetric spheres evaluated from outside).
• Dipole Approximation: The formulas $\mathbf{E}_{axis}$, $\mathbf{E}_{eq}$, and $V_{dipole}$ explicitly rely on the condition $r \gg a$.
• Parallel Plate Capacitor Approximation: $C = \epsilon_0 A / d$ fundamentally ignores the "fringing" of electric field lines at the plate edges, strictly holding true only when $d^2 \ll A$.
• Continuous Charge Distribution: Treating charge as a continuous density function ($\lambda, \sigma, \rho$) is a macroscopic approximation. Microscopically, charge is discrete and quantized.
• Dielectrics: The linear relationship $\mathbf{P} = \chi_e \epsilon_0 \mathbf{E}$ assumes a linear isotropic dielectric. Very strong electric fields will cause dielectric breakdown and non-linear behavior.

Important Graphs & Diagrams

1. Variation of $E$ and $V$ with distance $r$ for a Point Charge:
   • $E \propto 1/r^2$: Steeper curve.
   • $V \propto 1/r$: Flatter curve.
   • Graph intercepts at $r=1$ (in arbitrary units). $V$ lies above $E$ for $r > 1$, while $E$ lies above $V$ for $r < 1$.
2. Equipotential Surfaces vs Electric Field Lines:
   • Point Charge: Concentric spheres, field lines are radial.
   • Uniform Field: Parallel planes, perpendicular to parallel straight field lines.
   • Dipole: Figure-eight-like lobes surrounding the two charges, they crowd closer in the region between the charges due to field addition.
   • Two Identical Positive Charges: Two separate spherical surfaces close to the charges, merging into a peanut shape, and then a larger distorted sphere far away.
3. Solid Non-Conducting Sphere (JEE Adv):
   • $E$ vs $r$: Linearly increasing from $0$ to $R$ ($E \propto r$), then exponentially decaying ($E \propto 1/r^2$) for $r > R$.
   • $V$ vs $r$: Parabolic inside ($V \propto R^2 - r^2$), and $1/r$ outside.

Standard Derivations & Step-by-Step Problem Solving

1. Work done to assemble charges (Potential Energy Formulation):

• Step 1: Bring $q_1$ from infinity to position $r_1$. Work $W_1 = 0$ (no field).
• Step 2: Bring $q_2$ from infinity to $r_2$. Work $W_2 = q_2 V_1(r_2) = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$.
• Step 3: Bring $q_3$ from infinity to $r_3$. Work $W_3 = q_3 [V_1(r_3) + V_2(r_3)] = \frac{1}{4\pi\epsilon_0} (\frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}})$.
• Step 4: Total $U = W_1 + W_2 + W_3$.

2. Relation between E and V:

• Step 1: Consider two equipotential surfaces A ($V$) and B ($V+dV$) separated by $dl$.
• Step 2: Work done moving unit positive charge from B to A against field $\mathbf{E}$ is $dW = |\mathbf{E}| dl$.
• Step 3: Potential difference is $V_A - V_B = V - (V+dV) = -dV$.
• Step 4: Equate work to potential diff: $|\mathbf{E}| dl = -dV \Rightarrow |\mathbf{E}| = -dV/dl$.

3. Finding Field inside a Conductor Cavity (Electrostatic Shielding):

• Step 1: Take a Gaussian surface completely inside the solid material of the conductor, enveloping the cavity.
• Step 2: Since $\mathbf{E} = 0$ everywhere inside the conductor material, total flux $= 0$.
• Step 3: Thus, $q_{enc} = 0$. If there is no charge in the cavity, the inner surface charge must be zero, rendering field inside the cavity strictly zero.

Previous Year JEE Topics

• Gauss's Law: Flux calculations through cube faces when a charge is placed at corners/edges/centers.
• Capacitors with Dielectrics: Battery connected vs disconnected scenarios. Inserting dielectrics partially (creating parallel/series equivalent capacitors).
• Concentric Conducting Shells: Grounding/Earthing the inner or outer shell and calculating the final charge distribution and potential of each shell.
• Dipole Interactions: Force between a point charge and a dipole, or between two dipoles (proportional to $1/r^3$ or $1/r^4$ respectively), as well as work done to rotate dipoles.
• Energy Conservation in Electrostatics: Combining $\frac{1}{2} mv^2 + U_{elec} = \text{constant}$ to find minimum distance of approach (Rutherford scattering analog).
• Transients and Heat Loss: Calculating the heat generated when charged capacitors are connected to uncharged ones or when a dielectric is inserted abruptly.

Top 10 JEE MCQ Traps

• [JEE TIP] Trap 1 - The Conservative Work Sign Flip:

  • Misconception: The mechanical work performed by an internal electrostatic field equals the absolute change in the potential energy ($\Delta U$) of the system.
  • Correct Understanding: Because the electrostatic force is conservative, the work performed by the field itself is the negative of the change in potential energy ($W_{\text{elec}} = -\Delta U = U_i - U_f$). Conversely, it is the work done by an external agent against the field that equals the positive change in potential energy ($W_{\text{ext}} = \Delta U$), assuming no change in kinetic energy. Mixing these up causes fatal sign errors in conservation of energy loops.

• [JEE TIP] Trap 2 - The Zero Field Potential Fallacy:

  • Misconception: If the net electric field ($\mathbf{E}$) at a specific spatial coordinate is exactly zero, the electric potential ($V$) at that same location must also collapse to zero.
  • Correct Understanding: The electric field is the spatial gradient of the potential ($\mathbf{E} = -\frac{dV}{dr}$). A zero electric field implies that the potential is strictly constant with respect to distance, not necessarily zero. For example, inside a hollow, uniformly charged spherical conducting shell, the internal electric field is zero ($\mathbf{E} = 0$), but the potential at every internal point is a non-zero constant equal to its value at the surface ($V = \frac{kQ}{R}$).

• [JEE TIP] Trap 3 - Field Line Vector Intersection:

  • Misconception: Electric field lines can intersect or cross one another if they originate from multiple distinct point charges positioned closely together.
  • Correct Understanding: Electric field lines can never cross under any circumstances. The tangent to a field line represents the direction of the net electric field vector at that point. If two field lines intersected, a single point in space would simultaneously possess two distinct directions for the net electric force, which is physically impossible.

• [JEE TIP] Trap 4 - Dipole vs. Point Charge Radial Decay:

  • Misconception: The electric potential generated by a localized electric dipole decays as a function of distance at the standard point-charge rate of $\frac{1}{r}$.
  • Correct Understanding: Due to the structural pairing of equal and opposite charges, a dipole's field attributes cancel out much faster than a single isolated point charge. Along a position vector $r$ at a large distance, the dipole potential falls off as $\frac{1}{r^2}$, while its corresponding electric field decays as $\frac{1}{r^3}$.

• [JEE TIP] Trap 5 - The Uniform Charge Distribution Blanket:

  • Misconception: Any excess static charge deposited onto a clean, solid conducting block will naturally distribute itself uniformly across the entire outer boundary surface.
  • Correct Understanding: Charge distributes itself uniformly if and only if the conductor is a perfect sphere. On an irregularly shaped conductor, the surface charge density ($\sigma$) varies dynamically based on local surface geometry. It accumulates preferentially at sharp corners and points where the radius of curvature ($R_{\text{curvature}}$) is smallest, obeying the relation: $\sigma \propto \frac{1}{R_{\text{curvature}}}$.

• [JEE TIP] Trap 6 - Equipotential Boundary Overlap:

  • Misconception: Equipotential surfaces mapped across a multi-charge system can intersect or cross each other at locations where the field vectors are aligned.
  • Correct Understanding: Equipotential surfaces can never intersect. An intersection line would imply that a single spatial coordinate simultaneously possesses two different scalar values of electric potential, which violates the fundamental definition of a unique potential function.

• [JEE TIP] Trap 7 - Dielectric Capacitor Thermodynamic Split:

  • Misconception: Inserting a dielectric slab into a parallel-plate capacitor always reduces the total amount of electrostatic potential energy stored inside the system.
  • Correct Understanding: The energy shift depends strictly on whether the system is closed or open. If the battery is disconnected (Charge $Q$ is constant), the dielectric reduces the field strength and energy drops: $U' = \frac{U}{K}$. However, if the battery remains connected (Voltage $V$ is constant), the battery must perform work to pump extra charge onto the plates, causing the stored energy to increase: $U' = K \cdot U$.

• [JEE TIP] Trap 8 - The Net Capacitor Charge Illusion:

  • Misconception: The common textbook phrase "the charge on a capacitor" refers to the total algebraic sum of the net electric charge residing across both conducting plates.
  • Correct Understanding: The absolute net charge of any standard parallel-plate capacitor is always exactly zero, because the plates hold equal and opposite charges ($+Q$ and $-Q$). The technical term "charge on a capacitor" specifically isolates and refers only to the absolute magnitude $Q$ residing on the positive plate.

• [JEE TIP] Trap 9 - The Cavity Penetration Fallacy:

  • Misconception: A powerful, intensely concentrated external static electric field can force its way through a thin conducting shell and enter a hollow inner cavity.
  • Correct Understanding: This is the principle of Electrostatic Shielding. Under electrostatic equilibrium, induced charges relocate across the outer surface of a hollow conductor to perfectly cancel out the external field inside the bulk material. Consequently, the electric field within a closed, hollow uncharged cavity inside a conductor remains strictly zero, completely isolating the interior from external static fields regardless of their strength.

• [JEE TIP] Trap 10 - Dipole Equilibrium Angular Inversion:

  • Misconception: The electrostatic potential energy of an electric dipole placed within a uniform electric field is minimized when its dipole moment vector ($\mathbf{p}$) points in the exact opposite direction ($180^\circ$) to the field vector ($\mathbf{E}$).
  • Correct Understanding: The potential energy function of a dipole is defined by the dot product: $U = -\mathbf{p} \cdot \mathbf{E} = -pE\cos\theta$. The absolute minimum (most stable) energy occurs at $\theta = 0^\circ$ ($U = -pE$), where the dipole aligns parallel to the field. At $\theta = 180^\circ$, the cosine yields $-1$, maximizing the potential energy: $U = +pE$. While the net torque is zero at $180^\circ$, this state represents an highly unstable equilibrium point.