⚡ Electrostatics
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PART A electric charges & fields
1Electric Charges — Conservation of Charge
No formula — Charge is conserved (can neither be created nor destroyed).
TRICK In any process, net charge before = net charge after.
2Coulomb's Law
$$F = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2}$$
\(F\) = Electrostatic force (N) |
\(\varepsilon_0\) = Permittivity of free space (\(8.85\times10^{-12}\)) |
\(q_1, q_2\) = Charges (C) |
\(r\) = Distance between charges (m)
$$k = \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9 \, \text{N·m²/C²}$$
TRICK Like charges repel, unlike charges attract. Force is along the line joining the charges.
3Superposition Principle
$$\vec{F} = \vec{F}_{12} + \vec{F}_{13} + \vec{F}_{14} + \cdots = \sum_{i} \vec{F}_{1i}$$
\(\vec{F}\) = Net force on a charge due to multiple charges
TRICK Forces add as vectors — treat x and y components separately.
4Electric Field — Point Charge
$$\vec{E} = \frac{\vec{F}}{q_0} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2} \hat{r}$$
\(\vec{E}\) = Electric field (N/C or V/m) |
\(Q\) = Source charge |
\(q_0\) = Test charge |
\(\hat{r}\) = Unit vector from source to point
TRICK Direction: Away from positive charge, towards negative charge.
5Electric Field Lines
No formula — Field lines start at positive charge and end at negative charge.
TRICK Density of field lines = Strength of electric field.
6Electric Dipole
$$\vec{p} = q \times 2a$$
\(\vec{p}\) = Dipole moment (C·m) |
\(q\) = Charge |
\(2a\) = Distance between charges
Direction: From \(-q\) to \(+q\).
7Electric Field due to Dipole — On Axis
$$E_{\text{axis}} = \frac{1}{4\pi\varepsilon_0} \frac{2p}{r^3} \qquad (r \gg a)$$
\(r\) = Distance from centre of dipole
8Electric Field due to Dipole — On Equatorial (Perpendicular Bisector)
$$E_{\text{equator}} = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3} \qquad (r \gg a)$$
TRICK \(E_{\text{axis}} = 2 \times E_{\text{equator}}\) at the same distance \(r\).
9Torque on Dipole in Uniform Electric Field
$$\tau = pE\sin\theta$$
\(\tau\) = Torque (N·m) |
\(\theta\) = Angle between \(\vec{p}\) and \(\vec{E}\)
TRICK \(\tau_{\text{max}} = pE\) (when \(\theta = 90°\)), \(\tau = 0\) (when \(\theta = 0°\) or \(180°\)).
10Electric Flux
$$\Phi_E = \vec{E} \cdot \vec{A} = EA\cos\theta$$
\(\Phi_E\) = Electric flux (N·m²/C) |
\(\vec{A}\) = Area vector
11Gauss's Theorem
$$\oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}$$
\(q_{\text{enc}}\) = Net charge enclosed by Gaussian surface
12Application — Infinite Straight Wire
$$E = \frac{\lambda}{2\pi\varepsilon_0 r}$$
\(\lambda\) = Linear charge density (C/m) |
\(r\) = Distance from wire
13Application — Infinite Plane Sheet
$$E = \frac{\sigma}{2\varepsilon_0}$$
\(\sigma\) = Surface charge density (C/m²)
TRICK \(E\) is independent of distance from the sheet.
14Application — Thin Spherical Shell
$$E_{\text{inside}} = 0$$
$$E_{\text{outside}} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2} = \frac{\sigma R^2}{\varepsilon_0 r^2}$$
\(R\) = Radius of shell |
\(Q\) = Total charge |
\(\sigma\) = Surface charge density
TRICK Inside a conductor, electric field is always zero (electrostatic condition).
PART B electrostatic potential & capacitance
15Electric Potential
$$V = \frac{U}{q_0} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r}$$
\(V\) = Electric potential (V) |
\(U\) = Potential energy (J) |
\(q_0\) = Test charge
16Potential Difference
$$V_A - V_B = \frac{W_{AB}}{q_0}$$
\(W_{AB}\) = Work done to move charge from A to B
17Relation between Electric Field and Potential
$$E = -\frac{dV}{dr} \qquad \text{or} \qquad \vec{E} = -\nabla V$$
TRICK Electric field points from higher potential to lower potential.
18Potential due to Point Charge
$$V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$$
19Potential due to Dipole
$$V = \frac{1}{4\pi\varepsilon_0} \frac{p\cos\theta}{r^2}$$
\(\theta\) = Angle between dipole moment and position vector
20Potential Energy of System of Two Charges
$$U = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r}$$
21Potential Energy of Dipole in Electric Field
$$U = -pE\cos\theta = -\vec{p} \cdot \vec{E}$$
TRICK \(U_{\text{min}} = -pE\) (stable at \(\theta = 0°\)), \(U_{\text{max}} = +pE\) (unstable at \(\theta = 180°\)).
22Capacitance
$$C = \frac{Q}{V}$$
\(C\) = Capacitance (F) |
\(Q\) = Charge on capacitor |
\(V\) = Potential difference
23Parallel Plate Capacitor
$$C = \frac{\varepsilon_0 A}{d}$$
\(A\) = Area of plates |
\(d\) = Distance between plates
24Parallel Plate Capacitor with Dielectric
$$C = \frac{K\varepsilon_0 A}{d} = K C_0$$
\(K\) = Dielectric constant (relative permittivity) |
\(C_0\) = Capacitance without dielectric
TRICK Dielectric increases capacitance by factor \(K\).
25Spherical Capacitor — Solid Sphere
$$C = 4\pi\varepsilon_0 R$$
\(R\) = Radius of sphere
26Spherical Capacitor — Hollow Sphere
$$C = 4\pi\varepsilon_0 \frac{ab}{b-a}$$
\(a\) = Inner radius |
\(b\) = Outer radius
27Capacitors in Series
$$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots$$
TRICK For series: \(Q\) same, \(V\) divides. \(C_{\text{eq}} <\) smallest individual capacitor.
28Capacitors in Parallel
$$C_{\text{eq}} = C_1 + C_2 + C_3 + \cdots$$
TRICK For parallel: \(V\) same, \(Q\) divides. \(C_{\text{eq}} >\) largest individual capacitor.
29Energy Stored in Capacitor
$$U = \frac{1}{2} CV^2 = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} QV$$
\(U\) = Energy stored (J)
30Energy Density in Electric Field
$$u = \frac{1}{2} \varepsilon_0 E^2$$
\(u\) = Energy per unit volume (J/m³)
TRICK Energy is stored in the electric field between the plates.
📐 Units & Dimensions
| Quantity | SI Unit | Alternative / Equivalent | Dimension |
|---|---|---|---|
| Charge \(q\) | C (coulomb) | A·s | \(T\,A\) |
| Electric Field \(E\) | N/C | V/m | \(M\,L\,T^{-3}\,A^{-1}\) |
| Electric Potential \(V\) | V (volt) | J/C | \(M\,L^2\,T^{-3}\,A^{-1}\) |
| Electric Flux \(\Phi_E\) | N·m²/C | V·m | \(M\,L^3\,T^{-3}\,A^{-1}\) |
| Dipole Moment \(p\) | C·m | — | \(L\,T\,A\) |
| Capacitance \(C\) | F (farad) | C/V | \(M^{-1}\,L^{-2}\,T^4\,A^2\) |
| Permittivity \(\varepsilon_0\) | F/m | C²/(N·m²) | \(M^{-1}\,L^{-3}\,T^4\,A^2\) |
| Energy \(U\) | J (joule) | N·m | \(M\,L^2\,T^{-2}\) |
| Force \(F\) | N (newton) | kg·m/s² | \(M\,L\,T^{-2}\) |
| Torque \(\tau\) | N·m | J | \(M\,L^2\,T^{-2}\) |
⚡ Quick Recall
| Coulomb's Law | \(F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}\) |
| Electric Field (Point charge) | \(E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r^2}\) |
| Dipole Moment | \(p = q \times 2a\) |
| Dipole Field — Axis | \(E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}\) |
| Dipole Field — Equator | \(E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}\) |
| Torque on Dipole | \(\tau = pE\sin\theta\) |
| Electric Flux | \(\Phi_E = EA\cos\theta\) |
| Gauss's Theorem | \(\oint \vec{E}\cdot d\vec{A} = \dfrac{q_{\text{enc}}}{\varepsilon_0}\) |
| Infinite Wire | \(E = \dfrac{\lambda}{2\pi\varepsilon_0 r}\) |
| Infinite Sheet | \(E = \dfrac{\sigma}{2\varepsilon_0}\) |
| Spherical Shell (Inside) | \(E = 0\) |
| Spherical Shell (Outside) | \(E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r^2}\) |
| Electric Potential | \(V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r}\) |
| Relation E and V | \(E = -\dfrac{dV}{dr}\) |
| Parallel Plate Capacitor | \(C = \dfrac{\varepsilon_0 A}{d}\) |
| Capacitor with Dielectric | \(C = \dfrac{K\varepsilon_0 A}{d}\) |
| Series Combination | \(\dfrac{1}{C_{\text{eq}}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \cdots\) |
| Parallel Combination | \(C_{\text{eq}} = C_1 + C_2 + \cdots\) |
| Energy in Capacitor | \(U = \dfrac{1}{2}CV^2\) |
| Energy Density | \(u = \dfrac{1}{2}\varepsilon_0 E^2\) |
📘 Electrostatics · complete formula sheet with symbol meanings · updated 10 Aug 2026
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