Physics Quick Notes – Electric Field, Gauss’s Law & Capacitors

  • Electric Field and Potential (HC Verma – Ch 29)

• Charges

  • Property of matter, analogous to mass (gravitational) – two kinds: positive (proton) & negative (electron)

  • Unit: coulomb, e=1.602×10−19 Ce = 1.602\times 10^{-19}\text{ C}

  • Quantization: any charge Q=neQ = n e

  • Conservation & frictional electricity (triboelectric): electrons transfer on rubbing → induce + / – charges.

• Coulomb’s Law

  • Force between two point charges: F⃗=kq<em>1q</em>2r2r^=14πε<em>0q</em>1q2r2r^\vec F = k \frac{q<em>1 q</em>2}{r^{2}} \hat r = \frac{1}{4\pi\varepsilon<em>0} \frac{q</em>1 q_2}{r^{2}} \hat r

  • k=8.9875×109 N⋅m2/C2,  ε0=8.854×10−12 C2/N⋅m2k = 8.9875\times10^{9}\, \mathrm{N·m^{2}/C^{2}},\;\varepsilon_0 = 8.854\times10^{-12}\,\mathrm{C^{2}/N·m^{2}}

• Electric Field E⃗\vec E

  • Defined via test-charge E⃗=F⃗/q\vec E = \vec F/q

  • Field of point charge: E⃗=14πε0Qr2r^\vec E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^{2}}\hat r

  • Superposition principle for multiple/continuous distributions; integration if needed.

• Lines of Force

  • Tangent gives field direction; density ∝ |E|; originate on +, terminate on –.

• Potential Energy (PE)

  • Work done by electric force = –ΔU. For two charges: U(r)=14πε<em>0q</em>1q2rU(r)=\frac{1}{4\pi\varepsilon<em>0}\frac{q</em>1 q_2}{r} with reference U(∞)=0.

• Electric Potential V

  • Scalar: V=U/qV = U/q, difference V<em>B−V</em>A=−∫ABE⃗⋅dr⃗V<em>B-V</em>A = - \int_A^B \vec E\cdot d\vec r

  • Potential of point charge: V=14πε0QrV = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}; of system: add scalars.

• Relation E–V

  • E⃗=−∇V\vec E = -\nabla V; in Cartesian: Ex=−∂V/∂xE_x = -\partial V/\partial x etc.

  • Equipotential surfaces: ⟂ to E; no work moving along surface.

• Electric Dipole (charges ±q separated by d⃗\vec d)

  • Dipole moment p⃗=qd⃗\vec p = q \vec d (direction –→+).

  • Potential on axis: V=14πε02pcos⁡θr2V = \frac{1}{4\pi\varepsilon_0}\frac{2p\cos\theta}{r^{2}} (far field r≫dr \gg d).

  • Field components (far): E<em>r=14πε</em>02pcos⁡θr3,  E<em>θ=14πε</em>0psin⁡θr3E<em>r = \frac{1}{4\pi\varepsilon</em>0}\frac{2p\cos\theta}{r^{3}},\;E<em>\theta = \frac{1}{4\pi\varepsilon</em>0}\frac{p\sin\theta}{r^{3}}.

  • Torque in uniform field: τ⃗=p⃗×E⃗\vec \tau = \vec p \times \vec E; potential energy U=−p⃗⋅E⃗U = -\vec p\cdot\vec E.

• Conductors vs Insulators vs Semiconductors

  • Conductors: free electrons; inside conductor E⃗=0\vec E=0 (static); charge resides on surface; induced charges cancel internal field.

• Gauss’ Law (Ch 30)

• Flux definition: Φ=∬E⃗⋅dS⃗\Phi = \iint \vec E\cdot d\vec S (sign via outward normal).

• Solid angle Ω: Ω=S/r2\Omega = S/r^{2} subtended by surface area S at distance r.

• Gauss: ∮E⃗⋅dS⃗=Q<em>in/ε</em>0\oint \vec E\cdot d\vec S = Q<em>{in}/\varepsilon</em>0 – for any closed surface; useful with symmetry.
Cases:

  • Infinite line: E=λ2πε0rE = \frac{\lambda}{2\pi\varepsilon_0 r}.

  • Infinite plane: E=σ/2ε<em>0E = \sigma/2\varepsilon<em>0 (each side); near conducting charged surface E=σ/ε</em>0E = \sigma/\varepsilon</em>0 outward.

  • Uniform sphere: outside behaves like point charge; inside E∝rE \propto r.

• Capacitors (Ch 31)

Capacitance C=Q/VC=Q/V; units farad.

– Standard geometries:

  • Parallel plates: C=ε<em>0A/dC = \varepsilon<em>0 A/d; with dielectric C=Kε</em>0A/dC = K\varepsilon</em>0 A/d.

  • Cylindrical: C=2πε<em>0lln⁡(R</em>2/R1)C = \frac{2\pi\varepsilon<em>0 l}{\ln(R</em>2/R_1)}.

  • Spherical concentric: C=4πε<em>0R</em>1R<em>2R</em>2−R<em>1C = 4\pi\varepsilon<em>0 \frac{R</em>1 R<em>2}{R</em>2-R<em>1}; isolated sphere C=4πε</em>0RC=4\pi\varepsilon</em>0 R.

– Combinations:
• Series: 1/C=∑1/C<em>i1/C = \sum 1/C<em>i (charge equal). • Parallel: C=∑C</em>iC = \sum C</em>i (potential equal).

– Energy:
U=12CV2=Q22C=12QVU = \frac{1}{2}CV^{2}=\frac{Q^{2}}{2C}=\frac{1}{2}QV
Energy density in field: u=12εE2u = \tfrac{1}{2}\varepsilon E^{2} (vacuum), =12εKE2=\tfrac{1}{2}\varepsilon K E^{2} with dielectric.

– Force between parallel plates (vacuum): F=Q22ε0AF = \frac{Q^{2}}{2\varepsilon_0 A} (attractive).

– Dielectrics: polarization P⃗\vec P, bound surface charge σ<em>b=P⃗⋅n^\sigma<em>b = \vec P \cdot \hat n. Displacement vector D⃗=ε</em>0E⃗+P⃗=ε<em>0KE⃗\vec D = \varepsilon</em>0 \vec E + \vec P = \varepsilon<em>0 K \vec E; Gauss in dielectrics: ∮D⃗⋅dS⃗=Q</em>free\oint \vec D\cdot d\vec S = Q</em>{free}.

– Dielectric breakdown strength: max field (kV/mm) before conduction.

• van de Graaff Generator

  • Moving insulating belt transfers charge to large sphere → high V (≈ MV), limited by breakdown EmaxE_{max}.

Key Formula Sheet

$\bullet$ Coulomb: k=1/4πε<em>0k=1/4\pi\varepsilon<em>0 $\bullet$ Potential of point: V=kQ/rV= kQ/r $\bullet$ Field–Potential: E⃗=−∇V\vec E=-\nabla V $\bullet$ Dipole torque: τ=pEsin⁡θ\tau=pE\sin\theta $\bullet$ Gauss: Φ</em>E=Q<em>in/ε</em>0\Phi</em>E = Q<em>{in}/\varepsilon</em>0
$\bullet$ Capacitor forms: PP ε<em>0A/d\varepsilon<em>0 A/d, Cyl 2πε</em>0l/ln⁡(R<em>2/R</em>1)2\pi\varepsilon</em>0 l/\ln(R<em>2/R</em>1), Sphere 4πε<em>0R</em>1R<em>2/(R</em>2−R<em>1)4\pi\varepsilon<em>0 R</em>1 R<em>2/(R</em>2-R<em>1). $\bullet$ Combinations: series/parallel. $\bullet$ Energy: 12CV2\tfrac12 CV^{2}, density 12εE2\tfrac12 \varepsilon E^{2}. $\bullet$ Force plates: F=Q2/(2ε</em>0A)=12ε<em>0AE2F=Q^{2}/(2\varepsilon</em>0 A)=\tfrac12 \varepsilon<em>0 A E^{2}. $\bullet$ Dielectric constant K=C/C</em>0K=C/C</em>0.

Problem-Solving Tips

  1. Use symmetry for Gauss surfaces: choose shape where E⃗\vec E constant on surface.

  2. For capacitor with dielectric partially filled, treat as series combination of filled and empty regions.

  3. In series: charge same; in parallel: voltage same.

  4. When battery disconnected, charge remains constant.

  5. Energy conservation to compute heat when reconnecting capacitors/batteries.

  6. Displacement vector simplifies Gauss in dielectrics.

End of concise notes for HC Verma Volume II, Ch 29–31 (Electric Field, Gauss’s Law, Capacitors).