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Electrostatics II: Potential & Capacitance revision notes

Every idea in the Electrostatics II: Potential & Capacitance chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 26 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Electric potential

Potential of point charges
6× asked, last 2025
V=14πε0∑iqiri,VA−VB=∫ABE⃗⋅dl⃗V = \frac{1}{4\pi\varepsilon_0}\sum_i \frac{q_i}{r_i},\qquad V_A - V_B = \int_A^B \vec E\cdot d\vec l
A scalar: add with signs, no directions. Zero at infinity. Unit volt = J/C.
  • Not a vector sum. VV has no direction, so each charge's term just adds with its sign. Two points that mirror each other across the bisecting plane of +Q+Q and −Q-Q sit at ±V1\pm V_1: difference 2V12V_1, not V1V_1.
Ring and disc on the axis
3× asked, last 2024
Vring=kQR2+x2,Vdisc=σ2ε0(R2+x2−x)V_{\text{ring}}=\frac{kQ}{\sqrt{R^2+x^2}},\qquad V_{\text{disc}}=\frac{\sigma}{2\varepsilon_0}\left(\sqrt{R^2+x^2}-x\right)
xx along the axis from the centre. At the centre of any ring or arc every bit is at RR, so V=kQ/RV=kQ/R.
xxORR2R2R3R3RkQ/RkQ/Rcentre: V=kQ/RV=kQ/R, E=0E=0VVEE
Ring: VV is largest at the centre, exactly where E=−dV/dxE=-dV/dx is zero
  • Not kλ/Rk\lambda/R for an arc. Use its charge λ×\lambda\times arc length. A half ring holds λπR\lambda\pi R, so its centre is at kλπ=λ/4ε0k\lambda\pi=\lambda/4\varepsilon_0, independent of RR.
Spherical shell or conducting sphere
7× asked, last 2025
V=kQR  (r≤R),V=kQr  (r≥R)V=\frac{kQ}{R}\;(r\le R),\qquad V=\frac{kQ}{r}\;(r\ge R)
Isolated thin shell or metal sphere (solid/hollow); flat inside as E=0E=0. A charge qq at dd from the centre adds kq/dkq/d there.
rrVVORRkQ/RkQ/Rflat
Shell or metal sphere: flat inside, then kQ/rkQ/r
rrVVORRkQ/RkQ/R1.5 kQ/R1.5\,kQ/R
Insulating solid sphere: a parabola inside
  • Concentric shells: add kQi/rkQ_i/r for each, using the larger of the shell's radius and the point's distance. In V(a)−V(b)V(a)-V(b) a shell enclosing both points cancels; charge inside aa still adds kq(1/a−1/b)kq(1/a-1/b).
  • Uniformly charged insulating solid sphere: V=kQ(3R2−r2)/2R3V=kQ(3R^2-r^2)/2R^3 inside, kQ/rkQ/r outside; centre V=1.5 kQ/RV=1.5\,kQ/R. Not flat inside, unlike a shell or metal sphere: the VV–rr graph inside is a parabola peaking at the centre (Q>0Q>0).
  • Not zero potential inside a shell. E=0E=0 there, but VV stays at the surface value kQ/RkQ/R. The VV–rr graph is flat inside, then falls as 1/r1/r; moving a charge inside needs no work.
Dipole potential
3× asked, last 2025
V=kpcos⁡θr2,Vaxial=kpr2,Veq=0V=\frac{kp\cos\theta}{r^2},\qquad V_{\text{axial}}=\frac{kp}{r^2},\qquad V_{\text{eq}}=0
Short dipole, r≫ar\gg a; θ\theta from p⃗\vec p. VV falls as 1/r21/r^2, the field as 1/r31/r^3.
  • Not zero field where V=0V=0. On the equatorial line V=0V=0 but E=kp/r3E=kp/r^3. Zero potential never implies zero field, nor the reverse.
Infinite line and sheet
1× asked, last 2025
V1−V2=2kλln⁡r2r1,V1−V2=σ2ε0(x2−x1)V_1-V_2=2k\lambda\ln\frac{r_2}{r_1},\qquad V_1-V_2=\frac{\sigma}{2\varepsilon_0}(x_2-x_1)
VV cannot be zero at infinity, so only differences are defined. rr = distance from the line; xx = from the sheet, either side.
  • Quick method. Several identical line charges λ\lambda: V=−2kλ∑ln⁡ri+V=-2k\lambda\sum\ln r_i+ const, so an equipotential has a constant product of distances. The distance from the xx-axis is y2+z2\sqrt{y^2+z^2}, not xx.

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Practise Electrostatics II: Potential & Capacitance

21 more concepts in 5 topics

Field, equipotentials and energy; Conductors and capacitance; Combinations and circuits; Energy and charge sharing; and Dielectrics. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Field, equipotentials and energy

  • Field from potential3×
  • Equipotential surfaces4×
  • Potential energy of a system3×
  • Work and energy conservation6×
  • Dipole in a uniform field: energy4×

Conductors and capacitance

  • Connected conductors6×
  • Merging drops5×
  • Capacitance7×
  • Parallel plate capacitor8×
  • Force between the plates2×

Combinations and circuits

  • Series and parallel12×
  • Networks and the balanced bridge9×
  • Capacitor in a DC circuit21×

Energy and charge sharing

  • Energy stored8×
  • Energy density2×
  • Work done by the battery2×
  • Charge sharing between capacitors18×

Dielectrics

  • Dielectric and polarisation9×
  • Slab of thickness t13×
  • Dielectrics in series and parallel17×
  • Inserting a dielectric: battery on or off11×

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