Electrostatic Potential and Capacitance-Revision Notes

 CBSE Class 12 Physicsh

Revision Notes
Chapter-2
Electrostatic Potential and Capacitance

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• Electrostatic Potential at a Point: It is the work done per unit charge by an external agency, in bringing a charge from infinity to that point.

• Electrostatic Potential due to a Charge at a Point:
  V(r)=14πε0Qr$V(r)=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r}$
• The electrostatic potential at a point with position vector r due to a point dipole of dipole moment p place at the origin is:
  V(r)=14πε0p→.r^r2$V(r)=\frac{1}{4\pi {\epsilon }_0}\frac{\overrightarrow{p}.\hat{r}}{r^2}$
  The result is true also for a dipole (with charges –q and q separated by 2a) for r >> a.
• For a charge configuration q1, q2, …… qn with position vectors r1, r2, ….rn, the potential at a point P is given by the superposition principle,
  V=14πε0(q1r1p+q2r2p+.....+qnrnp)$V=\frac{1}{4\pi {\epsilon }_0}\left(\frac{q_1}{r_{1p}}+\frac{q_2}{r_{2p}}+.....+\frac{q_n}{r_{np}}\right)$
  where r1p is the distance between q1 and P, as and so on.
• Electrostatic Potential Energy Stored in a System of Charges: It is the work done (by an external agency) in assembling the charges at their locations.
• Electrostatic Potential Energy of Two Charges q1, q2, at r1, r2:
  U=14πε0q1q2r12$U=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r_{12}}$ where r12 is distance between q1 and q2
• Potential Energy of a Charge q in an External Potential V(r) is q V(r)
• Potential Energy of a Dipole of Dipole Moment p in a Uniform Electric Field E is –p.E.
• Equipotential Surface: 
  (i) An equipotential surface is a surface over which potential has a constant value.
  (ii) For a point charge, concentric spheres centered at a location of the charge are equipotential surfaces.
  (iii) The electric field E at a point is perpendicular to the equipotential surface through the point.
  (iv) E is in the direction of the steepest decrease of potential.
• Capacitance C of a System of Two Conductors Separated by an Insulator: It is defined as, C=QV$C=\frac{Q}{V}$ where Q and – Q are the charges on the two conductors V is the potential difference between them.
• Capacitance is determined purely geometrically, by the shapes, sizes, and relative positions of the two conductors.
• Capacitance C of a parallel plate capacitor (with the vacuum between the plates): C=ε0Ad$C={\epsilon }_0\frac{A}{d}$  where A is the area of each plate and d the separation between them.
• For capacitors in the series combination: The total capacitance C is  1C=1C1+1C2+1C3+....$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+....$
• For capacitors in the parallel combination: The total capacitance C is C=C1+C2+C3+......$C=C_1+C_2+C_3+......$ where C1, C2, C3...$C_1,C_2,C_3...$ are individual capacitances.
• The energy U stored in a capacitor of capacitance C, with charge Q and voltage V:  U=12QV=12CV2=12Q2C$U=\frac{1}{2}QV=\frac{1}{2}C{V}^2=\frac{1}{2}\frac{Q^2}{C}$
• The electric energy density (energy per unit volume) in a region with electric field = 12ε0E2$\frac{1}{2}{\epsilon }_0{E}^2$
• The potential difference between the conductor (radius ro) inside & outside spherical shell (radius R): V(r0)−V(R)=q4πε0(1r0−1R)$V(r_0)-V(R)=\frac{q}{4\pi {\epsilon }_0}\left(\frac{1}{r_0}-\frac{1}{R}\right)$ which is always positive.
• When the medium between the plates of a capacitor filled with an insulating substance changes observed are as follows:
  (i) Polarization of the medium gives rise to a field in the opposite direction.
  (ii) The net electric field inside the insulating medium is reduced.
  (iii) Potential difference between the plates is thus reduced.
  (iv) Capacitance C increases from its value Co when there is no medium (vacuum), C = KCo where K is the dielectric constant of the insulating substance.
• Electrostatic Shielding: Whatever be the charge and field configuration outside, any cavity in a conductor remains shielded from outside electric influence, thus field inside the cavity is always zero. This is known as electrostatic shielding.