2.16 SUMMARY

1. Electrostatic force is a conservative force. Work done by an external force (equal and opposite to the electrostatic force) in bringing a charge q from a point R to a point P is U P – U R, which is the difference in potential energy of charge q between the final and initial points.

2. Potential at a point is the work done per unit charge (by an external agency) in bringing a charge from infinity to that point. Potential at a point is arbitrary to within an additive constant, since it is the potential difference between two points which is physically significant. If potential at infinity is chosen to be zero; potential at a point with position vector r vector due to a point charge Q placed at the origin is given by


V(r vector) = 1/4πε0 × Q/r


Please note that potential is a scalar quantity, & its SI units are J/C or volt; by Dr. T K Bansal.

3. The electrostatic potential at a point with position vector r vector due to a point dipole of moment p vector placed at the origin is


V(r vector) = (1/4πε0) × (p vector ⋅ r cap/r^2)

The result is true for a dipole (with charges −q and +q separated by a small distance, 2a) for r >> a.

4. For a charge configuration q1, q2, ..., qn with position vectors r1 vector , r2 vector, ... rn vector, the potential at a point P is given by the superposition principle


V = (1/4πε0) × {q1/r1p + q2/r2p + .. . + qn/rnp}


where r1P is the distance between q1 and P, as and so on.

5. An equipotential surface is a surface over which potential has a constant value. For a point charge, concentric spherical shells centered at a location of the charge are equipotential surfaces. The electric field E vector at a point is perpendicular to the equipotential surface through the point. E vector is in the direction of the steepest decrease of potential.

6. Potential energy stored in a system of charges is the work done (by an external agency) in assembling the charges at their locations. Potential energy of two charges q1, q2 at r1 vector, r2 vector is given by


U = (1/4πε0) (q1q2/r12)


where r12 is distance between q1 and q2.

7. The potential energy of a charge q at a point with an external potential V(r vector) is qV(r vector).

The potential energy of a dipole p vector in an uniform electric field E vector is given by


U = −p vector ⋅ E vector.

8. Electrostatic field E vector is zero in the interior of a conductor; just outside the surface of a charged conductor, E vector is normal to the surface given by


E vector = σ/ε0 × n cap


where n cap is the unit vector along the outward normal to the surface and σ is the surface charge density.

Charges in a conductor can reside only at its surface.


Potential is constant within and on the surface of a conductor.


In a cavity within a conductor (with no charges), the electric field is zero.

9. A capacitor is a system of two conductors separated by an insulator. Its capacitance is defined by


C = Q/V,


where +Q and −Q are the charges on the two conductors and V is the potential difference between them.


C is determined purely geometrically, by the shapes, sizes and relative positions of the two conductors.


The unit of capacitance is farad: 1 F = 1 CV^−1.

For a parallel plate capacitor (with vacuum between the plates),


C = ε0 × A/d


where A is the area of each plate and d the separation between them.

10. If the medium between the plates of a capacitor is filled with an insulating substance (dielectric), the electric field due to the charged plates induces a net dipole moment in the dielectric. This effect, called polarization, gives rise to a field in the opposite direction. The net electric field inside the dielectric and hence the potential difference between the plates is thus reduced. Consequently, the capacitance C increases from its value C0 when there is no medium (vacuum), the increased value is given by,


C = KC0


where K is the dielectric constant of the insulating substance.

11. For capacitors in the series combination, the total capacitance C is given by


1/C = 1/C1 + 1/C2 + 1/C3 + ...

In the parallel combination, the total capacitance C is:


C = C1 + C2 + C3 + ...


where C1, C2, C3... are individual capacitances.

12. The energy capital U, stored in a capacitor of capacitance C, with charge Q and voltage V is


U = 1/2 × QV = 1/2 × CV^2 = 1/2 Q^2/C

The electric energy density (energy per unit volume) in a region with electric field is (1/2) ε0 E^2.