1. Electric potential V is a scalar.
2. The electric potential
difference between two given points in an electric field is equal to the amount
of work done against the electric field in order to bring a unit positive test
charge (without acceleration) from one point to the other. Mathematically, the potential difference between the points B and A, i.e., VB – VA is given by \[V_B - V_A= \frac{W_{A \to B}}{q_0}\] Where $ {W_{A \to B}} $ is the work done in order to carry a test charge q0 from
point A to point B. The value of the test charge should be as small as possible.
3. The electric potential at a point in an electric field is equal to the amount of work done by the external
force (against the electric field) in order to bring a unit positive test
charge (without acceleration) from infinity to that point. Mathematically,
electrostatic potential at a point A is given by \[V_A = \frac{W_{\infty \to A}}{q_0}\] Where \[{W_{\infty \to A}}\] is the amount of work done in order to
carry a test charge q0 from to the point, A. value of the test charge should be as small as possible.
4. Electrostatic potential, as well as potential differences, are scalar quantities and their SI unit is
volt.
5. Thus, the electric
potential at a point is said to be 1 volt, if 1 J of work is being done in
order to move a positive test charge of 1 C from infinity to that point.
6. The electric potential at a
point situated at a distance r from a point charge q is given by \[V = \frac{q}{4 \pi \epsilon r}\]. Thus, potential due to
a free, independent +ve point charge is positive because work is being done
against the repulsive force experienced by + ve test charge. However, potential
due to a -ve point charge is negative
because here work is done by the attractive force acting on the +ve test
charge.
7. It is possible to
maintain a positively charged body at a negative or zero potential and a
negatively charged body at zero or even positive potential.
8. Electric potential of
earth is considered to be zero. Thus, electric potential of any body connected
to earth will also be zero.
9. If a number of point
charges are present then electric potential at a given point is equal to the
algebraic sum of potentials due to different charges. Thus, if a number of
charges q1, q2, q3 …. are present at
distance r1, r2, r3 ….etc. respectively from a
given point then total electric at that point is \[V = V_1 + V_2 + V_3 + ......... = \frac{1}{4 \pi \epsilon }[{\frac{q_1}{r_1}+\frac{q_2}{r_2}+\frac{q_3}{r_3}+..........}] = \frac{1}{4 \pi \epsilon } \Sigma_{i=1}^{N} \frac{q_i}{r_i}\].
10. Electrostatic force is
a conservative force and electric field is an example of a conservative
field. It means that amount of work done in carrying a test charge in an
electric field depends only upon the positions of initial point and the final
point and is independent of the path followed. Moreover, the work done
in carrying a charge in an electric field along a complete cyclic path is zero.
Mathematically, \[\oint \vec{E}\vec{dl} = 0\].
11. Electric field at a
point due to a given point charge is inversely proportional to the square of
distance of given point from the point charge i.e., \[E \propto \frac{1}{r^2}\].
However, electric potential due to a given point charge is inversely
proportional to the distance i.e., \[V \propto \frac{1}{r}\].
12. Electrostatic potential
difference between two points in an electric field may also be defined as
negative of the line integral of the electric field between the given points i.e., \[V_B-V_A= - \int_A^B \vec{E}.\vec{dl}\]. The electric potential at a the point in an electric field is equal to negative of the line integral of the electric field from infinity to that point i.e. \[V_A = - \int_{\infty}^A \vec{E}\vec{dl} \]
13. Electric field at a
point may, thus, be defined as the negative of the rate of change of electric
potential with position (i.e., the negative of the potential gradient) at that
point i.e., \[\vec{E} = -\frac{dV}{dl}\] and \[|E| = \frac{dV}{dl}\]. Moreover, the direction of the electric field is the direction in which electric
potential is decreasing at a maximum rate i.e., where the decrease of potential is
steepest.
14. The electric potential
at a point situated at a distance ‘r’ from the mid-point of a short electric the dipole of dipole moment p inclined at an angle θ from the axial line of
dipole is given by \[V = \frac{1}{4 \pi \epsilon} \frac{pcos\theta}{r^2}\].
15. At a point situated on
the axial line of electric dipole θ = 0o or π and hence \[V = \pm \frac{1}{4 \pi \epsilon}\frac{p}{r^2}\]. Here +ve sign is taken
for θ
= 0o and -ve sign is taken for θ = π. Electric potential at
any point situated on the equatorial line of an electric dipole ( θ
= 900 ) is zero i.e., V = 0.
16. Electric potential due
to an electric dipole, in general, is inversely proportional to the square of the distance of the point from mid-point of the dipole, i.e., \[V \propto \frac{1}{r^2}\].
17. An equipotential
surface is a surface with a constant value of the potential at all points on the
surface. For any charge configuration, the equipotential surface through a point is
normal to the electric field at the point.
18. For a point charge, equipotential surfaces are concentric spheres with a given charge point as the
center. For a uniform electric field, equipotential surfaces are planes
perpendicular to the direction of the electric field.
19. As potential at all
points of an equipotential surface is the same, hence, work done in moving a charge
from one point to another along the equipotential surface is always zero.
20. The electric potential the energy of a system of point charges is equal to the amount of work done in assembling
the given system of charge by bringing them to their respective positions from
infinity. A point to be noted is that the potential energy is characteristic of
the present state of assembly (or configuration ) and not the way the state is
achieved. SI unit of electric
potential energy is joule ( J ). For atomic and sub-atomic particals a unit
“electron volt” (1 ev) is frequently used, where 1 eV = 1.60 *10-19J.
21. For a system of two
point charges q1 and q2 separated by a distance r the
potential energy is given by \[U = \frac{q_1 q_2}{4 \pi \epsilon r}\]. If two charges are like
charges then the force between them is repulsive. Work is being done against this
repulsive force while bringing the charges to their present position and hence
electric potential energy of the system will be positive. If two charges are
unlike one, the force between them is attractive and work is being done by the
attractive force. Consequently, the potential energy of the system will be negative.
22. For a system of n point
charges the total electric potential energy of the system is given by \[U = \frac{1}{2}[ \frac{1}{4 \pi \epsilon } \Sigma_{i=1}^{n} \Sigma_{j = 1, i \neq j}\frac{q_i q_j}{r_{ij}}]\]. Here the factor $ \frac{1}{2} $ has
been incorporated on account of the that in the summation each term has been
counted twice ij and ji in the above expression.
23. The electrostatic potential
energy of a single charge q in an external electric field E is given by U(r) = qV(r), Where V(r) is the
potential at the given point due to the external electric field.
24. Electrostatic potential
energy of a system of two charges q1 and q2 located at $ \vec{r_1} $ and $ \vec{r_2} $ respectively in an external electric field is given by \[U = q_1V(r_1) + q_2V(r_2) + \frac {q_1 q_2}{4\pi \epsilon r_{12}}\].
25. Work done for rotating
an electric dipole of dipole moment p in a uniform electric field E from
orientation θ1 to orientation θ2
is given by \[W = - pE(cos \theta_2 - cos \theta_1 ) = pE(cos \theta_1 - cos \theta_2 )\]. The electrostatic
potetial energy of an electric dipole in a uniform electric field E is given by \[W = - \vec{p}.\vec{E} = - pEcos\theta\] Where $ \theta $ is
the angle which axis of given dipole makes with the direction of electric field
E.
26. An electric dipole is
in a state of stable equilibrium when dipole is placed along the direction of
external electric field (i.e., p and E are in same direction) because in that orientation
torque on dipole is zero and potential energy of dipole in minimum having a
value -pE. On the other hand
electric dipole is in a state of unstable equilibrium when p and E are in
mutually opposite directions because in that orientation torque on dipole is
zero and potential energy of dipole is maximum having a value +pE.
27. Conductors are the
materials which possess large number of free electron and, therefore, allow
flow of electric charge through them easily. Silver, copper, aluminium and
other metals, mercury etc., are example of good conductors of electricity.
28. When a conductor is
placed in an electric field, it exhibits the following properties :
(i). Net electric field
inside the conductor is zero.
(ii). Electric charge always
reside on the outer surface of the conductor only.
(iii). Net electric charge in
the interior of the conductor is zero in equilibrium state.
(iv). Just outside the
surface of a conductor, the electric field is perpendicular (normal) to the
surface at every point.
(v). Electric field lines do
not pass through the interior of the conductor.
(vi). Electric potential at
all points of the conductor, situated inside as well as on its surface, is
uniform. Moreover, it has the same value as on its surface.
(vii). Electric field at the
surface of a charged conductor is given by \[\vec{E} = \frac{\sigma}{\epsilon_0}\widehat{n}\], Where σ
is the surface charge density and n is a unit vector normal to the
surface. For σ > 0, electric field is normal to the
surface outward but for σ < 0, electric field is normal to the
surface inward.
29. As electric charge, as well as electric field inside a cavity of any conductor, is zero, the cavity
remains shielded from outside electric influence. It is known as electrostatic
shielding. Thus, electrostatic shielding (or screening) is the phenomenon of maintaining a certain region in space completely free from external electric fields. Property of electric
shielding is made use of in protecting sensitive instruments from outside
electric influences.
30. Insulators are those
materials that cannot conduct electricity. Insulators possess a negligibly
small number of free electrons.
31. Dielectrics are
non-conducting substances. When a dielectric is held in an electric field. Small
induced charges appear on the surface of the dielectric. However, there is no free
movement of charges inside a dielectric. Dielectrics are of two
types: polar and non-polar dielectrics. Non-polar dielectrics
e.g., O2, N2, H2, CO4, etc.,
consist of non-polar molecules in which the centre of positive charge exactly coincides
with the centre of negative charge and dipole moment of a molecule is zero. Polar dielectrics e.g. H2O, HCl, NH3, alcohol are made of polar molecules in which centre of positive charge does not coincides
with the centre of negative charge and each molecule has some intrinsic electric
dipole moment.
32. When a dielectric is
placed in an external electric field, the field induces dipole moment by stretching
or re-orienting molecules of the dielectric. As a collective effect of these molecular
dipole moments, some net charge is developed on the surface of the dielectric which
produces a field that opposes the external electric field. The opposing field so
induced reduce the external field. The electric dipole moment
developed per unit volume in a dielectric when placed in an external electric
field E is called “polarisation” or polarisation vector P. For linear isotropic
dielectrics, \[\vec{p} = \chi_e \vec{E}\] where $ \chi_e $ is a
constant, known as the electric susceptibility of the given dielectric, whose
value depends on the nature of the dielectric and is a characteristic of the dielectric.
33. The potential of a given
charged conductor is directly propositional to its charge. The ratio of charge
Q of an isolated conductor to its potential V is called a capacitance C of the given
conductor. Thus, \[C = \frac{Q}{V}\]. Alternately capacitance
of a given conductor is equal to the amount of charge given to the conductor in
order to raise its electric potential by unity. SI unit of capacitance
is farad (F).
34. Capacitance of a conductor depends upon
its dimensions and shape. Capacitance of an isolated spherical conductor is
given by \[C = 4 \pi \epsilon_0 R\] in free space, where R is
the radius of conductor. Capacitance of a conductor is independent of the amount of charge
given to it. However, capacitance of a conductor depends on the nature of the
surroundings.
35. Capacitance of a system of two-conductor, besides their geometrical
configuration (shape and size), also depends on (i) the separation between the
two conductors, and (ii) the nature of the dielectric separating the two conductor.
36. A parallel plate capacitor is simplest type of capacitor.it consists
of two parallel metal plates separated by a thin layer of dielectric. Capacitance
of a parallel plate capacitor is \[C = \frac{\epsilon_0 A}{d}\] if free space is the intervening
dielectric. Here, A = surface area
of either plate and d = separation between the two plates of capacitor.
37. When capacitors is connected in series
then the resultant capacitance Cs is given by \[\frac{1}{C_S} = \frac{1}{C_1} +\frac{1}{C_2}+\frac{1}{C_3}+............\] where C1, C2, C3…
etc. are the capacitances of individual capacitors.In series combination charge on all
the capacitors is same but potential difference between the plates of
difference capacitors is inversely proportional to their capacitances. In a series combination of capacitors,
resultant capacitance is less then the capacitance of anyone capacitor. However,
the combination may withstand a higher potential difference (voltage).
38. In parallel grouping of capacitors
resultant capacitance is the sum of individual capacitance of capacitors joined
in parallel i.e., \[C_P = C_1+C_2+C_3+........\]. In parallel arrangement potential
difference across all the capacitors is same but charges on individual
capacitors are directly proportional to their capacitances.
40. On filling a dielectric medium of
dielectric constant K between the plates of a parallel plate capacitor, due to
polarisation of the dielectric, the net electric field and hence the potential
difference between the plates of capacitor is reduced to $ \frac{1}{K} $ times its precious
value. Consequently, the capacitance of the capacitor increases
K times. Thus, $ K = \frac{Capacitance in presence of dielectric medium Cm}{Capacitance when free space is the medium C_0 }$.
41. The capacitance of a parallel plate capacitor with a dielectric
medium introduced between the plates is given by \[C = \frac{K\epsilon_0 A}{d} \]
42. If a dielectric medium of dielectric constant K and thickness t (t
< d) is filled between the plates of a capacitor then its capacity is given
by \[C = \frac{\epsilon_0 A}{d - t - \frac{t}{K}}\].
43. If a conducting sheet of thickness t (t < d) is introduced
between the plates of a capacitor, without touching either plate of a capacitor,
then the capacitance of the arrangement is given by \[C = \frac{\epsilon_0 A}{d-t} .\]
44. The energy stored in a charged capacitance is \[U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}\] where Q = charge
given to capacitance C, and V = potential difference between the plates of
capacitor.
46. The energy density (energy per unit volume) of electric field in a
capacitor is \[E' = \frac{1}{2} \epsilon_0 E^2\] electric field between the plates of capacitor.
47. When two charged capacitor are joined together, they share their
charges till they acquire same “common potential” V, which is given by \[V = \frac{Total Charge}{Total Capacitance} = \frac{C_1V_1+C_2V_2}{C_1+C_2}\]. Total charge remains the same during this process.
48. The sharing of charges between two capacitors is always accompanied by
some loss of electrical energy. Loss of electrical energy is given by \[\Delta U = U_2 - U_1 = \frac{C_1C_2(V_1-V_2)^2}{2(C_1+C_2)}\].
49. Dielectric strength of a dielectric is the maximum value of
electric field (or potential gradient) which it can tolerate without its
electric break-down. For reasons of safety a maximum electric field equal to 10%
of dielectric strength of the material is actually applied.
Video Lectures:
1. Electric Potential energy, Electric potential definition, Electric potential unit, Electric force conservative nature watch video
2. Electric potential due to group of the charges, Electric potential due to electric dipole watch video
3. Equipotential surfaces and relation between Electric field and electric potential watch video
4. Conductor properties, Capacitor and Capacitance, spherical capacitor watch video
5 Parallel plate capacitor with conducting slab and dielectric slab watch video
6 Grouping of the capacitors watch video
7 Energy stored in capacitor, Common potential and Energy loss in charge sharing watch video
