Electrostatic force of interaction acting between two stationary charges is given by
F = 1 / 4π ε^{o} q1q2 / r2
where q1, q2 are magnitude of point charges, r is the distance between them and εo is permittivity of free space.
Here, 1 / 4πε_{o} = (10-7 N – s_{2} / C_{2})C_{2}
Substituting value of c = 2.99792458 X 108 m/s, We get 1 / 4πεo = 8.99 x 109N-m2/C2 In examples and problems we will often use the approximate value,
1 / 4πεo = 9 * 10^{9}N-m^{2}/C^{2} The value of εo is 8.85 * 10^{-12} C^{2} / N-mC^{2}. If there is another medium between the point charges except air or vacuum, then εo is replaced by εoK or εoεr or ε.
where K or εr is called dielectric constant or relative permittivity of the medium. K = εr = ε / εo where, ε = permittivity of the medium.
For air or vacuum, K = 1 For water, K = 81 For metals, K = ∞
Coulomb’s Law in Vector Form
Force on q2 due to q1,
The above equations give the Coulomb’s law in vector form.
Force on q1 due to q2 = – Force on q2 due to q1
F12 = – F21 F12 = q1q2 / 4πε . r1 – r2 / |r1 – r2|3
The forces due to two point charges are parallel to the line joining point charges; such forces are called central forces and electrostatic forces are conservative forces.
Electric Field
The space in the surrounding of any charge in which its influence can be experienced by other charges is called electric field.
An electric field line is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction of the electric field vector at that point. The relative closeness of the lines at some place give an idea about the intensity of electric field at that point.”
Two lines can never intersect.
Electric field lines always begin on a positive charge and end on a negative charge and do not start or stop in mid space.
The electrostatic force acting per unit positive charge on a point in electric field is called electric field intensity at that point.
Electric field intensity E = Its SI unit is NC-1 or Vim and its dimension is [MLT-3 A-1]. It is a vector quantity and its direction is in the direction of electrostatic force acting on positive charge. Electric field intensity due to a point charge q at a distance r is given by E = 1 / 4π εo q / r2
Electric potential at any point is equal to the work done per positive charge in carrying it from infinity to that point in electric field. Electric potential, V = W / q Its SI unit is J / C or volt and its dimension is [ML2T-3A-1]. It is a scalar quantity. Electric potential due to a point charge at a distance r is given by v = 1 / 4π εo q / r
Potential Gradient
The rate of change of potential with distance in electric field is called potential gradient.
Potential gradient = dV / dr
Its unit is V / m.
Relation between potential gradient and electric field intensity is given by
E = – (dV / dr)
Equipotential surface is an imaginary surface joining the points of same potential in an electric field. So, we can say that the potential difference between any two points on an equipotential surface is zero. The electric lines of force at each point of an equipotential surface are normal to the surface.
Electric flux over an area is equal to the total number of electric field lines crossing this area. Electric flux through a small area element dS is given by
φE = E. dS
where E= electric field intensity and dS = area vector.
Its SI unit is N – m2C-1.
The electric flux over any closed surface is 1 / εo times the total charge enclosed by that surface, i.e.,
CBSE Class 11 Physics Notes Electrostatics
If a charge q is placed at the centre of a cube, then
total electric flux linked with the whole cube = q / εo
electric flux linked with one face of the cube = q / 6 εo
(i) Electric Field at Any Point on the Axis of a Uniformly Charged Ring A ring-shaped conductor with radius a carries total charge Q uniformly distributed around it. Let us calculate the electric field at a point P that lies on the axis of the ring at distance x from its centre.
Ex = 1 / 4π εo * xQ / (x2 + a2)3/2
The maximum value of electric field
Ex = 1 / 4π εo (2Q / 3√3R2)
(a) At an extreme point (r > R)
V = 1 / 4π εo q / r
(b) At the surface of a shell (r = R)
V = 1 / 4π εo q / R
(c) At an internal point (r < R)
V = 1 / 4π εo q / R
Therefore potential inside a charged conducting spherical shell equal to the potential at its surface.
(iv) Electric Field and Potential due to a Charged Non-Conducting Sphere
(v) Electric Field Intensity due to an Infinite Line Charge
E = 1 / 2 π εo λ / r
where λ is linear charge density and r is distance from the line charge.
(vi) Electric Field Near an Infinite Plane Sheet of Charge
E = σ / 2 εo
where σ = surface charge density.
If infinite plane sheet has uniform thickness, then
E = σ / εo
An electric dipole consists of two equal and opposite point charges separated by a very small distance. e.g., a molecule of HCL, a molecule of water etc.
(i) On Axial Line
When electric dipole is parallel to electric field, it is in stable equilibrium and when it is anti-parallel to electric field, it is in unstable equilibrium.
Potential Energy
Potential energy of an electric dipole in a uniform electric field is given by U = – pE cos θ.
Dipole in Non-uniform Electric Field
Potential Energy of Charge System
Two point charge system, contains charges q1 and q2 separated by a distance r is given by U = 1 / 4 π εo * q1q2 / r
Three point charge system
U = 1 / 4 π ε_{o} * [q_{12} / r_{1} + q_{2}q_{3} / r_{2} + q_{3}q_{2} / r_{3}
Important Points
Behaviour of a Conductor in an Electrostatic Field
Electrostatic Shielding
Dielectric
Dielectrics are of two types Non-polar Dielectric The non-polar dielectrics (like N2, O2, benzene, methane) etc. are made up of non-polar atoms/molecules, in which the centre of positive charge coincides with the centre of negative charge of the atom/molecule.
Polar Dielectric
The polar dielectric (like H2O, CO2, NH3 etc) are made up of polar atoms/molecules, in which the centre of positive charge does not coincide with the centre of negative charge of the atom.
Capacitor
Capacitance of an Isolated Spherical Conductor
Parallel Plate Capacitor
Capacitors in series and parallel combinations
For practical applications , two or more capacitors are often used in combination and their total capacitance C must be known.To find total capacitance of the arrangement of capacitor we would use equation
Q=CV
(i) Parallel combination of capacitors
Q=Q1+Q2
=V(C1+C2) and
Q/V=C1+C2where,
When capacitors are connected in parallel their resultant capacitance C is the sum of their individual capacitances.
The value of equivalent capacitance of system is greater then the greatest individual one. If there are number of capacitors connected in parallel then their equivalent capacitance would be
C=C1+C2+ C3...........
Series combination of capacitors
V_{AR}=Q/C_{1}
and,
V_{RB}=Q/C_{2}
Sum of VAR and VRB would be equal to applied potential difference V so,
V=V_{AB}=V_{AR}+V_{RB}
=Q(1/C_{1} + 1/C_{2})
or,
where
i.e., resultant capacitance of series combination C=Q/V, is the ratio of charge to total potential difference across the two capacitors connected in series.
So, from equation 12 we say that to find resultant capacitance of capacitors connected in series, we need to add reciprocals of their individual capacitances and C is always less then the smallest individual capacitance.
Result in equation 12 can be summarized for any number of capacitors i.e.,