If `q_1` and `q_2` be two stationary point charges in free space separated by a distance `r`, then the force of attraction or repulsion between them is given by
` F = (k|q_1| |q_2|)/r^2`

`=1/(4 pi epsilon_o) . (|q_1| |q_2|)/r^2 \ \ \ \ \ \ [ ∵ k = 1/(4 pi epsilon_o) ]`

` = ( 9 xx 10^9 xx | q_1 | | q_2 |)/r^2 \ \ \ \ \ \ [ ∵ 1/(4 pi epsilon_o) = 9 xx 10^9 ]`

The term `epsilon_o` is called the electric permittivity of free space having a value of `8.85 xx 10^(-12) C^2 N^(-1) m^( -2)`. Its dimensional formula is `[M^(-1) L^(-3) T^4 A^2 ]`.
If some dielectric medium is completely tilled between the given charges, then the Coulomb's force between them becomes
` F_m = 1/(4 pi epsilon) (q_1 q_2)/r^2 `

` = 1/(4 pi epsilon_0 epsilon_r) (q_1 q_2)/r^2 `

` = 1/(4 pi K epsilon_0 ) . (q_1 q_2)/r^2 `

`epsilon/epsilon_0 = epsilon_r` or ` K`

` = F_text(Free space)/K\ \ \ \ \ \ \ \ \ \ \ [ ∵ F_text(Free space) = (q_1 q_2)/(4 pi r^2 epsilon_0) ]`

Here , `epsilon_o =` absolute electric permittivity of the given medium, `K` dielectric constant and `epsilon_r` is the relative permittivity of the given medium.

`text(NOTE)` Force of two charges exerts on each other is not changed by the presence of a third charge.
Coulomb's force between two protons is `10^(36)` times the gravitational force between them.

According to the superposition principle, forces on any charge due to number of other charges is the vector sum of all the forces on that charge due to other charges, taken one at a time. The individual forces are unaffected due to the presence of other charges. Consider a system of n point charges `q_1, q_2, q_3 , ... , q_n` be distributed in space in a discrete manner. The charges are interacting with each other. Let the charges be `q_2, q_3, ... , q_n` exert forces `F_(12) , F_(13) , ... , F_(1n)`, respectively on charge `q_1`.
Then, according to the principle of superposition, the total force on charge `q_1` is given by
`F_1 = F_(12) + F_(13) + ... + F_(1n)` ... (i)
If the distance between the charges `q_1` and `q_2` is denoted as `r_(12)` and `hat r_(21)` is unit vector from charge
`q_2` to `q_1`, then

`F_(12) = 1/(4 pi epsilon_o) . ( q_1 q_2)/r_(21)^2 hat r_(21)`

Similarly, the force on charge `q_1` due to other charges is given by

` F_(13) = 1/(4 pi epsilon_o) . ( q_1 q_3)/r_(31)^2 hat r_(31)`

`F_(1n) = 1/(4 pi epsilon_o) . ( q_1 q_n)/r_(n1)^2 hat r_(n1)`

Substituting these values in Eq.(i), we get

` F_(1) = 1/(4 pi epsilon_o)( (q_1 q_2)/r_(21)^2 hat r_(21) + (q_1 q_3)/r_(31)^2 hat r_(31) + .... + (q_1 q_n)/r_(n1)^2 hat r_(n1))`

` F_(1) = q_1/(4 pi epsilon_o) sum_(i = 2)^n (q_i)/r_(i1)^2 hat r_(i1)`

♦ The interaction must be on the charge which is to be studied due to other charges.

The space surrounding an electric charge `q` in which another charge `q_o` experiences a force of attraction or repulsion is called the electric field of charge `q. ` Electric field vector E (also known as the electric field intensity) at any point is given by

` E = lim_(q_0 -> 0) F/q_0`

where, `q_o` is a small positive test charge which experiences a force `F` at a given point.

The SI unit of electric field is `NC^(-1)` and it is also known as `Vm^(-1)` . the dimensional formula for electric field is `[MLT^(-3) A^( -1) ]`.

`text (Electric Field Intensity)`

The electric field intensity at any point is the strength of electric field at that point . It is defined as the force experienced by unit positive charge placed at that point. If `F` is the force acting on a small test charge `+ q_o` at any point r, then electric field intensity at that point is given by

` E(r) = (F(r))/q_0`

The SI unit of electric field intensity is newton per coulomb (N/C).

`text (Electric Field Lines)`

An electric field line in general is a curve drawn in such a way that the tangent to it at each point is in the direction of the electric field at the point. A field line is a space curve , i.e. a curve in three - dimensions.

Electric field lines follow some important properties which are discussed below
`(i)` Electric field lines start from positive charges and end at negative charges. In the case of a single charge, they may start or end at infinity.
`(ii)` Tangent to any point on electric field lines shows the direction of electric field at that point.
`(iii) ` Two field lines can never intersect each other because, if they intersect, then two tangents drawn at that point will represent two-directions of field at that point, which is not possible.
`(iv)` In a charge free region, electric field lines can be taken to be continuous curves without any breaks.
`(v) ` Electric field lines do not form closed loops (because of conservative nature of electric field).
`(vi) ` Electric field lines are perpendicular to the surface of a charged conductor.
`(vii)` Electric field lines contract lengthwise to represent attraction between two unlike charges.
`(viii)` Electric field lines exert sideways pressure to represent repulsion between two like charges.
`(ix) ` The number `Delta N` of lines per unit cross-sectional area perpendicular to the field lines (i.e. density of lines of force) is directly proportional to the magnitude of the intensity of electric field in that region .
` (Delta N)/(Delta A) prop E`

Electric potential at a point in an electric field is defined at the amount of work done in bringing a unit positive charge, without any acceleration, from infinity to that point, along any arbitrary path. If `W` work is to be done to being a test charge `q_0` from infinity to a point, then the potential of that point, is

` V = W/q_0`

SI unit of potential is volt, where

` 1 V = (1 J)/(1 C)`

`text(Potential Difference)`

Potential difference between two points in an electric field is equal to the work done per unit charge in carrying a positive test charge from one point to the other point. Potential difference between two points A and B is given as, `Delta V = V_B - V_A = W/q_0` . Its SI unit is volt. It can also be joule/ coulomb.

Potential difference is that physical quantity, which decides the direction of flow of charge between two points in the electric field. Positive charge always tends to move from higher potential towards lower potential.

The surface drawn in an electric field, at which each point has same potential, is called equipotential surface.
• Equipotential surfaces are always normal to lines of force.
• The work done in moving a charge on equipotential surface is zero because potential difference is zero.
• Two equipotential surfaces never intersect each other. Electric field `(E)` and electric patential `(V)` are related as
`dV = - E· dr`
where, `dr` is small displacement.
• If `theta` is the angle between `dr` and `E`, then `dV = - E dr cos theta`

` text (Electric Potential Energy)`

Electric potential energy is a potential energy that results from conservative coulomb forces and is associated with the configuration of particular set of point charge within a defined system. Electric potential energy `(U) = (q_1 q_2)/( 4 pi epsilon _o r)`

Unit of electric potential energy is joule.

` text (Energy Density)`

Energy density is the amount of energy stored in a given system or region of space per unit volume or mass, through the latter is more accurately termed specific energy ` V_e = 1/2 epsilon_0 E^2`

An electric dipole consists of a pair of equal and opposite point charges separated by some small distance.

`text(Dipole Moment of an Electric Dipole)`

The strength of an electric dipole is measured by a vector quantity known as electric dipole moment `(p)` which is the product of the charge `(q)` and separation between the charges `(2 l)`.

i.e. ` p = q xx 2 l ` or ` | p | = q (2l )`

It is a vector quantity and its direction is always from negative charge to positive charge. The `SI` unit of dipole moment is coulomb-metre (`C-m`). lf charge `q` gets larger and the distance `2l` gets smaller and smaller, keeping the product `| P| = q xx 2l` = constant, we get what is called an ideal dipole or point dipole. Thus, an ideal dipole is the smallest dipole having almost no size .

Electric flux over an area is equal to the total number of electric field lines crossing through that area. Its `SI` unit is `Nm^2 C^(- 1)`.

electric flux ` Delta phi_E` crossing through a small area element `Delta S` is given by

` Delta phi_E = E Delta S cos theta`

where, E = electric field intensity

and ` Delta S =` area vector

`text(Special Cases)`

(i) For `0 ° < theta < 90 °, Delta phi_E` is positive.
(ii) For `theta = 90 °, Delta phi_E` is zero.
(iii) For `90° < theta < 180°, Delta phi_E` is negative.

The electric flux over any closed surface is `1/epsilon_0` times the total charge enclosed by that surface.

i.e. ` phi_E = (sum q)/epsilon_o`

where, `sum q =` total charge enclosed (i.e. sum of all the charges inside a closed surface). If total enclosed surface is zero, then total electric flux is zero.


`"Application Of Gauss Law : "`

`1.text( Electric field due to a point charge:)`

`ointvecE.vecA = oint E dA = E(4pir^2)`

`sumq_text(en) = Q`

From gauss law

`E(4pir^2) = Q/epsilon_0`

`E = Q/((4pi epsilon_0)r^2)`


`"2.Electric field due to uniformly charged spherical shell : "` (charge Q):

`text(Case 1)` `text(At external point)` (r > R)

`E(4pir^2) = Q/epsilon_0`

`E = Q/((4pi epsilon_0)r^2)`


`text(Case 2)` `text(At internal point)` (r < R)

`E(4pir^2) = 0/epsilon_0`

`E = 0`

`"3. Electric field due to uniformly charged spherical volume"` (charge Q):

`text(Case 1)` `text(At external point)` (r > R)

`E(4pir^2) = Q/epsilon_0`

`E = Q/((4pi epsilon_0)r^2)`

`text(Case 2)` `text(At internal point)` (r < R)

`E(4pir^2) = (Q/(4/3 pi R^3)(4/3 pi r^3))/epsilon_0`

`E = Q/(4 pi epsilon_0 R^3)r = rho/(3epsilon_0)r`

`text (Conductors)`
Conductors are the materials through which electric charge can flow easily. Most of the metals arc conductors of electric charge. Silver is the best conductor of electric charge.

`text(Insulators)`
Insulators are the materials through which electric charge cannot flow, glass, rubber, wood etc. Insulators are also called dielectrics, when an electric field is applied, induced charges appear on the surface of the dielectric. Hence, it can be said that dielectric are the insulating materials which transit electric effect without conducting.

`text(Dielectrics)`
In a dielectric under the effect of an external field, a net dipole moment is induced in the dielectric. Due to molecular dipole moments, a net charge appears on the surface of the dielectric.

These induced charges produce a field opposing the external field. Induced field is lesser in magnitude than the external field. So, field inside the dielectric gets reduced.
`E = | E_o | - | E_(i n) |`
where, `E =` resultant electric field in the dielectric
`E_0 =` external electric field between two plates and
`E_(i n)=` electric field inside the dielectric.
A net dipole moment is developed by an external field in either case, whether a polar or non-polar dielectric.


The ratio of the strength of the applied electric field to the strength of the reduced value of the electric field on placing the dielectric between the plates of a capacitor is called the dielectric constant of the dielectric medium. It is also known as relative permittivity or specific inductive capacity and is denoted by `K` (or `epsilon_r` ). Therefore, dielectric constant of a dielectric medium is given

` K = E_0/E`

♦ Note The value of K is always greater than 1 .

`text(Polarisation)` (P)

The induced dipole moment developed per unit volume in a dielectric slab on placing it in an electric field is called polarisation. It is denoted by P. If p is induced dipole moment acquired by an atom of the dielectric and N is the number of atoms per unit volume, then polarisation is given by `P = Np`

Capacitance of a conductor is the amount of charge needed in order to raise the potential of the conductor by unity. Mathematically, capacitance, `C = Q/V`

Electrical capacitance is a scalar. `SI` unit of capacitance is `1` farad `(1 F)`.

where, `1 F = (1 C)/(1 V)`

Capacitance of an solated spherical conductor (solid or hollow) is given by

`C = 4 pi epsilon_0 R = R/( 9 xx 10^9)`

`text(Capacitor)`

A capacitor is a device which stores electrostatic energy. Net charge on a capacitor is zero. However, ordinarily we talk in terms of charge on either plate of a capacitor and that is finite.

`text(Capacitance of Capacitor)`

Capacitance of capacitor is the ratio of charge `(q)` and applied potential `( V)` i.e. ` C = q/V`

`"Parallel Plate Capacitor:"`


`"Parallel Plate Capacitor:"`
A parallel-plate capacitor consist of two large plane metallic plates A and B each of area A separation `d`. Plate A is positively charged and plates B is earthed. If K is the dielectric constant of the material medium and E is the field that exist between the two plates, then

`E = sigma/epsilon = sigma/(kepsilon_0)`

`V/d = q/(Kepsilon_0A),` `{ E = V/d `and` sigma= q/A }`

`C=q/V = (Kepsilon_0A)/d`

If medium between the plates is air or vaccume , then K=1

`C_0 = epsilon_0A/d`



`text(Spherical Capacitor Capacitance)`

A spherical capacitor consists of a solid or hollow spherical conductor surrounded by another concentric hollow spherical conductor.
let's consider a spherical capacitor which consists of two concentric spherical shells of radii `a` and `b`, as shown in Figure. The inner shell has a charge `+Q` uniformly distributed over its surface, and the outer shell an equal but opposite charge `-Q`

The potential difference between the two conducting shells is :

`DeltaV = V_b - V_a = -int_a^b E_r dr = - Q/(4pi epsilon_0)int_a^b dr/r^2 = - Q/(4pi epsilon_0)(1/a - 1/b) `

`= - Q/(4pi epsilon_0)(b-a)/(ab)`

which yields

`C = Q/|DeltaV| = 4pi epsilon_0 (ab)/(b-a)`

Again, the capacitance C depends only on the physical dimensions, a and b. An " isolated" conductor(with the second conductor placed at infinity) also has a capacitance. In the limit where `b -> oo,` the above equation becomes.

`lim_(b->oo) C = lim_(b->oo) 4pi epsilon_0 (ab)/(a-b) = lim_(b->oo) 4pi epsilon_0 a/(1-a/b) = 4 pi epsilon_0a`

Thus, for a single isolated spherical conductor of radius R, the capacitance is

`C = 4pi epsilon_0R`


(i) When external sphere is connected to earth then,

`C = 4 pi epsilon_o ( (ab)/(b - a))` [In air]

`C = ( 4 pi epsilon_o K ab)/(b - a) ` [In medium]

(ii) When internal sphere is connected to earth, then

` C = (4 pi epsilon_o b^2)/( b - a)` [In air]

`C = ( 4 pi epsilon_o K ab)/(b - a)` [In medium]

There are two common methods of grouping of capacitors

`text(1. Series Grouping)`

In a series arrangement, the charge on each plate of each capacitor has the same magnitude, equal to the charge supplied by the battery. The potential difference is distributed inversely in the ratio of capacitors,

i,e. `V = V_1 + V_2 + V_3 + ...`
and `V_1 : V_2 : V_3 ... = 1/C_1 : 1/C_2 : 1/C_3 : ...`

The equivalent capacitance `C_s` is given by

`1/C_s = 1/C_1 + 1/C_2 + 1/C_3 + ...`

` = sum_(i = 1)^( i = n) 1/C_i`

`text(2. Parallel Grouping)`

In a parallel arrangement, the potential across each of the capacitor is, exactly same. Charges on different capacitors are different. In fact, the charge is distributed in the ratio of capacitance,
i.e. ` Q = Q_1 + Q_2 + Q_3 + ...`
and `Q_1 : Q_2 : Q_3 ... = C_1 : C_2 : C_3 ......`
The equivalent capacitance is given by

`C_p = C_1 + C_2 + C_3 + ... = sum_(i = 1)^(i = n) 1/C_i`

The energy of a charged capacitor is measured by the total work done in charging the capacitor to a given potential.
Let us assume that initially both the plates are uncharged.
Now, we have to repeatedly move small positive charges from one plate and transfer them to the other plate. Now,
when an additional small charge (`dq`) is transferred from one plate to another plate, then the small work done is given by

`dW = V' dq = (d')/C dq`

[ `∵` charge on plate when `dq` charge is transferred be q' ]
The total work done in transferring charge Q is given by

`W = int_0^Q (q')/C dq = 1/C int_0^Q q' dq = 1/C [ (q')^2/2 ]_0^Q = Q^2/(2C)`

This work is stored as electrostatic potential energy U in the capacitor.

`U = Q^2/(2C) = (CV)^2/(2C) = 1/2 CV^2 \ \ \ \ \ \ \ \ \ [ ∵ Q = CV ]`

The energy stored per unit volume of space in a capacitor is called energy density.

Energy density, `U = 1/2 epsilon_0 E^2`

Total energy stored in series combination or parallel combination of capacitors is equal to the sum of energies stored in individual capacitors.

i.e. `U = U_1 + U_2 + U_3 + ...`