1. ELECTRIC CHARGES AND FIELDS

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1. ELECTRIC CHARGES AND FIELDS

1. Electric charge: Electric charge is a property possessed by matter that makes it possible for them to exert electrical force and to respond to electrical force.

Note: SI unit of charge is coulomb (C).

Electrification: It is the process of charging a body Types of Electrification (methods of charging)

1. Charging by friction:

When two bodies of different material are rubbed against each other, one body loses electrons and other body gains electrons. The body which loses electrons becomes positively charged and the body which gains electrons becomes negatively charged. The opposite charges acquired by the two bodies are equal in magnitude.

2. Charging by conduction:

An uncharged conductor can be charged by touching it with a charged body. The charge acquired by the uncharged body is of same nature as that of the charging body

3. Charging by Induction:

When a charged body is brought near an uncharged conductor, charges of opposite polarity are induced on the nearer surface of the conductor & equal amount of charges of same polarity are induced on the farther surface of the conductor. By connecting the farther surface to the ground charge on the body can be retained.

Properties of charges:

1. Additivity of charges: Total charge of a system is the algebraic sum of all individual charges in the system.

2. Quantization of charge: Total charge (q) of a body is always an integral multiple of basic unit of charge (e). Therefore, the charge q on a body is always given by 𝑞 = ±𝑛𝑒 where n = 1, 2, 3... e - magnitude of charge on one electron or proton = 1 .6 x 10^-19 C

3. Conservation of charge: The total charge of an isolated system is always conserved.

i.e. Charge can neither be created nor be destroyed

4. Like charges repel each other and unlike charges attract each other.

5. Electric charge is a scalar quantity.

6. The magnitude of charge on a body is independent of the speed of the body Note: 1. Mass can exist without charge but charge cannot exist without mass.

2. Charge density is more at curved (or sharp) region of a conductor.

3. There are 6.25 x 10¹⁸ electrons in -1C.

Note: Gold leaf electroscope is used to detect and measure charge.

Note: Charge can be distributed on a body in three ways.

1. Charge distributed uniformly on unit length is called linear charge density. It is given by the expression 𝜆 = ^𝑞

𝑙 . Its unit is Cm^-1.

2. Charge distributed uniformly on unit area is called surface charge density. It is given by expression 𝜎 = ^𝑞

𝐴. Its unit isCm^-2.

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3. Charge distributed uniformly in unit volume is called volume density of charge. It is given by the expression 𝜌 = ^𝑞

𝑉. Its unit is Cm^-3. 2. Coulomb's Law:

Statement: The electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them & and it acts along the line joining the two charges.

Explanation: Consider two point charges q1 and q2 separated by a distance r in vacuum. Then the magnitude of force (F) between them is given by

𝐹𝛼 |𝑞₁ 𝑞₂| 𝑎𝑛𝑑 𝐹 𝛼 ¹

𝑟²

𝐹 𝛼

^|𝑞¹^𝑞²^|

𝑟²

OR

𝐹 =

^{𝐾 |𝑞}¹^𝑞²^|

𝑟²

Where K is a constant of proportionality and is given by K = ¹

4𝜋𝜀_𝑜 = 9 x 10⁹ for free space Where, 𝜀_𝑜 = 8.854x 10^-12 C²N^-1m^-2 called - absolute permittivity or permittivity of free space

∴ 𝐹 =

¹

4𝜋𝜀_𝑜

|𝑞₁ 𝑞₂| 𝑟²

Coulomb's La w in vector form:

The force on q2 due to q1 is

𝐹⃗

₂₁

=

¹

4𝜋𝜀_𝑜

|𝑞₁ 𝑞₂| 𝑟₂₁² 𝑟̂₂₁ The force on q1 due to q2 is

𝐹⃗

₁₂

=

¹

4𝜋𝜀_𝑜

|𝑞₁ 𝑞₂|

𝑟₁₂²

𝑟̂

₁₂

= - 𝐹⃗

₂₁

Thus Coulomb's law agrees with Newton's third law.

Note: 1. Force between two like charges is repulsive and the product q1q2> 0 i.e, either both positive and both negative.

2. Force between two unlike charges is attractive and the product q1q2 < 0 , i.e., one charge is positive & other is negative

3. Definition of 1C: One Coulomb is that charge which when placed at a distance of 1m from another charge of the same magnitude in vacuum experiences an electrical force of repulsion of magnitude 9 x 10⁹ N

4. Principle of superposition: Force on any charge due to number of other charges is the vector sum of all the forces on that charge due to the individual charges.

Forces between Multiple charges

Consider a system of n charges q1, q2, q3.... qn placed in vacuum. Let r1, r2, r3, ... rn be their position vectors w.r.t. origin.

According to principle of superposition, net force on q1 is 𝐹⃗₁ = 𝐹⃗₁₂+ 𝐹⃗₁₃+ 𝐹⃗₁₄ … … . + 𝐹⃗_1𝑛

Where 𝐹⃗₁₂, 𝐹⃗₁₃, 𝐹⃗₁₄ are forces exerted on q1 by individual charges

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3 q1, q2, q3.... qn

From Coulomb's law, force exerted on q1 by q2 is given by 𝐹⃗₁₂ = ¹

4𝜋𝜀𝑜

|𝑞₁ 𝑞₂| 𝑟122 𝑟̂₁₂

Similarly, force exerted on q1 by q3 is given by 𝐹⃗₁₃= ¹

4𝜋𝜀𝑜

|𝑞₁ 𝑞₃| 𝑟132 𝑟̂₁₃ Net force ∴ 𝐹⃗₁ = ¹

4𝜋𝜀𝑜

|𝑞₁ 𝑞₂|

𝑟122 𝑟̂₁₂+ ¹

4𝜋𝜀𝑜

|𝑞₁ 𝑞₃|

𝑟132 𝑟̂₁₃+ … … … . . + ¹

4𝜋𝜀𝑜

|𝑞₁ 𝑞_𝑛| 𝑟1𝑛2 𝑟̂_1𝑛 In general

𝐹⃗

₁

=

^𝑞

4𝜋𝜀_𝑜

∑

^𝑞^𝑖

𝑟_1𝑖²

𝑟̂

_1𝑖

𝑛𝑖=2

3. Electric field

Electric field (or Electric field intensity) at a point is the force exerted on unit positive charge kept at that point.

Electric field, 𝐸 = ^𝐹

𝑞 where F- force on charge q

Note: 1. Electric field is a vector quantity. Its direction is always from a positive charge and is directed towards a negative charge.

2. SI unit of electric field is NC^-1 or Vm^-1

3. Dimensional formula of Electric field is [MLA^-1 T^-3]

4. Force experienced by a charge q in an electric field E is 𝐹 = 𝑞𝐸. Hence acceleration produced in the charge is 𝑎 = ^𝑞𝐸

𝑚

5. Electric field is a conservative field i.e. work done in moving a charge in this field is independent of the path, which is similar to gravitation field.

Electric field due to a point charge:

Consider a point charge +Q placed at O.

Another small positive charge q is placed at P at a distance r from O.

The force acting on q due to +Q is 𝐹 = ¹

4𝜋𝜀𝑜 𝑄𝑞

𝑟² r

Since, electric field 𝐸 = ¹

4𝜋𝜀𝑜 𝑄𝑞

𝑟² x ¹

𝑞 +Q q

∴ 𝐸 =

¹

4𝜋𝜀_𝑜 𝑄 𝑟²

In vector form,

𝐸⃗⃗ =

¹

4𝜋𝜀_𝑜 𝑄 𝑟²

𝑟̂

where

𝑟̂

is a unit vector in the direction of

𝐸⃗⃗

. Note: Small positive charge q is called test charge.

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4 Electric field due to a system of charges:

Consider a point P. Let q1, q2 and q3 be the charges kept at distances of r1, r2, r3 from P respectively.

Let r1, r2, r3 be the unit vectors from q1 to P, q2 to P, and q3 to P respectively The electric field at P due to q1 is ,

𝐸⃗⃗

= ¹

4𝜋𝜀_𝑜 𝑞₁

𝑟₁²

𝑟̂

₁

Electric field at P due to q2 is

𝐸⃗⃗

= ¹

4𝜋𝜀_𝑜 𝑞₂

𝑟₂²

𝑟̂

₂

Electric field at P due to qís

𝐸⃗⃗

= ¹

4𝜋𝜀_𝑜 𝑞₃

𝑟₃²

𝑟̂

₃

Resultant field at P is

𝐸⃗⃗

=

𝐸⃗⃗

₁+

𝐸⃗⃗

₂ +

𝐸⃗⃗

₃ = ¹

4𝜋𝜀_𝑜

[

^𝑞¹

𝑟₁²

𝑟̂

₁+ ^𝑞²

𝑟₂²

𝑟̂

₂ + ^𝑞³

𝑟₃²

𝑟̂

₃

]

OR in general

𝐸⃗⃗ =

¹

4𝜋𝜀_𝑜

[∑

^𝑞^𝑖

𝑟_𝑖²

𝑟̂

_𝑖

𝑛𝑖=1

]

4. Electric dipole

An electric dipole is a pair of equal and opposite point charges +q and -q, separated by a distance 2a.

Note: Net charge of a dipole is zero Electric Dipole moment (p):

The product of magnitude of one of the charges and the distance between the charges is called electric dipole moment.

i.e. p =2aq

Note: 1. Electric dipole moment is a vector quantity. Its direction is from negative to positive charge. In vector form 𝑝⃗ = 𝑞 2𝑎 𝑝̂

2. Its SI unit is Cm (Coulomb-metre)

Expression for electric field at a point on the axis of an electric dipole (or axial line)

Consider an electric dipole consisting of charges -q and +q, separated by a distance 2a and placed in vacuum

Let P be a point on the axial line at a distance r from centre O of the dipole.

The electric field at P due to charge +q is

𝐸⃗⃗

_+𝑞

=

¹

4𝜋𝜀_𝑜 𝑞 (𝑟−𝑎)²

𝑝̂

The electric field at P due to charge –q is

𝐸⃗⃗

_−𝑞

=

¹

4𝜋𝜀_𝑜 𝑞

(𝑟+𝑎)²

(−𝑝̂) 𝐸⃗⃗

_−𝑞

=

⁻¹

4𝜋𝜀_𝑜 𝑞 (𝑟+𝑎)²

𝑝̂

The resultant electric field at P is 𝐸 = 𝐸_+𝑞+ 𝐸_−𝑞

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𝐸⃗⃗ =

¹

4𝜋𝜀_𝑜 𝑞

(𝑟−𝑎)²

𝑝̂ −

¹

4𝜋𝜀_𝑜 𝑞

(𝑟+𝑎)²

𝑝̂

𝐸⃗⃗ =

^𝑞

4𝜋𝜀_𝑜

[

¹

(𝑟−𝑎)²

−

¹

(𝑟+𝑎)²

] 𝑝̂

𝐸⃗⃗ =

^𝑞

4𝜋𝜀_𝑜

[

^(𝑟+𝑎)²^{(𝑟−𝑎)}²

(𝑟−𝑎)²(𝑟+𝑎)²

] 𝑝̂

𝐸⃗⃗ =

^𝑞

4𝜋𝜀_𝑜

[

^𝑟²^{+ 𝑎}²^{+ 2𝑟𝑎− 𝑟}²^{− 𝑎}²^{+ 2𝑟𝑎}

(𝑟²−𝑎²)²

] 𝑝̂

𝐸⃗⃗ =

^𝑞

4𝜋𝜀_𝑜

[

^4𝑟𝑎

(𝑟²−𝑎²)²

] 𝑝̂

Since

𝑝⃗

= 2aq 𝑝̂

𝐸⃗⃗ =

¹

4𝜋𝜀_𝑜

2𝑟𝑝⃗⃗

(𝑟²−𝑎²)²

𝑝̂

At very large distance from the dipole, i.e. r >> a, r² >> a², (𝑟²− 𝑎²)² ≈ 𝑟⁴

∴ 𝐸⃗⃗ =

¹

4𝜋𝜀_𝑜 2𝑝⃗⃗

𝑟³

𝑝̂

Expression for electric field at a point on the equatorial line of an electric dipole

Consider an electric dipole consisting of charges -q and +q, separated by a distance 2a and placed in vacuum

Let P be a point on the equatorial line of the dipole at a distance 'r’ from the centre O' of the dipole

The magnitude of the electric field a P due to the charge +q and -q are

given by

𝐸

_+𝑞

=

¹

4𝜋𝜀_𝑜 𝑞

(𝑟²+ 𝑎²)

= 𝐸

₁

𝐸

_−𝑞

=

¹

4𝜋𝜀_𝑜 𝑞

(𝑟²+ 𝑎²)

= 𝐸

₂

The components perpendicular to the dipole axis ( 𝐸₁sin 𝜃) and (𝐸₂ sin 𝜃 ) cancel out as they are same magnitude and opposite in direction.

The component parallel to the dipole axis (𝐸₁cos 𝜃) and (𝐸₂cos 𝜃 ) add up

𝐸⃗⃗ = ( E₁ cos 𝜃 + E₂ cos 𝜃 )(– p̂) (Here ‘ - ve' symbol indicates E and p are opposite)

𝐸⃗⃗ = [

¹

4𝜋𝜀_𝑜 𝑞

(𝑟²+ 𝑎²)

+

¹

4𝜋𝜀_𝑜 𝑞

(𝑟²+ 𝑎²)

] cos 𝜃 𝑝̂

From the figure

cos 𝜃 =

^𝑎

(𝑟²+ 𝑎²)¹^⁄²

𝐸⃗⃗ = −2 [

¹

4𝜋𝜀_𝑜 𝑞

(𝑟²+ 𝑎²)

]

^𝑎

(𝑟²+ 𝑎²)¹^⁄²

𝑝̂

𝐸⃗⃗ =

⁻¹

4𝜋𝜀_𝑜

2𝑎𝑞

(𝑟²+ 𝑎²)³^⁄²

𝑝̂

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𝐸⃗⃗ =

⁻¹

4𝜋𝜀_𝑜

𝑝

(𝑟²+ 𝑎²)³^⁄²

𝑝̂

At large distance, r >>a, (𝑟

²

+ 𝑎

²

)

³^⁄²

≈ (𝑟

²

)

³^⁄²

= 𝑟

³

∴ 𝐸⃗⃗ =

⁻¹

4𝜋𝜀_𝑜 𝑝 𝑟³

𝑝̂

Dipole in a uniform external field:

An electric dipole placed in a uniform external electric field, it experiences a torque Expression for torque experienced by an electric dipole in a uniform electric field:

Consider a dipole of dipole moment P in a uniform external electric field E. The force acting on +q and -q are q

𝐸⃗⃗

and -q

𝐸⃗⃗

Two equal and opposite forces acting at different points, hence a torque acts on the dipole.

Torque = Force x perpendicular distance Torque 𝜏 = (𝐹)(𝐴𝐶)

𝜏 = (𝑞𝐸)(𝐴𝐶) --- (1) From Δ ABC, sin 𝜃 = ^𝐴𝐶

𝐴𝐵

Therefore AC = AB sin 𝜃 = 2a sin 𝜃 --- (2) Substituting (2) in (1),

𝜏 = (𝑞𝐸)(2a sin 𝜃 ) 𝜏 = (2𝑎𝑞)(E sin 𝜃 ) 𝜏 = 𝑝𝐸 𝑠𝑖𝑛𝜃

In vector form 𝜏 = 𝑝 ⃗⃗⃗⃗ ×

𝐸

⃗⃗⃗

Note: 1.If an electric dipole is kept in a uniform electric field then net force on it is zero 2. Torque turns the dipole to align p in the direction of electric field.

3. Torque is maximum (τ = PE) When p is perpendicular E i, e. θ = 90°

And Torque is minimum ( τ=0) when p is parallel to E i.e. θ = 0° or θ = 180°

5. Electric Field lines

An electric field line is a line such a way that the tangent to it at any point gives the direction of the net electric field at that point. OR

An electric field lines are imaginary path along which a unit positive charge tends to move.

Properties of electric field lines:

1. Field lines starts from positive charge and end at negative charge 2. Two field lines never intersect each other

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7 3. Electric field lines do not form any closed path

4. Electric field lines are absent within a charged conductor

5. Electric field lines are always normal to the surface of charged conductor Pictorial representation of electric field:

6. Electric Flux

The number of electric field lines of force passing normally through a given surface is called the electric flux.

Flux is a measure of flow of electric field through a given surface.

If Δ∅ is the electric flux through an area ΔS around a point where E is the electric field.

Then

∆∅ = 𝐸⃗⃗. ∆𝑆⃗ = 𝐸 ∆𝑆 cos 𝜃

where 𝜃 is the angle between area vector and electric field vector.

Note: 1. Electric flux is a scalar quantity.

2. SI unit of electric flux is Nm² C^-1

3. When 𝜃 = 0 [∆∅ = 𝐸 ∆𝑆 ] i.e. electric flux is maximum when the plane of the surface is normal to the direction of the electric field.

4. When 𝜃 = 90° [∆∅ = 0 ] i.e, electric flux is zero or minimum when the plane of the surface is parallel to the direction of the electric field

7. Gauss's Law

Statement: The total electric flux through a closed imaginary surface is ¹

𝜀0 times the total charge enclosed by the surface.

If q is the total charge enclosed by the closed surface, then the total electric flux through the surface is

∅ = ¹

𝜀0 (q) , where 𝜀₀ - permittivity of free space.

Expression for electric field at a point due to an infinitely long straight uniformly charged wire:

Consider a straight conducting wire of infinite length of uniform linear charge density λ. Let P be a point at a distance r from the wire. Let E be the electric field at P

Imagine a Gaussian cylinder of radius r and length l about the wire.

Electric field is parallel to end circular faces and perpendicular to curved face Electric flux through the end circular faces is zero.

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8 Total electric flux Gaussian surface

∅ = flux through curved faces + flux though end circular faces ∅ = flux through curved faces

∴ ∅ = ∑ 𝐸 ∆𝑆 = 𝐸 ∑ ∆𝑆

∴ ∅ = 𝐸 (2𝜋𝑟𝑙) --- ( 1 ) where ∑∆𝑆 = 2𝜋𝑟𝑙 , area of curved surface According to Gauss's law

Total electric flux,

∅ =

¹

𝜀₀

(q)

OR

∅ =

¹

𝜀₀

λl

--- ( 2 ) From (1) and (2)

𝐸(2𝜋𝑟𝑙 ) =

¹

𝜀₀

λl ⇒ 𝐸 =

^λ

2𝜋𝑟𝜀₀

Expression for electric field due to uniformly charged infinite plane sheet:

Consider a plane conducting sheet of infinite area of uniform surface charge density 𝜎. Let P be a point at a distance r from the sheet. Let E be the electric field at P.

Imagine a Gaussian rectangular parallelepiped of cross-sectional area A with end surfaces at equal distance r from the sheet.

Electric field is parallel to surface is in x-direction and perpendicular to the surfaces are in the y and z- directions. Therefore electric flux though surface along y and z-direction is zero.

Therefore total electric flux ∅ = flux through the surface along x-directio

∅ = ∑ 𝐸∆𝑆 = 𝐸 ∑ ∆𝑆

∴ ∅ = E (2A) --- ( 1 )

where ∑∆𝑆 = 2𝐴, total area of the end surfaces.

According to Gauss's law Total electric flux,

∅ =

¹

𝜀₀

(q)

OR

∅ =

¹

𝜀₀

σA

--- ( 2 ) From (1) and (2)

E (2A) =

¹

𝜀₀

σA

∴ 𝐸 =

¹

2𝜀₀

σ

Expression for electric field at any point due to charged spherical shell:

Consider a conducting spherical shell of radius R carrying a charge +Q. The electric field at different points can be found as follows:

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9 1) At a point outside the surface:

Let P be a point at a distance r from O. Consider a Gaussian sphere with centre O and radius r The magnitude of E is same at all points on the Gaussian sphere.

Through the Gaussian surfaceTotal electric flux

∅ = ∑ 𝐸∆𝑆 = 𝐸 ∑ ∆𝑆 = 𝐸 ( 4𝜋𝑟² ) --- ( 1 ) According to Gauss law

Total electric flux,

∅ =

¹

𝜀₀

(q)

--- ( 2 ) From (1) and (2)

𝐸 ( 4𝜋𝑟

²

) =

¹

𝜀₀

(q)

∴ 𝐸 =

¹

4𝜋𝜀₀

^q

r²

2) At any point inside the sphere:

Let P be a point inside the sphere at a distance r from O. Consider a Gaussian sphere with centre O and radius r.

Gaussian surface does not enclose any charge, because charges are present on the shell.

Through the Gaussian surface total electric flux

∅ = ∑ 𝐸∆𝑆 = 𝐸 ∑ ∆𝑆 = 𝐸 ( 4𝜋𝑟² ) --- ( 1 ) According to Gauss law

Total electric flux, ∅ = ¹

𝜀0 (q) = 0 --- ( 2 ) From (1) and (2)

𝐸 ( 4𝜋𝑟² ) = 0 ⇒ E = 0

Electric filed does not exist inside spherical shell.