Electric Charges and Fields

Charge (q)

Charge is an entity possessed by an object
Two types - +ve and -ve

Methods to Create Charge
1) By Friction

Two different bodies rubbed against each other

when glass rod and silk are rubbed together,
glass rod => +ve
silk => -ve

when plastic and wool (or) fur are rubbed together,
wool => +ve
plastic => -ve

The ones that become positive have lower work functions than the other

2) By Induction

No direct contact

Step 1: Two metal spheres A and B kept in contact with a glass rod (+ve charged) kept next to A. -ve charge comes to A and +ve goes to B

Step 2: Separate the spheres, A is -ve charged and B is +ve charged

Step 3: Remove the glass rod, the charge is still intact

Properties of Charge

1) Additive Nature
2) Conservation of Charge
3) Quantisation of Charge

Additive Nature

qₙₑₜ = q₁ + q₂ + q₃ + ... + qₙ

The net charge of an isolated system equals the algebraic sum of all charges

Conservation of Charge

- Charge can neither be created nor destroyed
- The total charge of the isolated system is always conserved

Quantisation of Charge

q = ne

n -> integer
e -> charge of electron/proton = 1.6 × 10⁻¹⁹ C

1C of charge contains 1/e number of electrons = 6.25 × 10¹⁸

Q) Find number of positive and negative charge in x g of water

First find number of moles i.e given mass/molar mass

Then multiple by avagrado number = Number of molecules

Then use, q=ne where n is the number of molecules and e is 1.6 × 10⁻¹⁹ C

Note: Multiply e with the number of protons in the molecule, in this case, 10 (8 from the oxygen atom and 1 from each of the two hydrogen atoms)

Conductors and Insulators

Conductors
- Allows flow of charges
- Free electron
- Eg: Copper, Human Body, Earth

Insulators
- Doesn't allow flow of charges
- Bound electron
- Eg: Rubber, Plastic

Coulomb's Law

The force of attraction or repulsion between 2 point charges is directly proportional to the product of the charges and inversely proportional to the square of the distances between them.

F = Kq₁q₂/r²

Force Constant, K = 9 × 10⁹ Nm²C⁻²

K = 1/4πε₀

ε₀ = 8.85 × 10⁻1²

- Coulomb's law is only valid for point charges
- Electrostatic force is conservative (path independent)

Vector Form of Coulomb's Law

See cw

Force Between Multiple Charges

See cw

Force in a Medium

The ratio of force in medium to force in vacuum is known as relative permissivity (εᵣ)

ε = ε₀ × εᵣ

ε -> Permittivity of medium
ε₀ -> Permittivity of free space (8.85 × 10⁻¹²)
εᵣ -> Relative Permittivity

Electric Field

A space around the charge in which a test charge will experience a force of attraction or repulsion

Electric Field due to a Point Charge

F = Qq/4πr²ε₀ r̂

E = Q/4πr²ε₀ r̂

Electric Field Intensity

Force per unit charge

E = F/q₀

- Only depends on source charge (Q) and r
- Outward for +ve Q
- Inward for -ve Q

Electric Field due to Multiple Charges

see c.w

Continuous Charge Distribution

On a macroscopic level, quantisation of charge can be ignored and it can be distributed in 3 possible ways:

1) Linear Charge Distribution
2) Surface Charge Distribution
3) Volume Charge Distribution

Linear Charge Distribution

- Charge is distributed along the length of the material
- Linear charge density, λ = q/l C/m
- The electric field at a point due to a small element is given by,
dE= dq/4πr²ε₀
- Total electric field,
E = 1/4πr²ε₀ ∫ λdl

Surface Charge Distribution

σ = q/A C/m²
E = 1/4πr²ε₀∫σdA

Volume Charge Distribution

ρ = q/V C/m³
E = 1/4πr²ε₀∫ρdV

Electric Field Lines

An imaginary line drawn to represent the path of a unit +ve charge when kept in an electric field

1) Isolated +ve charge - Radially outward
2) Isolated -ve charge - Radially inward
3) Two like charges - cw
4) Two unlike charges - cw
5) In presence of metal - cw

Properties of Field Lines

- Originates at +ve charge and ends at -ve charge
- They will not intersect each other
- Tangent at any point on the field line gives direction of electric field
- They don't form closed loops
- Closer the lines stronger the field strength
- Field lines are continuous curves without a break for a charge free region

Electric Dipole

A pair of equal and opposite charges separated by a very small distance

p = q x 2a

Dipole Moment

- It is a vector quantity whose magnitude is given by the product of one of the charges and distance of separation
- The direction of the dipole moment is along -q to +q and distance is 2a

P = q × 2a Cm

Electric Field due to a Dipole
1) Axial Point

check cw

E = 2Kpr/(r²-a²)² p-cap

For a short dipole, a<<r

E = 2Kp/r³ p-cap

2) Equatorial Point

check cw

E = Kp/((r²+a²)³/² (-p) cap

For a short dipole, a<<r

E = Kp/r³ (-p) cap

Electric Flux

• It is the number of electric field lines crossing normally in a given area

• The rate of flow of liquid is given by the volume crossing the area per unit time represents the flux of liquid flowing across the plane

Φ = Escosθ

Electric Dipole

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

Dipole In A Uniform External Field

𝜏 = p × E

Continuous Charge Distribution

Surface Charge Density,

σ = q/A

Linear Charge Density,

λ = q/l

Volume Charge Density/Charge Density:

ρ = q/V

Gauss’s Law

The total electric flux over a closed surface equals 1/ε₀ times the net charge enclosed

Φ = q/ε₀

Applications Of Gauss’s Law

below flashcards

Field Due To An Infinitely Long Straight Uniformly Charged Wire

E = λ/2πε₀r

Field Due To A Uniformly Charged Infinite Plane Sheet

E = σ/2πε₀

Field Due To A Uniformly Charged Thin Spherical Shell

• Field outside the shell:

  E = Kq/r² = sigmaR²/epsilonnot r²

• Field inside the shell:

  E = 0

• Field on the surface of the shell:

  E = Kq/R² = sigma/epsilonnot