Electric charge and field

Electric charge:

• Intrinsic properties of particles

• It can be Positive or negative in nature.

• Electrically neutral: Amount of positive and negative charge are same on Particles.

• Electrically Charged: Amount of positive and negative charge are not same on Particles.

• Particles with the same nature repel each other.

• Particles with opposite nature attract each other.

Methods of charging:
1.By friction:

• Two bodies will be initially neutral.

• One body rubs with another.

• Finally Opposite charge will appear on both bodies.

2.By conduction:

• One will be charged and another neutral.

• Physical contact is required.

• Same types of charges will appear.

• Charge will transfer from one body to another, hence the sum of the charge on bodies will be constant.

3.Induction:

• One body will be charge and another will be neutral

• Initially a neutral body would be earthed.

• Keep both bodies closer. i.e: Physical contact is not required.

• Opposite charge induced on the neutral body.

• Charged body will not lose its charge.

Properties of electrical charge:
1.Quantization of charge:

• Charge on the body is the integer multiple of charge on an electron.
  i.e: $Q=n\times e$ where n=1,2,.........$e=1.6\times 10^{-16}$

2.Conservation of charge:

• Total charge on an isolated system is constant.

• Isolated system means: the system through its boundary change is not allowed to escape.

3.Additivity of charge:

• Total charge on the body is an algebraic sum of total charge located anywhere on it.

• While adding signs taken into consideration.

Q.Total charge on this body is
Q= 2+4+6-5= 7C

4.Charge is Invariant:

• Charge does not depend on the speed of the body.

Coulomb's law:
Force between point charge is proportional to the product of magnitude of charge and inversely proportional to square of distance between them.
ie, $F_e=\frac{K{q}_1{Q}_2}{r^2}$
Where, $K=\frac{1}{4\pi {ϵ}_0{ϵ}_r}$$K=9\times 10^{9}N{m}^2{C}^{-2}$${ϵ}_r=1$ in air
.Note:

• Applicable for point charge only.

• Not applicable for distance less then $10^{-15}$ m.

• Electrostatic force is consevative force.

• Comparatively stronger than gravitational force.

• Coulomb's law obey newton's third law.

Electric field:
Space around a charge in which its influence can be felt by any other charged particle.
Electric field intensity: Force experienced by test charge in presence of other charge particle.

• Test charge is always considered positive.

• Let a charged particle 'q' is experiencing a force 'F'

$E={\lim }_{q\to 0}\frac{F}{q}$
Note:

• Test charge is always considered small because large magnitudes may disturb the original charge distribution. And then we get electric field disturbed configuration.

• For our convenience we took a magnitude of test charge of 1 C.

Direction of electric field due to point charge:

• Positive point charge:

$E=\frac{KQ}{r^2}$ away from the charge

• Negative point charge

$E=\frac{KQ}{r^2}$ towards the charge

Note:

• Electric field due to point charge is spherically symmetrical.

• Force experienced by point charge 'q' in electric field 'E',

$\vec{F}=q\vec{E}$.

• Electric field due to system of charge, $\vec{E}={\vec{E}}_1+{\vec{E}}_2+{\vec{E}}_3+.........{\vec{E}}_n$

Lines of force:

• Imaginary lines

• Tangent at any point on this line gives direction to the electric field.

Properties of Electrical lines of forces:

• Originate from positive charge.

• Converge at negative charge.

• Numbers of lines originating or terminating from charge 'q' C is $q/{ϵ}_o$

• Two lines never cross each other.

• These lines can never be closed loops. ie: originating and converging at same point.

• From surface conductor electrical lines of force start or end normally.

• Tangent on it gives direction of electric field vector.

• Closer lines show more intensity points.

Point 2 will have more intensity in comparison to point 1.

Electric field from charged body

• Charged ring

• Infinite line charge

• Uniformly charged disc

• Oppositely charged sheet

Electric dipole:

• Two equal and opposite charges at small distance.

• Dipole moment $P=q\times 2l$

• Direction of dipole moment is from negative charge to positive charge.

• Unit of dipole moment 'Coulomb-metre'

Electric field intensity due to electrical dipole:

• Along axial line:

$\vec{E}=\frac{1}{4\pi {ϵ}_o}\frac{2\vec{P}}{r^3}$
Direction: Same as direction of dipole vector

• Along equatorial line:

$\vec{E}=\frac{1}{4\pi {ϵ}_o}\frac{\vec{P}}{r^3}$
Direction: Opposite of direction of dipole vector

• At any general point:

$\vec{E}=\frac{1}{4\pi {ϵ}_o}\frac{\vec{P}}{r^3}\sqrt{3{\cos }^2\theta +1}$
Direction:$\tan \varphi =\frac{1}{2}\tan \theta$

Net Force on dipole

• Due to point charge:

From newton's third law,
Force on dipole due to charge = Force on charge due to dipole.
Let, the electric field due to the dipole at a given point of charge is E.Force on charge 'q' due to dipole,
${\vec{F}}_q=q\vec{E}$
Hence, Force on dipole, ${\vec{F}}_d=-{\vec{F}}_q=-q\vec{E}$

• Due to uniform electric field:

There will no force experienced by the dipole.
$F_{net}=qE-qE=0N$

• Due to non-uniform electric field:

$|\vec{F}|=\left|P\frac{d\vec{E}}{dx}\right|$

• Due to another dipole:

$|{\vec{F}}_{12}|=\left|P\frac{d\vec{E}}{dx}\right|$
Where, $F_{12}$ = Force on dipole 1 due to 2.
$E$ = Electric field due to 2 nd dipole.

Torques on dipole:
$\tau =\vec{P}\times \vec{E}$

Electric flux

• Measure the amount of field line passing through the cross section.
• Electric flux $\varphi =\vec{E}\cdot d\vec{S}$
• Direction:
• Consider Negative: when flux inter to the surface.
• Consider Positive: when flux leaves the surface.

Gauss's law:

• Gives relation between electric field and charge.
• Useful to calculate electric fields due to symmetric charge distribution.

According to Gauss's law
Flux through any closed surface $\varphi =\frac{q_{inclosed}}{{ϵ}_o}$
Also, $\varphi =\varphi =\vec{E}\cdot d\vec{S}=\frac{q_{inclosed}}{{ϵ}_o}$
Where, $d\vec{S}$ Elementary area of gaussian surface.

Selection of Gaussian surface

Charge Distribution	Gaussian Surface	Electric field
Point charge	Spherical	Radial
Spherical charge	Spherical	Radial
Linear charge	Cylindrical	Radial
Planer charge	Planer, Parallel to charged distribution	Normal to the surface