## Chapter : 1

## What Are Electric Charges and Fields ?

## Electric Charges

Charge is the property associated with matter due to which it produces and experiences electrical and magnetic effects. There are mainly two types of charge : positive charge and negative charge. Like charges repel each other whereas unlike charges attract each other. The basic unit of charge is coulomb (C).

- • Charge is a scalar quantity.
- • Charge is transferable.
- • Charge is quantized.
- • Charge is invariant i.e., independent of the frame of reference.

## Conservation Of Charge

Total charge of an isolated system remains unchanged with time. In other words, charge can neither be created nor destroyed. Conservation of charge is found to hold good in all types of reactions either chemical or nuclear.

## Coulomb’s Law

F =(kq_{1}q_{2}/r^{2}), where k =(1/4πε_{0})=9×10^{9}Nm^{2}C^{-2} is a proportionality constant

and ε_{0} = 8.854 × 10^{-12} C^{2} N^{-1} m^{-1} is permittivity of free space.

• Coulomb force and gravitational force follow the same inverse square law.

• Coulomb force can be attractive or repulsive whereas gravitational force is always attractive.

• Coulomb force is a central force as it acts along a line joining the two point charges.

• Coulomb force is a conservative force.

## Superposition Principle

Total force on any charge due to a number of other charges, is the vector sum of the forces exerted on it by all other charges.

$$\text{F}=\text{K}\frac{\text{q}_0\text{q}_1}{\text{r}^2_1}+\text{k}\frac{\text{q}_0\text{q}_2}{\text{r}^2_2}+——–+\text{k}\frac{\text{q}_0\text{q}_\text{n}}{\text{r}^2_\text{n}}\space\text{or} \space\text{F}=\text{kq}_0 \displaystyle\sum_{\text{i}=1}^n\frac{\text{q}_\text{i}\hat {\text{r}}_\text{i}}{\text{r}^2_1}$$

## Continuous Charge Distribution

**Linear Charge Density:** Charge per unit length is known as linear charge density.

λ**=**(Charge/Length)

**Surface Charge Density:** Charge per unit area is known as surface charge density.

σ =(Charge/Area)

• Unit of λ is Cm^{-1} whereas unit of σ is Cm^{-2}.

## Electric Field

The space around the charge in which electrostatic force of attraction or repulsion due to the charge can be experienced by any other charge. Electric field intensity at any point is the strength of electric field at that point. It can be defined as the force experienced by unit positive charge placed at that point.

$$\text{E}=\frac{\vec{\text{F}}}{\text{q}_0}, \text{where} \space\vec{\text{F}} \space\text{is the force acting on the test charge q}_0.$$

## Electric Field Due To A Point Charge

Electric field due to a point charge is the three-dimensional space around the charge in which its influence (force) can be experienced by a unit positive test charge placed at any point in that region.

$$\text{E}= \lim_{\text{q}_{0} \to 0} \frac{\vec{\text{F}}}{\text{q}_0}=\frac{1}{4\pi \epsilon_0}.\frac{\text{q}}{|\vec{\text{r}}|^2}\hat r $$

_{0}is a test charge which is fictitious charge that exerts no force on nearby charges but experiences force due to them.

## Electric Field Lines

Electric field lines are the pictorial mapping of the electric field around a configuration of charges. Electric field line is a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point.

Electric field due to a positive point charge is represented by straight lines originating from the charge.

Electric field due to a negative point charge is represented by straight lines terminating at the charge.

• Electric field lines are purely geometrical construction. They have no physical existence.

• Field lines start from positive charges and end at negative charges.

• Two field lines can never intersect each other.

• Electric field lines do not form any closed loops.

## Electric Dipole And Dipole Moment

An electric dipole is a system formed by two equal and opposite point charge placed at a small distance apart. Electric dipole moment is a vector quantity whose magnitude is equal to the product of the magnitude of either charge and distance between the charges.

$$|\vec{\text{p}}| = \text{q} × 2\text{a}$$

• By convension, the direction of dipole moment is from negative charge to positive charge.

• The SI unit of dipole moment is Cm.

## Electric Field Due To A Dipole

Electric field due to a dipole is the space around the dipole in which the electric effect of the dipole can be experienced.

**Electric field at a point on the axial line of a dipole (End-on-position)**

_{0}) (2pr/(r

^{2}-a

^{2})

^{2})

At very large distance i.e., r >> a ,

E = (2p/4πε_{0}r^{3})

• The direction of the electric field on axial line of the electric dipole is along the direction of the dipole moment (i.e., from –q to q)

**Electric field at a point on the equatorial line of a electric dipole ****(Broadside-on-position)**

At a distance r from the centre of the electric dipole, E=(1/4πε_{o}) (P/r^{2}+a^{2})^{3/2}

At very large distance i.e., r >> a,

E =(1/4πε_{0})(P/r^{3})

The direction of electric field on equatorial line of the electric dipole is opposite to the direction of the dipole moment (i.e., from q to -q)

## Torque On A Dipole In Uniform Electric Field

When a dipole is placed in a uniform external electric field, then net force (+qE –qE) on it is zero, but it experiences a torque,

$$\vec τ=\vec{\text{p}}×\vec{\text{E}} =\text{pE sin}\space\theta$$

where θ is the angle between p and E.

• Torque acting on a dipole is maximum when dipole is perpendicular to the field, i.e., θ = 90°

∴ τ _{max} = pE sin 90° = pE

• Torque acting on a dipole is minimum when dipole is parallel or antiparallel to the field, i.e., θ = 0° or 180°

∴ τ_{min} = pE sin 0° = 0

## Electric Flux

Electric flux over an area in an electric field is a measure of the number of electric field lines crossing this area. Electric flux linked with small area element on the surface of the body.

$$\text{d}\phi =\vec{\text{E}}.\text{d}\vec{\text{s}}$$

$$\text{where}\space \text{d}\vec{\text{s}}\space \text{is the area vector of the small area element. Total flux linked with whole body will be}$$

$$\phi=\oint\vec{\text{E}}\space.\space \text{d}\vec{\text{s}}=\oint \text{Eds} \space \text{cos}\space \theta$$

• Electric flux is a scalar quantity.

• Electirc flux over a closed surface can be positive, negative or zero for θ<90°, θ > 90° or θ = 90° respectively.

## Gauss Theorem

The total flux linked with a closed surface is (1/ε_{0})

times the charge enclosed by the closed surface (Gaussian surface) i.e.,

$$\oint \vec{\text{E}}.\text{d}\vec{\text{s}}=\frac{\text{q}}{\epsilon_0}$$

• This theorem is suitable for symmetrical charge distribution and valid for all vector fields obeying inverse square law.

## Application Of Gauss Theorem To Find Field

## Due To Infinitely Long Straight Wire

_{0}r)

where r is the perpendicular distance of the observation point from the wire.

• The variation of E with r for a uniformly charged infinitely long straight wire is as shown in the graph.

## Application Of Gauss Theorem To Find Field

## Due To Uniformly Charged Plane Sheet

Electric field due to an infinitely thin plane sheet of uniform surface charge density σ is E = (σ/ε_{0})

Here, E is independent of the distance of the point from the sheet i.e., r.

- Variation of E with r for a uniformly charged infinitely thin plane sheet is as shown in the graph.

## Application Of Gauss Theorem To Find Field

## Due To Uniformly Charged Thin Spherical Shell

Electric field due to a uniformly charged thin spherical shell of uniform surface charge density σ and radius

R at a point distance r from the centre of the shell is:

At a point outside the shell i.e., r > R,E =(1/4πε_{0}.(q/r^{2})

At a point on the surface of the shell, i.e., r = R, E =(1/4πε_{0}).(q/r^{2})

At a point inside the shell, i.e., r < R, E = 0. Here q = 4πR^{2} × σ

- Variation of E with r for a uniformly charged thin spherical shell is as shown in the graph below.