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Electrostatics deals with the study of forces, fields, and potentials arising from static charge

1. Electric charge

• Discovered by Thales of Miletus, Greece.
• Electricity is coined from the Greek word elektron meaning amber.

Rubbing experiment

• Glass rods rubbed with wool or silk cloth repel each other when brought close.
• Same case with plastic rods when rubbed with fur.
• But when one plastic and one glass rod after rubbing brought together, attracts each other.
• Thus the glass and plastic rods are electrified.
• The glass rod and fur acquired positive charge and the plastic rod and silk acquires negative charges.
• Initially they were electrically neutral, ie. their charges are exactly balanced.
• To electrify any body, we have to remove one kind of charge.
• When a body is charged, it means there is an excess or deficit of charges.
• No new charges are created in the process of rubbing.
• To charge positive, the object loses electrons.
• To charge negative, the object gains electrons.

Two kinds of electrification

1. like charges repel
2. unlike charges attract This is also known as polarity of charges.

• Charges are acquired after rubbing are lost when the charged bodies are brought in contact. The charges were named as positive and negative by the American Scientist Benjamin Franklin.

Apparatus to detect charge

Gold leaf electroscope

It consists of vertical metal rod, box, two thin gold leaves attached to its bottom

Experiment

When the charged object touches the metal knob at the top of the rod, charge flows on to the leaves and they diverge. The degree of divergence is an indicator of amount of charge.

2. Conductors and Insulators

Conductors

• Have electrons that are comparatively freer to move inside the material.
• Charge acquired, is distributed over the entire surface.

Insulators

• Have electrons, but aren't free enough inside the material.
• Charge acquired, stays at the same place.

Semi conductors

3. Basic properties of electric charge

Point charges

• sizes of charged bodies are very small
• concentrated at one point in space

Additivity of charges

• Two point charges q1 and q2. Then, Total charge = adding algebraically q1 and q2
• charges are always scalar quantity, that is, it have only magnitude and no direction, like mass.
• But the difference between the two is, mass is always positive but charge can be both positive and negative.
• Hence it is always written with sign + or -
• If there are n charges in a system, then total charge = q1 + q2 + … + qn

Charge is conserved

The total charge of the system is always conserved.

Sometimes, nature creates charged particles: a neutron turns into a proton and an electron. The proton and electron thus created have equal and opposite charges and the total charge is zero before and after the creation.

Subatomic particles

Quantisation of charge

• All free charges are integral multiples of a basic unit of charge, "e" ie. q = ne, n is any integer

• Charge of electron is -e and charge of proton is +e e = 1.602192 * 10^-19 C = basic unit of charge

• This was first suggested by experimental law of electrolysis by English scientist Faraday.

• This was experimentally demonstrated by Millikan in 1912.

SI unit of charge

• Coulomb - symbol "C"

C in terms of A

• One coulomb is the charge flowing through a wire in 1 second is 1 ampere.
• Q = It

Number of electrons in 1C

• n=Q / e
• n= -1C / -1.6×10^-19 C​​
• There are about 6 * 10^18 electrons in a charge of -1C.

Smaller units

• 1μC (micro coulomb) = 10${^{-6}}$ C
• 1mC (milli coulomb) = 10${^{-3}}$ C
• 1nC (nano coulomb) =10${^{−9}}$ C

Charge of body with electrons and protons

• If a body contains n1 electrons and n2 protons, then the total amount of charge on the body is n2 * e + n1 * -e = (n2 - n1) e

• The charge on any body is equal to the integral multiple of e and can be increased or decreased also in steps of e.

• At macroscopic level, the quantisation of charge has no practical consequence and can be ignores but not at microscopic level.

4. Coulomb's law

It is about the forces between 2 point charges.

In case of linear sized charges, then the distance can be ignored.

The law states that the force between 2 point charges are: varied inversely as the square of the distance between the charges directly proportional to the product of magnitude of the two charges acted along the line the joining the 2 charges the charges being placed in a vacuum

• F ∝ | q1 and q2 |
• F ∝ 1 / r²
• F = k { |q1 q2| } / r²
• k = arbitrary constant and the value of k is, k= 1/ 4πε0 = = 8.987 × 10^9 Nm²/C²
  • The value of k comes from gauss law and maxwell equation
• ε0 is the permittivity of free space, the value of ε0​ = 8.854×10^−12 C² / N^1 . m^2

Gauss law

• Gauss’s Law states that the total electric flux (ΦE) through a closed surface is proportional to the total charge enclosed within that surface.

• The electric field lines passing through a closed surface depend only on the total charge inside, not the charge distribution.

Maxwell equation

• Maxwell’s Equations unify electricity, magnetism, and light as electromagnetic waves.

Why does the value of k matters?

• This value of k ensures that the unit of charge (Coulomb, C) is defined in such a way that the force between two 1 C charges placed 1 meter apart is 9 × 10⁹ Newtons, a very large force.

What is ε0?

• Permittivity (ε0​) is a measure of how easily an electric field can form in a vacuum (or free space).
• It is a fundamental physical constant that appears in Maxwell’s equations and determines the strength of electric fields in space.
• Describes how electric fields behave in a vacuum. It determines how much electric field is generated per unit charge in free space.

How Coulomb arrive at this law?

• The charges on the spheres were unknown. So, he thought that the charge on the sphere be q. If the sphere is put in contact with another sphere of identical uncharged, then by symmetry, the charge on each sphere will be q/2.

• This is because, the assumption that additivity of charges and conservation of 2 charges, ie. q/2 + q/2 = q.

• Repeating this process, he got the charges as q/2, q/4, q/8, etc...

• Initially

  • he varied the charges in pairs, keeping the distance constant,
  • then he kept the charges same and varied the distance of separation between them.

• The experiment is established both at macroscopic level and sub atomic level. (r = 10^-10)

Coulomb's law in vector notation

Since, force is a vector quantity, it is better to write coulomb's law in vector notation.

• F12 = force on q1 due to q2

• F21 = force on q2 due to q1

• r21 = leading from q1 to q2

• r12 = leading from q2 to q1

• F21 = 1/4πε0 . { (q1. q2) / r21 ^2 } ^r21​ — (i) unit vector ^r21 = r21 / r21

• The unit vector r^21​ gives the direction of the force.

Unit Vector

A unit vector is a vector that has a magnitude of 1 and is used to indicate direction. It does not have units of measurement and is dimensionless. A^= A / ∣A∣ ​

1. If both charges are of same sign, then F21 is along r^21, that is the force is repulsive
2. If both charges are of opposite sign, then F21 is along r^12, that is the force is attractive So, it proves that F12 = -F21, because r21 = - r^12

Coulomb's law and Newton's third law

For every action, there is an equal and opposite reaction.

5. Forces between multiple charges

• System of n stationary charges q1, q2, q3, … , qn are in a vacuum.

• Force on q1 due to all other forces cannot be answered using Coulomb's law.

• This can answered using Parallelogram law of addition and Principle of super position.

Parallelogram law of addition

If two vectors acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then the resultant vector is given by the diagonal of the parallelogram starting from the same point.

Principle of super position

• Force on any charge due to a number of other charges is the vector sum of all 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.

• Each charge acts on the test charge as if the other charges were not present.

• The order in which we calculate the forces does not matter.

Forces between multiple charges

force on

• q1 due to q2 = F12
• q1 due to q3 = F13 Substitute in (i) Total force on q1 = F12 + F13
  
  F1 = q1/4πε0 . { (q2) / r12 ^2 } ^r12 + ​(q3) / r13 ^2 } ^ r13 }
  

6. Electric field

• If a point charge Q placed in a vacuum at origin O.

• Another point charge q is placed at point P.

• OP = r

• Then Q exerts a force on q, as per coulomb's law.

• When q is removed, there is no force. Then what caused the force come into picture, as Q is still placed there.

• Answer is Field.

• Charge Q produces an electric field everywhere in its surroundings. Another charge q is brought at some point P, the field acts on it and produces a force.

• Electric field by a point charge Q is given by E(r)

  

Both the charges Q and q, exerts force on each other. That is, the electrostatic force between the charges Q and q can be said as an interaction between the charge q, and the electric field Q and vice versa. Q = source charge q = test charge The source charge must remain at its original location r is the vector denoting the position of q Therefore, it is established that, F(r) = q .E(r)

• E(r) = F(r) / q

SI unit of electric field

• N / C

One unit of Electric field

• It is the force that a unit positive charge would experience if placed at that point.

Make q negligibly small

Q itself is bound to experience an electrical force due to q, and will tend to move. To avoid this, make q negligibly small. then the electric field becomes,

Electric field due to system of charges

In case of a system of charges, q1, q2, q3, …, qn with position vectors r1, r2, r3, … relative to some origin O.

Force experienced by a unit test charge q, due to all these charges can be calculated by coulomb's law and principle of super position.

Physical significance of electric field

• Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point.
• It is independent of the test charge
• The term "field" refers to a quantity that is defined at every point in space and may vary from point to point.

Time dependent electromagnetic phenomena

• Force between 2 distant charges q1 and q2. The greatest speed to move from q1 to q2 is c, speed of light.
• The time delay between the effect (force on q2) and the cause (motion of q1).
• The accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed of light and reach q2.
• This field elegantly accounts for the time delay.
• The electric and magnetic fields can be detected only by their effects on charges, they are regarded as physical entities.
• The concept of field was introduced by Faraday.

Example 1.8

refer note for the solution.

7. Electric field lines

The magnitude of electric field at a point decreases inversely as the square of the distance between that point from the charge. That is, the vector gets shorter as one goes away from the origin, always pointing radially outwards. Each arrow indicates the electric field. ![obsidian|350][Electric-Field-Due-To-Point-Electric-Charges.png]

The magnitude of the field is represented by the density of the field lines. E is strong near the charge, so the density of field lines is more near the charge.

Away from the charge, the field gets weaker and the density of the field lines is less, resulting in well separated lines.

Infinite number of lines can be drawn from a charge.

Relation between the electric field, distance and the number of lines

To estimate the density of field lines, consider the number of lines per unit cross sectional area, perpendicular to the lines.

The number of field lines crossing the enclosed area remains constant, whatever be the distance of the area from the charge.

Solid angle

A solid angle is the 3D equivalent of a 2D angle. In two dimensions, an angle is measured in radians (rad). In three dimensions, a solid angle is measured in steradians (sr).

Consider a sphere of radius r. If a small surface area ΔS is cut out on the sphere, the solid angle ΔΩ subtended by this area at the center of the sphere is given by: ΔΩ = ΔS / r^2

Difference between a normal and perpendicular lines

• Two lines meet each other at 90 degree are perpendicular to each other.
• Line which is perpendicular to a plane or surface, it called as normal to that plane or surface.

Experiment

Two equal and small elements of area places at points R and S normal to the field lines.

ΔS = ΔΩ * r^2

In a given solid angle, the number of radial field lines is the same. For two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ΔΩ, ΔS = r1^2 ΔΩ at P1 ΔS = r2^2 ΔΩ at P2.

Since the number of lines (n) are same, the unit area element is n / (r1^2 ΔΩ ) at P1 and n / (r2^2 ΔΩ ) at P2.

This clearly indicates that the strength of the field depends on 1/r^2

Faraday called these lines as line of force. A field line is a space curve, a curve in three dimensions.

The field lines of a single positive charge are radially outwards single negative charge are radially inward two positive charges - mutual repulsion one positive and one negative - mutual attraction, but the lines never cross each other.

Properties of field lines

• start from positive charges and end at negative charges
• in case of single charge, it starts and ends at infinity
• field lines are taken to be continuous curves without breaks
• two field lines never cross each other.
• filed lines do not form closed loops

8. Electric flux

9. Electric dipole

10. Dipole in a uniform external field

11. Continuous charge distribution

12. Gauss law

13. Applications of gauss law