Electric charges and fields

static electricity

the accumulation of excess electric charge on an object

Electrostatics

study of forces, fields and potentials arising from static charges.

Thales of Miletus

A Greek natural philosopher (ca. 624-ca. 547 B.C.E.), noted for his application of reason to astronomy and for his questioning of the fundamental nature of the universe.

amber rubbed with wool or silk cloth attracts light objects

Elektron

Greek word for amber

Electric charges

tiny particles that carry units of electricity;

can be positive (+) or negative (-) {Benjamin Franklin}

specific charge

Charge/mass value of a charged particle

q/m

where, m= mass at rest/ √1- v^2/c^2

m is the dynamic mass

glass rod or cat fur

positive by convention

silk or plastic rod

negative charges by convention

Like charges

repel each other

Unlike charges

attract each other

UNIFICATION OF ELECTRICITY AND MAGNETISM

-In 1820 Danish scientist Oersted found that a compass needle is deflected by passing an electric current through a wire placed near the needle.

-Ampere and Faraday supported this observation by saying that electric charges in motion produce magnetic fields and moving magnets generate electricity.

-The unification was achieved when the Scottish physicist Maxwell and the Dutch physicist Lorentz put forward a theory where they showed the interdependence of these two subjects.

-This field is called electromagnetism.

Gold leaf electroscope

A device with a metallic stem and a gold leaf that can be used to identify and measure electric charge — a device that was historically used as a voltmeter for measuring large voltages

It consists of a vertical metal rod housed in a box, with two thin gold leaves attached to its bottom end. When a 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 the amount of charge.

Conductors

materials that allow electric charges to flow through them easily

They have electric charges (electrons) that are comparatively free to move inside the material.

When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor.

Metals, human and animal bodies and earth are conductors.

Insulators

materials that prevent electric charges from flowing through them easily

if some charge is put on an insulator, it stays at the same place.

Most of the non-metals like glass, porcelain, plastic, nylon, wood

Grounding (Earthing)

Allowing charges to move freely along a connection between a conductor and the ground.

Earthing provides a safety measure for electrical circuits and appliances.

live wire

The brown wire in a cable or plug.

Neutral wire

a wire that carries current away from the component; it is coated in blue plastic

Earth wire

a wire through which current only flows when there is a leak of current in an appliance; it is coated in yellow and green plastic

When any fault occurs or live wire touches the metallic body, the charge flows to the earth without damaging the appliance and without causing any injury to the humans

charging by induction

process of rearranging electrons on a neutral object by bringing a charged object close to it

-When electrified rods are brought near light objects, a similar effect takes place. The rods induce opposite charges on the near surfaces of the objects and similar charges move to the farther side of the object.

-the magnitude of force depends on the distance between the charges and in this case the force of attraction overweighs the force of repulsion

charging by contact (conduction)

energy is transferred by the movement of electrons or ions from opposite charged objects or charged to neutral objects.

Point charges

spatial size negligible compared to other distances

Basic properties of Electric charge

- Additivity of charge: Charge has magnitude but no direction, similar to the mass. However, there is one difference between mass and charge. Mass of a body is always positive whereas a charge can be either positive or negative. Proper signs have to be used while adding the charges in a system.

q1+q2+...= total charge

-Conservation of charge: the total charge of the isolated system is always conserved. Conservation of charge has been established experimentally.

-Quantisation of charge: The fact that electric charge is always an integral multiple of e is termed as quantisation of charge

q = ne

Quantisation of charge

The fact that electric charges are only observed in integer multiples of e (charge on the electron)

-The quantisation of charge was first suggested by the experimental laws of electrolysis discovered by English experimentalist Faraday. It was experimentally demonstrated by Millikan in 1912.

-SI UNIT: one coulomb is the charge flowing through a wire in 1 s if the current is 1 A (ampere)

-if a body contains n1 electrons and n2 protons, the total amount of charge on the body is n2 × e + n1 × (-e) = (n2 - n1) e.

-the charge on any body is always an integral multiple of e and can be increased or decreased also in steps of e.

quanta

The bundle of electromagnetic energy that is absorbed or emitted by matter

1 quanta is smallest dicrete value of charge that can exist in nature and is equal to chatge of electron

Quantisation of charge at macroscopic level

The step size e is, however, very small because at the macroscopic level, we deal with charges of a few μC. At this scale the fact that charge of a body can increase or decrease in units of e is not visible.

This situation can be compared with the geometrical concepts of points and lines. A dotted line viewed from a distance appears continuous to us but is not continuous in reality. As many points very close to each other normally give an impression of a continuous line, many small charges taken together appear as a continuous charge distribution.

at the macroscopic level, the quantisation of charge has no practical consequence and can be ignored

Quantisation of charges at microscopic level

the charges involved are of the order of a few tens or hundreds of e, i.e., they can be counted, they appear in discrete lumps and quantisation of charge cannot be ignored.

1 stat coulomb/ esu/ frankline

cgs electrostatic system

1 esu= 3.36*10^-10 C

Coulomb

the SI unit of electric charge, equal to the quantity of electricity conveyed in one second by a current of one ampere.

e = 1.602192 × 10-19 C

there are about 6 × 1018 electrons in a charge of -1C. In electrostatics, charges of this large magnitude are seldom encountered and hence we use smaller units 1 μC (micro coulomb) = 10-6 C or 1 mC (milli coulomb) = 10-3 C.

Coulomb's Law

the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges.

F=K q₁*q₂/r², magnitude of force between two charges

Formation of coulomb's law

-Coulomb used a torsion balance* for measuring the force between two charged metallic spheres

-Suppose the charge on a metallic sphere is q. If the sphere is put in contact with an identical uncharged sphere, the charge will spread over the two spheres. By symmetry, the charge on each sphere will be q/2*. Repeating this process, we can get charges q/2, q/4, etc. Coulomb varied the distance for a fixed pair of charges and measured the force for different separations. He then varied the charges in pairs, keeping the distance fixed for each pair. Comparing forces for different pairs of charges at different distances, Coulomb arrived at the relation

-While the original experiments established it at a macroscopic scale, it has also been established down to subatomic level (r ~ 10-10 m).

1 Coulomb

1 C is the charge that when placed at a distance of 1 m from another charge of the same magnitude in vacuum experiences an electrical force of repulsion of magnitude 9 × 109 N.

In SI units, the value of k is about 9 × 109. The unit of charge that results from this choice is called a coulomb which we defined earlier in Section 1.4. Putting this value of k in Eq

F= 9 × 109 N

Value of 'k' in Coulomb's law

The constant k in Eq. (1.1) is usually put as k = 1/4 πε 0 for later convenience, so that Coulomb's law is written as

F= 1/4πε0 (q1.q2)/r2

permittivity of free space

the charge per unit area in coulombs per square metre on oppositely charged parallel plates in a vacuum when the field strength between the plates is 1 Vm^-1

e0 (capacitance) = 8.85 x 10^-12 F/m (SI unit)

dielectric constant

a quantity measuring the ability of a substance to store electrical energy in an electric field

also called relative permittivity

Principle of Superposition

force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time. The individual forces are unaffected due to the presence of other charges.

The vector sum is obtained as usual by the parallelogram law of addition of vectors. All of electrostatics is basically a consequence of Coulomb's law and the superposition principle.

Limitations of coulomb's law

-valid only for static charges

- valid only for point sources and not for extended bodies due to induction in charged bodies

electric field

the space around a charged object in which another charged object experiences an electric force

the force experienced by a unit test charge placed at that point, without disturbing the original positions of charges

introduced by faraday

physical significance is calculating time delay in time -dependent electromagnetic phenomena.

e.g. the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2. The notion of field elegantly accounts for the time delay.

-independent dynamic, i.e. evolve according to their own laws

- can transport energy

- vector quantity

source charge

the charge that creates an electric field

test charge

A charge used to define the electric field in at some point. The charge must be positive and small enough so as to not disturb existing charges.

electric field lines

Lines that provide a picture of an electric field, indicate the field's strength by the spacing between the lines (relative density), never cross, and are directed toward negative charges and away from positive charges

-the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward.

-Electric field is strong near the charge, so the density of field lines is more near the charge and the lines are closer. Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines.

-The field lines crowd where the field is strong and are spaced apart where it is weak.

-contract longitudinally and expand laterally

why do conductors not have electric field lines?

there is no electric field

- line start or end normally from the surface of conductor

uniform electric field

Electric field strength is equal in magnitude and has the same direction at all points in the region.

definition of electric field lines

Electric field lines are a way of pictorially mapping the electric field around a configuration of charges.

- An electric field line is, in general, 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.

-An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve.

-A field line is a space curve, i.e., a curve in three dimensions.

Properties of electric field lines

-The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward.

(I) Field lines start from positive charges and end at negative charges. If there is a single charge, they may start or end at infinity.

(ii) In a charge-free region, electric field lines can be taken to be continuous curves without any breaks.

(iii) Two field lines can never cross each other. (If they did, the field at the point of intersection will not have a unique direction, which is absurd.)

(iv) Electrostatic field lines do not form any closed loops. This follows from the conservative nature of electric field

electric flux

the product of a surface area and the component of the electric field perpendicular to the surface

i.e. electric field passing through a given area

-The rate of flow is given by the volume crossing the area per unit time v dS and represents the flux of liquid flowing across the plane.

-If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle θ with it, the projected area in a plane perpendicular to v is v dS cos θ . Therefore the flux going out of the surface dS is v. ˆ nd s

there is no physical flow visible

Direction of planer area vector

along it's normal

Vector of a closed surface area

The vector associated with every area element of a closed surface is taken to be in the direction of the outward normal.

electric flux equation

Electric flux Δ φ through an area element ΔS is defined by

Δ φ = E.ΔS = E ΔS cos θ

which is proportional to the number of field lines cutting the area element.

angle is between ΔS and E

Unit of electric flux

N m^2/ C

the total flux φ through a surface S is

φ ~ Σ E.ΔS

The approximation sign is put because the electric field E is taken to be constant over the small area element. This is mathematically exact only when you take the limit ΔS → 0 and the sum is written as an integral.

electric dipole

a separation of equal and opposite charge by a small distance; can be seen in polar molecules.

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

the direction from -q to q is said to be the direction of the dipole.

The mid-point of locations of -q and q is called the centre of the dipole.

total charge of the electric dipole is obviously zero

Since the charge q and -q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out.

SI unit of electric dipole

coulomb- Metre

pratical unit is debye

Debye

The unit used to express dipole moments.

1 debye

3.3 * 10^-30 C-m

When does the electric fields of dipole charges cancel each other out?

at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and -q nearly cancel out. The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2

The field of an electric dipole

The electric field at any general point P is obtained by adding the electric fields E-q due to the charge -q and E +ve q due to the charge q, by the parallelogram law of vectors.

dipole moment

a property of a molecule whose charge distribution can be represented by a center of positive charge and a center of negative charge

dipole moment equation

p=qd

d is distance or

The dipole moment vector p of an electric dipole is defined by p = q × 2a p cap

stable equilibrium

the dipole moment is in the direction of electric field

Unstable equilibrium

dipole moment is opposite to the direction of the field

potential energy of dipole moment

U= -p E cos theta

Molecules which don't have dipole moment

CO2 and CH4

Molecules that have dipole moment

Water molecules

How to find the centre of positive point charges?

Centre of a collection of positive point charges is defined much the same way as the centre of mass

permanent dipole

permanent separation of electrical charge in a molecule due to unequal distributions of bonding and/or lone pairs of electrons

(By permanent dipole, we mean that dipole moment exists irrespective of E; it has not been induced by E

Torque

a turning or twisting force

When the net force is zero, the torque (couple) is independent of the origin.

- Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces). Magnitude of torque = q E × 2 a sin θ

= 2 q a E sin θ

Its direction is normal to the plane of the paper, coming out of it.

-The magnitude of p × E is also p E sin θ and its direction is normal to the paper, coming out of it.

Thus, τ = p × E

When p is parallel to E

when p is parallel to E, the dipole has a net force in the direction of increasing field.

When p is antiparallel to E

When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field.

A comb run through dry hair attracts pieces of paper. The comb, as we know, acquires charge through friction. But the paper is not charged. What then explains the attractive force?

the charged comb 'polarizes' the piece of paper, i.e., induces a net dipole moment in the direction of field. Further, the electric field due to the comb is not uniform. In this situation, it is easily seen that the paper should move in the direction of the comb

surface charge density

surface charge density σ at the area element by

σ= ΔQ/ΔS

σ , called the surface charge density.

-The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level.

-σ represents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ΔS which, as said before, is large microscopically but small macroscopically.

-The units for σ are C/m^2

linear charge density

when a charge is distributed along a line

λ= ΔQ/Δ l

where Δl is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and ΔQ is the charge contained in that line element.

The units for λ are C/m.

Volume Charge Density (rho)

ρ= ΔQ/ΔV

where ΔQ is the charge included in the macroscopically small volume element ΔV that includes a large number of microscopic charged constituents. The units for ρ are C/m3.

Gauss's Law

q/Eo = E*A (flux); where E is electric field, E0 is permittivity constant, and A is area

-the electric flux through a closed surface depends upon the charge enclosed in that surface

-The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface

whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero.

Important points of Gauss's Law

(i) Gauss's law is true for any closed surface, no matter what its shape or size.

(ii) The term q on the right side of Gauss's law includes the sum of all charges enclosed by the surface. The charges may be located anywhere inside the surface. (iii) In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side] is due to all the charges, both inside and outside S. The term q on the right side of Gauss's law, however, represents only the total charge inside S.

(iv) The surface that we choose for the application of Gauss's law is called the Gaussian surface. You may choose any Gaussian surface and apply Gauss's law. However, take care not to let the Gaussian surface pass through any discrete charge. This is because electric field due to a system of discrete charges is not well defined at the location of any charge. (As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution.

(v) Gauss's law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry. This is facilitated by the choice of a suitable Gaussian surface.

(vi) Finally, Gauss's law is based on the inverse square dependence on distance contained in the Coulomb's law. Any violation of Gauss's law will indicate departure from the inverse square law.

Gaussian surface

A hypothetical closed surface in the vicinity of an electric field or a charge distribution. The Gaussian surface may enclose a net charge.