Electric Charges and Fields

Introduction

Experiencing sparks or crackling sounds when removing synthetic clothes, lightning during thunderstorms, and electric shocks from car doors or bus bars are all examples of electric discharge. These phenomena are due to the accumulation and discharge of electric charges, often referred to as static electricity. Electrostatics is the study of forces, fields, and potentials arising from static charges.

Electric Charge

Historical Context

Thales of Miletus, around 600 BC, is credited with discovering that amber rubbed with wool or silk attracts light objects. The word "electricity" comes from the Greek word "elektron," meaning amber.

Properties of Electric Charge

Rubbing certain pairs of materials can cause them to attract light objects.
Like charges repel each other, while unlike charges attract each other.
The property that differentiates the two kinds of charges is called the polarity of charge.
Benjamin Franklin named the two types of charges as positive and negative.
By convention, the charge on a glass rod rubbed with silk or cat's fur is positive, and the charge on a plastic rod rubbed with fur or silk is negative.
An object with an electric charge is said to be electrified or charged, while an object with no charge is electrically neutral.

Gold-Leaf Electroscope

A gold-leaf electroscope is a simple apparatus used to detect charge on a body. 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, charge flows to the leaves, causing them to diverge. The degree of divergence indicates the amount of charge.

Microscopic Explanation of Charge

All matter is made up of atoms and/or molecules, which are normally electrically neutral because they contain an equal number of positive and negative charges. Forces holding molecules and atoms together are electrical in nature. To electrify a neutral body, we need to add or remove electrons. A body can be charged positively by losing electrons and negatively by gaining electrons. During rubbing, electrons are transferred from one body to another; no new charge is created.

Conductors and Insulators

Conductors

Substances that easily allow the passage of electricity through them are called conductors. They have electric charges (electrons) that are comparatively free to move inside the material. Examples include metals, human and animal bodies, and the earth.

Insulators

Substances that offer high resistance to the passage of electricity are called insulators. Examples include glass, porcelain, plastic, nylon, and wood.

Semiconductors

Semiconductors are a third category of materials that offer intermediate resistance to the movement of charges.

Charge Distribution

When charge is transferred to a conductor, it readily gets distributed over the entire surface. In contrast, charge put on an insulator stays at the same place.

Basic Properties of Electric Charge

Point Charges

If the sizes of charged bodies are very small compared to the distances between them, they are treated as point charges. The charge content of the body is assumed to be concentrated at one point in space.

Additivity of Charges

The total charge of a system is obtained by adding the individual charges algebraically, i.e., charges add up like real numbers or scalars.
If a system contains $n$ charges $q1, q2, q3, …, qn$ , then the total charge of the system is: $q1 + q2 + q3 + … + qn$ . Charge has magnitude but no direction, similar to mass. Mass is always positive, whereas charge can be positive or negative.

Conservation of Charge

When bodies are charged by rubbing, there is a transfer of electrons from one body to the other; no new charges are created or destroyed. Within an isolated system, charges may get redistributed, but the total charge of the isolated system is always conserved.

Quantisation of Charge

Experimentally, it is established that all free charges are integral multiples of a basic unit of charge denoted by $e$ .
The charge $q$ on a body is always given by: $q = ne$ where $n$ is any integer, positive or negative. This basic unit of charge is the charge that an electron or proton carries. The charge on an electron is taken to be negative ($-e $), and that on a proton is positive ($ +e).

The quantisation of charge was first suggested by Faraday's experimental laws of electrolysis and was experimentally demonstrated by Millikan in 1912.

In the International System (SI) of Units, the unit of charge is called a coulomb (C). One coulomb is defined as the charge flowing through a wire in 1 s if the current is 1 A (ampere). The value of the basic unit of charge is:
e = 1.602192 × 10^{-19} C

Therefore, there are about 6 × 10^{18} $electrons in a charge of$ -1 C $. In electrostatics, charges of this large magnitude are seldom encountered, and smaller units like$ 1 \mu C = 10^{-6} C $or$ 1 mC = 10^{-3} C are used.

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 step size e $is very small, and at the macroscopic level, the grainy nature of charge is lost, and it appears to be continuous. At the macroscopic level, the quantisation of charge has no practical consequence and can be ignored. However, at the microscopic level, where the charges involved are of the order of a few tens or hundreds of$ e, quantisation of charge cannot be ignored.

Examples

Example 1.1: If 10^9 electrons move out of a body to another body every second, the time required to get a total charge of 1 C on the other body is:
1 C ÷ (1.6 × 10^{-10} C/s) = 6.25 × 10^9 s = 198 years

Example 1.2: The amount of positive and negative charge in a cup of water (assuming 250 g) is:
(250/18) × 6.02 × 10^{23} × 10 × 1.6 × 10^{-19} C = 1.34 × 10^7 C

Coulomb’s Law

Coulomb’s law is a quantitative statement about the force between two point charges. The force varies inversely as the square of the distance between the charges and is directly proportional to the product of the magnitude of the two charges and acts along the line joining the two charges.

If two point charges q1 $and$ q2 $are separated by a distance$ r in vacuum, the magnitude of the force (F) between them is given by:
F = k \frac{q1 q2}{r^2}

where k is a constant.

Experimental Verification

Coulomb used a torsion balance to measure the force between two charged metallic spheres.

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.

SI Units

In SI units, the value of k $is about$ 9 × 10^9 Nm^2 C^{-2}.
The unit of charge that results from this choice is called a coulomb (C).
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 × 10^9 N.

The constant k $is usually put as$ k = \frac{1}{4 \pi \epsilon0} for later convenience, so that Coulomb’s law is written as: F = \frac{1}{4 \pi \epsilon0} \frac{q1 q2}{r^2}

where \epsilon0 $is called the permittivity of free space. The value of$ \epsilon0 in SI units is:
\epsilon_0 = 8.854 × 10^{-12} C^2 N^{-1} m^{-2}

Vector Notation

Let the position vectors of charges q1 $and$ q2 $be$ \vec{r1} $and$ \vec{r2} respectively.
We denote force on q1 $due to$ q2 $by$ \vec{F{12}} $and force on$ q2 $due to$ q1 $by$ \vec{F{21}}.
The two point charges q1 $and$ q2 $have been numbered 1 and 2 for convenience, and the vector leading from 1 to 2 is denoted by$ \vec{r{21}}: \vec{r{21}} = \vec{r2} - \vec{r1}
In the same way, the vector leading from 2 to 1 is denoted by \vec{r{12}}: \vec{r{12}} = \vec{r1} - \vec{r2} = - \vec{r{21}} The magnitude of the vectors \vec{r{21}} $and$ \vec{r{12}} $is denoted by$ r{21} $and$ r{12} $, respectively ($ r{12} = r{21}). The direction of a vector is specified by a unit vector along the vector. To denote the direction from 1 to 2 (or from 2 to 1), we define the unit vectors: \hat{r}{21} = \frac{\vec{r}{21}}{r{21}}, \quad \hat{r}{12} = \frac{\vec{r}{12}}{r_{12}}

Coulomb’s force law between two point charges q1 $and$ q2 $located at$ \vec{r1} $and$ \vec{r2}, respectively, is then expressed as
\vec{F}{21} = \frac{1}{4 \pi \epsilon0} \frac{q1 q2}{r{21}^2} \hat{r}{21}

Remarks

The equation is valid for any sign of q1 $and$ q2, whether positive or negative.
If q1 $and$ q2 $are of the same sign (either both positive or both negative),$ \vec{F{21}} $is along$ \hat{r}{21}, which denotes repulsion, as it should be for like charges.
If q1 $and$ q2 $are of opposite signs,$ \vec{F}{21} $is along$ -\hat{r}{21} (=\hat{r}_{12}), which denotes attraction, as expected for unlike charges.
The force \vec{F{12}} $on charge$ q1 $due to charge$ q2, is obtained by simply interchanging 1 and 2, i.e., \vec{F}{12} = -\vec{F}_{21}
Thus, Coulomb’s law agrees with Newton’s third law.
Coulomb’s law gives the force between two charges q1 $and$ q2 in vacuum. If the charges are placed in matter or the intervening space has matter, the situation gets complicated due to the presence of charged constituents of matter.

Examples

Example 1.3: Comparing the strength of electrostatic and gravitational forces:
The ratio of electric force to gravitational force between an electron and a proton is:
\frac{Fe}{FG} = \frac{e^2}{G mp me 4 \pi \epsilon_0} ≈ 2.4 × 10^{39}

The ratio of the magnitudes of electric force to the gravitational force between two protons is:
\frac{Fe}{FG} = \frac{e^2}{G mp^2 4 \pi \epsilon0} ≈ 1.3 × 10^{36}

Example 1.4: A charged metallic sphere A is suspended by a nylon thread. Another charged metallic sphere B held by an insulating handle is brought close to A. After touching spheres A and B with uncharged spheres C and D, respectively, and then removing C and D, the electrostatic force on A, due to B, remains unaltered if the separation between A and B is halved.

Forces Between Multiple Charges

The force on a 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. This is termed as the principle of superposition.

Consider a system of n $charges$ q1, q2, …, qn $in vacuum. The total force$ \vec{F1} $on$ q1 due to all other charges is given by: \vec{F1} = \vec{F{12}} + \vec{F{13}} + … + \vec{F{1n}} = \sum{i=2}^{n} \vec{F_{1i}}

Examples

Example 1.5: Consider three charges q1, q2, q_3 $each equal to$ q $at the vertices of an equilateral triangle of side$ l $. The force on a charge$ Q $(with the same sign as$ q) placed at the centroid of the triangle is 0.

Example 1.6: Consider the charges q, q, $, and$ -q $placed at the vertices of an equilateral triangle. The sum of the forces on the three charges is zero, i.e.,$ \vec{F1} + \vec{F2} + \vec{F_3} = 0

Electric Field

The electric field produced by a charge Q $at a point$ \vec{r} is given as
\vec{E}(\vec{r}) = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \hat{r}

The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position.

We obtain the force \vec{F} $exerted by a charge$ Q $on a charge$ q, as
\vec{F} = \frac{1}{4 \pi \epsilon_0} \frac{Qq}{r^2} \hat{r}

The electrostatic force between the charges Q $and$ q $can be looked upon as an interaction between charge$ q $and the electric field of$ Q and vice versa.

If we denote the position of charge q $by the vector$ \vec{r} $, it experiences a force$ \vec{F} $equal to the charge$ q $multiplied by the electric field$ \vec{E} $at the location of$ q. Thus,
\vec{F}(\vec{r}) = q \vec{E}(\vec{r})

Important Remarks

From the equation, we can infer that if q $is unity, the electric field due to a charge$ Q is numerically equal to the force exerted by it.
The electric field \vec{E} $due to$ Q $is independent of$ q.
The force \vec{F} $on the charge$ q $due to the charge$ Q $depends on the particular location of charge$ q $, which may take any value in the space around the charge$ Q.
For a positive charge, the electric field will be directed radially outwards from the charge. On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards.
At equal distances from the charge Q $, the magnitude of its electric field$ \vec{E} is the same.

Electric Field due to a System of Charges

Consider a system of charges q1, q2, …, qn $with position vectors$ \vec{r1}, \vec{r2}, …, \vec{rn} $relative to some origin O. The electric field$ \vec{E} $at a point$ \vec{r} due to the system of charges is
\vec{E}(\vec{r}) = \sum{i=1}^{n} \frac{1}{4 \pi \epsilon0} \frac{qi}{r{iP}^2} \hat{r}_{iP}

Physical Significance of Electric Field

Electric field is an elegant way of characterising the electrical environment of a system of charges. 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 (without disturbing the system). Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field.

Examples

Example 1.7: An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 10^4 N C^{-1} $. The time of fall for the electron is$ te = 2.9 × 10^{-9} s $. The time of fall for the proton is$ tp = 1.3 × 10^{-7} s.

Example 1.8: Two point charges q1 $and$ q2 $, of magnitude$ +10^{-8} C $and$ -10^{-8} C, respectively, are placed 0.1 m apart. The electric fields at points A, B, and C can be calculated using the superposition principle.

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.

Properties of Field Lines

Field lines start from positive charges and end at negative charges. If there is a single charge, they may start or end at infinity.
In a charge-free region, electric field lines can be taken to be continuous curves without any breaks.
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.)
Electrostatic field lines do not form any closed loops. This follows from the conservative nature of electric field.

Electric Flux

Electric flux is defined as the number of field lines crossing a surface. Electric flux is given by:
\Delta \phi = \vec{E} \cdot \Delta \vec{S} = E \Delta S \cos \theta

where \theta $is the angle between$ \vec{E} $and$ \Delta \vec{S}.

Electric Dipole

An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a. The line connecting the two charges defines a direction in space. By convention, the direction from –q to q is said to be the direction of the dipole.

The dipole moment vector \vec{p} of an electric dipole is defined by
\vec{p} = q × 2\vec{a}

Field of an Electric Dipole

At a point on the dipole axis:
\vec{E} = \frac{4 q a}{4 \pi \epsilon_0 r (r+a)^2} \hat{p}

For r >> a
\vec{E} = \frac{p}{4 \pi \epsilon_0 r^3} \hat{p}

At a point on the equatorial plane:
\vec{E} = - \frac{2 q a}{4 \pi \epsilon0 (r^2 + a^2)^{3/2}} \hat{p} For r >> a \vec{E} = - \frac{p}{4 \pi \epsilon0 r^3} \hat{p}

Physical Significance of Dipoles

In most molecules, the centres of positive charges and of negative charges lie at the same place. Therefore, their dipole moment is zero. However, they develop a dipole moment when an electric field is applied. In some molecules, the centres of negative charges and of positive charges do not coincide. Therefore they have a permanent electric dipole moment, even in the absence of an electric field. Such molecules are called polar molecules.

Examples

Example 1.9: Two charges \pm 10 \mu C $are placed 5.0 mm apart. The electric field at a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge is$ 2.7 × 10^5 N C^{-1} $. The electric field at a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, is$ 1.33 × 10^5 N C^{-1}.

Dipole in a Uniform External Field

In a uniform external electric field \vec{E} $, a dipole with dipole moment$ \vec{p} $experiences a torque$ \vec{\tau} given by
\vec{\tau} = \vec{p} × \vec{E}

Continuous Charge Distribution

For many purposes, it is impractical to work in terms of discrete charges, and we need to work with continuous charge distributions.

Charge Density

Linear charge density: \lambda = \frac{\Delta Q}{\Delta l}
Surface charge density: \sigma = \frac{\Delta Q}{\Delta S}
Volume charge density: \rho = \frac{\Delta Q}{\Delta V}

Using Coulomb’s law and the superposition principle, the electric field can be determined for any charge distribution, discrete or continuous, or part discrete and part continuous.

Gauss’s Law

Electric flux through a closed surface S = q/\epsilon_0

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.

Applications of Gauss’s Law

Field due to an infinitely long straight uniformly charged wire:
E = \frac{2 \lambda}{4 \pi \epsilon0 r} = \frac{\lambda}{2 \pi \epsilon0 r}
Field due to a uniformly charged infinite plane sheet:
E = \frac{\sigma}{2 \epsilon_0}
Field due to a uniformly charged thin spherical shell:
- Outside the shell (r > R):
 E = \frac{q}{4 \pi \epsilon_0 r^2}
- Inside the shell (r < R):
 E = 0$$