Electric Charges and Fields

There are two types of electric property, eventually called electric charge. The difference between the two types is in the direction of the electric forces that each type causes: these forces are repulsive when the same type of charge exists on two interacting objects (electrostatic repulsion), and attractive when the charges are of opposite types (electrostatic attraction).

The S.I unit of electric charge is Coulomb (C), named after the French scientist Charles-Augustin de Coulomb (1736–1806).

This force is an action-at-a-distance force because it does not require physical contact between the two objects in order to cause an acceleration. Franklin labeled the charges that flow as negative, while the positive charge remains largely motionless. Not all objects are affected by this force. The magnitude of the force decreases as the square of the distance between the two interacting objects increases.

Properties of Electric Charge

Quantization of charge: Charge exists in discrete packets, the smallest being

$e = 1.602 \times 10^{-19}C$

Equal magnitude: The smallest possible positive charge is $+e$, and the smallest possible negative charge is $-e$. Their magnitudes are exactly equal.

Conservation of charge: Charge cannot be created or destroyed; it can only be transferred between objects.

Local conservation: In a closed system, the total charge remains constant. Charges can move around or cancel in effect, but the net amount of charge does not vanish.

Conductors, Insulators and Charging by Induction

Conductors: These are mostly metals like copper, that allow free movement of electric charges (usually electrons) through them. This happens because conductors have a large number of free (or delocalized) electrons in their outer shells, which are not tightly bound to their atoms.

Insulators: are materials like rubber, glass, plastic, wood etc. that do not allow free movement of electric charges through them. Their electrons are tightly bound to their atoms and cannot move freely.

While the process of giving an uncharged body an electric charge without direct contact with a charged body, but by the influence of an electric field, is called charging by induction.

Coulomb's Law

States that the electric force $F$ between two charges is proportional to the magnitude of each charge, and inversely proportional to the square of the distance between them:

$F \propto \frac{q_1 q_2}{r^2}$

The magnitude of the electric force between two electrically charged particles is

$F = \frac{1}{4\pi\varepsilon_0} \frac{|q_1 q_2|}{r^2}$

Where:

• $F$ = Force
• $q_1$, $q_2$ = Charges
• $r$ = Distance apart
• $\varepsilon_0 = 8.85 \times 10^{-12} \dfrac{C^2}{N \cdot m^2}$ (the permittivity of free space)

Example

A hydrogen atom consists of a single electron. The proton has a charge of $+e$ and the electron has $-e$. In the ground state of the atom, the electron orbits the proton at a most probable distance of $5.29 \times 10^{-11}m$. Calculate the electric force on the electron due to the proton.

Solution

$q_1 = +e = +1.602 \times 10^{-19}C$

$q_2 = -e = -1.602 \times 10^{-19}C$

$r = 5.29 \times 10^{-11}m$

$F = \frac{1}{4\pi\varepsilon_0} \cdot \frac{|q_1 q_2|}{r^2}$

The magnitude of force must always be a positive number. Hence,

$F = \frac{1}{4\pi\varepsilon_0} \cdot \frac{|e|^2}{r^2}$

$F = \frac{1}{4\pi(8.85 \times 10^{-12})} \cdot \frac{(1.602 \times 10^{-19})^2}{(5.29 \times 10^{-11})^2}$

$F = 8.85 \times 10^{-8}N$

Example

Three different small charged objects are placed as shown in the diagram below. The charges $q_1$ and $q_3$ are fixed in place; $q_2$ is free to move. Given $q_1 = 2e$, $q_2 = -3e$ and $q_3 = -5e$, and that $d = 2.0 \times 10^{-7}m$, what is the net force on the middle charge $q_2$?

Solution

The net force is obtained from applying the Pythagorean theorem to its $x$ and $y$ components:

$F = \sqrt{F_x^2 + F_y^2}$

Where:

$q_1 = 2e = 3.204 \times 10^{-19}C$

$q_2 = -3e = 4.806 \times 10^{-19}C$

$q_3 = -5e = 8.01 \times 10^{-19}C$

$F_x = F_{23} = \frac{1}{4\pi\varepsilon_0}\frac{q_2 q_3}{r_{23}^2}$

$= (8.99 \times 10^{9}) \cdot \frac{(4.806 \times 10^{-19})(8.01 \times 10^{-19})}{(4.00 \times 10^{-7})^2}$

$= 2.16 \times 10^{-14}N$

$F_y = F_{21} = \frac{1}{4\pi\varepsilon_0}\frac{q_2 q_1}{r_{21}^2}$

$= (8.99 \times 10^{9}) \cdot \frac{(4.806 \times 10^{-19})(3.204 \times 10^{-19})}{(2.00 \times 10^{-7})^2}$

$= 3.46 \times 10^{-14}N$

$F = \sqrt{F_x^2 + F_y^2} = \sqrt{(2.16 \times 10^{-14})^2 + (3.46 \times 10^{-14})^2} = 4.08 \times 10^{-14}N \text{ (magnitude)}$

$\theta = \tan^{-1}\left(\frac{F_y}{F_x}\right) = \tan^{-1}\left(\frac{3.46 \times 10^{-14}}{2.16 \times 10^{-14}}\right) = 58°$

Electric Field

A field, in physics, is a physical quantity whose value depends on position relative to the source of the field. Electric field is a region of space around a charged particle in which another charged particle experiences a force. Its SI unit is $1 N/C = 1 V/m$.

$\overrightarrow{E} = \frac{\overrightarrow{F}}{q}$

Where:

• $\overrightarrow{E}$ = electric field (N/C or V/m)
• $\overrightarrow{F}$ = electric force on the test charge (N)
• $q$ = magnitude of the test charge (Coulombs)

For a point charge, we get from Coulomb's law that the electric field due to a point charge $Q$ at distance $r$ is:

$E = \frac{1}{4\pi\varepsilon_0} \cdot \frac{|Q|}{r^2}$

Where $\varepsilon_0 = 8.85 \times 10^{-12} \, C^2 N^{-1} m^{-2}$ is the permittivity of free space.

For positive charges, the field lines point away from the charge. For negative charges, the field lines point toward the charge.

Example

In an ionized helium atom, the most probable distance between the nucleus and the electron is $26.5 \times 10^{-12}m$. What is the electric field due to the nucleus at the location of the electron?

Solution

$\overrightarrow{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2}$

$q = 2e = 2(1.602 \times 10^{-19}C)$

$\overrightarrow{E} = (8.99 \times 10^{9}) \cdot \frac{2(1.602 \times 10^{-19})}{(26.5 \times 10^{-12})^2}$

$\overrightarrow{E} = 4.1 \times 10^{12}\frac{N}{C}$

Electric Field Lines

Electric field lines visualize how space is altered by a charge.

Direction: At any point, the field vector is tangent to the field line. Arrowheads show the field's direction.

Magnitude: Indicated by field-line density — the number of lines per unit area. Closer lines $\rightarrow$ stronger field. Farther apart $\rightarrow$ weaker field.

Field lines represent both the direction (tangent to the line) and strength (density of lines) of the electric field.

Electric Dipoles

A dipole is two equal and opposite charges separated by distance $d$.

When placed in an external electric field $E$:

• The dipole is assumed permanent (it exists even without the field).
• No net force acts on the dipole, because the forces on $+q$ and $-q$ are equal and opposite.
• However, a torque acts to align it with the field.

Torque ($\tau$):

$\tau = p \times E$

Dipole moment ($p$):

$p = qd$

— points from negative to positive.

An electric dipole in a uniform field doesn't move, but rotates to align with the field direction.

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