Electric Charges and Forces

Useful References:

• https://openstax.org/books/physics/pages/18-introduction

The Charge Model

• There are two types of charges: positive and negative
• Objects are charged by adding/removing electrons
• Amount of charge \(q = (N_p-N_e)e\)
• Object electrically neutral iff \(q=0\)
• Charge is conserved; it is neither created nor destroyed

Charge and Electrons

• Charge is quantized; it exists only in discrete (i.e. non-continuous) amounts
• Elementary charge $e$ is the fundamental unit of electric charge
• $e$ is approximately equal to \(1.60217663\times 10^{-19} Coulombs\)
• Charge of any object is an integer multiple of e, that is: \(q=ne\) where $q$ is the total charge, $n$ an integer, and $e$ the elementary charge
• Objects acquire positive charge by loss of electrons; this process is ionization

Charge Relationships

• Like charges exert repulsive force; opposites exert attractive force
• Force varies directly according to the magnitude of charge
• Force varies inversely according to the distance between charged objects

Bands

• Bands are closely spaced orbitals approximately indistinguishable in energy level
• Extension of molecular orbital theory; recall HOMO and LUMO

Valence Band

• The valence band is the highest occupied energy band in a material
• Electrons in the valence band are strongly associated with their individual atoms and are typically involved in bonding

Conduction Band

• The conductive band is the lowest unoccupied energy band in a material
• It lies directly above the valence band
• Electrons in the conduction band often are “excited” electrons; through a thermal process, these electrons gain energy and make the jump from the valence band to the conduction band

Band Gap

• The difference in energy between the valence band and conduction band is the band gap
• The greater the band gap, the less likely electrons are to be found in the conduction band; they lack the energy to make the jump to the conduction band

Conductivity

• Materials with electrons in the conduction band are good conductors of electricity, because these “free” electrons are able to to move easily through the subtrate
• In highly conductive materials, such as metals, the band gap overlaps; electrons therefore readily jump between the two bands
• Insulators are characterized by a prohibitively large band gap, resulting in an impossibly high amount of energy being required to move electrons into the conduction band to form a current

Charging

• Charging may occur via direct contact, whereby electrons are directly transferred between a neutral and a charged object

Charge Transfer by Rubbing

• Rubbing objects breaks intramolecular bonds in otherwise neutral molecules, resulting in the formation of polar ions.

Discharging

• Discharging is a process whereby a system is neutralized
• An object may be discharged by direct contact with the ground, into which excess electrons flow

Polarization

• Polarization is the displacement of charges in an object that remains neutral, resulting in an induced dipole moment
• Polarization is the result of the application of an external electric field

Electric Dipoles

• The electric dipole is a measure of a system’s overall polarity
• It comprises two electric charges of equal magnitude q and opposite sign, separated by some distance
• The vector form of the electric dipole moment is expressed as \(\mathbf{p}=q\mathbf{d}\) where $q$ is the magnitude of the dipole’s charge and $\mathbf{d}$ is the vector pointing from the negative charge to the positive charge

Coulomb’s Law

If two charged particles having charges $q_1$ and $q_2$ are separated by distance $r$, the magnitude of the forces they exert on each other is

\[|F|=F_\textrm{1 on 2}=F_\textrm{2 on 1}=\frac{k|q_1||q_2|}{r^2}\]

where $k$ is Coulomb’s constant. In SI units, $k\approx 8.99\times 10^9\textrm{ N}\cdot\textrm{m}^2/\textrm{C}^2$

• The forces are directed along the line joining the two particles
• Like forces repel, opposites attract
• The forces are attractive for like charges, repulsive for opposite charges

Coulomb’s law may be rewritten using the permittivity constant $\epsilon_0$:

\[F=\frac{1}{4\pi\epsilon_0}\frac{|q_1||q_2|}{r^2}\]

where $\epsilon_0 = \frac{1}{4\pi K}$.

Electric Fields

The spatial forces exerted by charges on the objects around them constitute a vector field. The magnitude of the force exerted by an electric field on an object with charge $q$ is

\[\vec F=q\vec E\]

where $\vec E$ is the force per unit charge ratio (in $\textrm{N}/\textrm{C}$) corresponding to

\[\vec E = \frac{\vec F}{q}\]

Electric Field of Point Charges

Given a single point charge $q$, we can calculate its electric field using a probe $q’$:

\[\vec E = \frac{\vec F_{\textrm{on }q'}}{q'}\]

• Assuming both charges are positive, the direction of $\vec E$ will be away from $q$.