Chapter 19: Electric Charges, Forces, and Fields

for lightning, you need…

• electric charge (many electrons)

• voltage (millions-billions of volts)

charge is…

• quantized

  • charge exists in integral multiple of a fundamental unit of charge, symbolized by e

  • quarks are an exception because they can exist in fractional charges

Fundamental Charge

• e = 1.6 x 10^-19 Coulombs (C)

• the charge carried by a single proton (+e) or the negative charge of a single electron (-e)

Electric charge is always…

• conserved

  • charge is not created, only exchanged

A positively charged atom has…

A negatively charged atom has…

• more protons than electrons

• more electrons than protons

Charge Equation

• q = ne

  • q: total charge

  • n: number of elementary charges

  • e: fundamental charge

• measured in Coulombs

• n = ±1, ±2, ±3, …

A metallic object holds a charge of -3.8×10^-6 C.

• What total number of electrons does this represent? (e = 1.6×10^-19 C is the magnitude of the electronic charge.)

• How about 1 C of charge?

• q = ne

• n = q/e

• n = -2.4×10^13 electrons

• n = 0.6 electrons

A defibrillator has a charge of 0.3 C. If the magnitude of the electronic charge is e = 1.6×10^-19 C, what is n?

• q = ne

• n = 1.8×10^18 electrons going to the heart

Conductor

• physical medium/object (copper wire or metallic object) that contains a sea of free electrons

• allows electric current to flow easily through them in response to electric force

• charges redistribute very easily

• 10^(28)/m³ = 10^(22)/cm³

  • 10²⁸ electrons per cubic meter is the same density as 10²² electrons per cubic centimeter

Do charges redistribute easily in conductors or insulators?

conductors

Current

• the flow of electric charge, typically measured in amperes (A). It represents the movement of electrons through a conductor.

• need a wire, for example

Insulator

• not many free electrons

• charges do not redistribute

• ex: plastic, glass, rubber

When a conductor is charged in a small region, the charge…

readily distributes itself over the entire surface of the material

When insulators are charged by rubbing…

• only the rubbed area becomes charged

• there is no tendency for the charge to move into other regions of the material

Semiconductor

• characteristics between conductors and insulators

• ex: silicon and germanium

Initially, sphere A has a charge of -50e and sphere B has a charge of +20e. The spheres are made of conducting material and are identical in size. If the spheres then touch, what is the resulting charge on sphere A?

• -50e + 20e = -30e

  • this is split equally between the two spheres because they are conductors of equal size, which then redistributes

• -15e

Copper vs. Silver vs. Gold as Conductors

• Silver

  • Highest electrical conductivity

  • Lowest resistivity

  • Free electrons move the easiest

  • BUT (tarnishes and expensive)

• Copper

  • Slightly less conductive than silver

  • WAY cheaper

  • Very ductile and flexible

  • Doesn’t corrode badly

• Gold

  • Conductivity slightly worse than copper

  • Does not oxidize (corrosion-resistant)

  • Extremely stable over time

  • EXPENSIVE

Charging by Conduction

• charged object (rod) is placed in contact (touching) with another object (sphere)

  • some electrons on rod can move to the sphere

  • when rod is removed, the sphere is left with a charge (same charge as rod)

In charging by conduction, the object being charges is always left with…

a charge having the same sign as the object doing the charging

Charging by Induction

• a negatively charged rubber rod is brought near an uncharged sphere

• the charges in the sphere are redistributed

  • some of the electrons in the sphere are repelled from the electrons in the rod

• a grounded conducting wire is connected to the sphere

  • allows some of the electrons to move from the sphere to the ground

• the wire to ground is removed, the sphere is left with an excess of induced positive charge

• charging by induction requires no contact with the object inducting the charge

Conduction vs. Induction

• Conduction:

  • physical contact

  • generate same polarity

• Induction:

  • no physical contact

  • do grounding

  • generate opposite polarity

Charge Polarization

• the charged object (on the left) induces charge on the surface of the insulator

• this realignment of charge on the surface of an insulator is known as: polarization

Charge Polarization: comb & paper

• a charged comb attracts bits of paper due to polarization of the paper

• if comb is positive, paper polarizes to be negative on closer side and positive on the far side, and continues on the other pieces of paper, forming a chain

Light: Animation

• blue wave: electric field

• red wave: magnetic field

• both self-generating fields

Light

• electromagnetic wave

• doesn’t need medium to travel

• self-generating electric and magnetic field

• speed = v = c = 3 x 10^8 m/s

  • also 300 × 10^6 m/s

Example of Polarization: Bees

• when the bee moves, its wings get a slight positive charge from interacting with the air (electrons free in clouds etc. from friction and air flow)

• this helps the bee collect pollen because the pollen will stick to the slightly charged wings (just like pieces of paper stick to a charged comb)

Polarizing Glass vs. 3D Glass

• polarizing glass blocks light and has vertical fences

• 3D glass has one vertical fence and one horizontal fence

Coulomb’s Law

• F = k​​_e( |q₁||q₂| )/ r²

  • k_e = Coulomb Constant = 8.9875×10⁹ N m²/C²

  • F → electric force (Newtons)

    • vector quantity: direction matters

  • q₁, q₂ → charges (Coulombs)

  • r → distance between charges (meters)

• Works best for point charges or spherical charges far apart: charges in the μC range

• Inverse square law

Force is proportional to…

• q₁q₂

• 1/r²

• mass leads to…

• charge leads to…

• magnet leads to…

• gravitational field

• electric field

• magnetic field

If body P, with a positive charge, is placed in contact with body Q (initially uncharged), what will be the nature of the charge left on Q?

• in contact = conduction

• conduction = same charge

• Q must be positive

According to Coulomb’s Law, force increases if…

• Charges get bigger

• Distance gets smaller

Coulomb Inverse Square Law

• Doubling distance = force ÷ 4

Gravitational Force

• F = G (m₁m₂)/r²

Electrical Force vs. Gravitational Force

• both are inverse square laws

• the mathematical form of both laws is the same

  • masses replaced by charges

• electrical forces can be either attractive or repulsive; gravitational forces are always attractive

• electrostatic force is stronger than the gravitational force

Two point charges are separated by a distance, d. q₁ = -5e, q₂ = +2e. Which charge experiences the strongest electric force (greatest magnitude)?

• the force is the same magnitude for both charges

• F = k_e (|q₁|q₂|)/r²

• think of Newton’s Law, equal and opposite reaction

Coulomb’s Law for Multiple Charges

• the resultant force on any one charge equals the vector sum of the forces exerted by the other individual charges that are present

• remember to add the forces as vectors

• Use Coulomb’s Law to find the force from each charge individually

• Treat each force as a vector (direction matters!)

• Add them vectorially → superposition (F_{1, net} = F₁₂ + F₁₃ + F₁₄ + F₁₅ + F_1n, …

Vector Equations

• A (QI)

• B (QII)

• C (QIII)

• D (QIV)

• A:

  • A_x = Acosθ

  • A_y = Asinθ

• B:

  • B_x = Bsinθ

  • B_y = -Bcosθ

• C:

  • C_x = -Ccosθ

  • C_y = -Csinθ

• D:

  • D_x = Dcosθ

  • D_y = -Dsinθ

If the vector is directly on the x-axis…

• D_x = Dcosθ = D

• D_y = 0

Adding Vectors Graphically

Adding Vector’s Mathematically

Which is the direction of the net force on the charge at the top?

What is the direction of the net force if the bottom left or bottom right charges are the point of interest?

A particle with charge 2microC is placed at the origin. An identical particle, with the same charge, is placed 2 m from the origin on the x axis, and a third identical particle, with the same charge, is placed 2 m from the origin on the y axis. The magnitude of the force on the particle at the origin is (k_e = 8.9875×10^9 N*m²/C²).

If you are asked to find the force of q³, what will happen?

• |F₂| = |F₃| based on where q’s are

• |F₂| = k(q₁q₂)/r² = 8.9×10^9 (2×10^-6)²/2²

  • 2² cancels on top and bottom

  • F = 8.99×10^-3 N

• q₁ = q₂, q₃ = 2×10^-6 C

• Find Resultant now:

  • R_x = -F₂

  • R_y = -F₃

  • R = sqrt( (-F₂)² + (-F₃)²) = 1.3×10^-2 N

• if asked to find force of q³, the F arrows will move and the Resultant will be different

The Electric Field

A region around a charged object where other charges experience a force. It is defined as the force per unit charge at a point in space.

• particle 2 pushes on particle 1 for example, despite a distance

• object 1 fills space around itself with a field, when object 2 is placed in field, field acts on object 2

• E = F/q₀ = k_eQ/r²

  • units: N/C

  • vector quantity

If an object has mass, it produces…

a gravitational field

If an object has charge, it produces…

an electrical field

The direction of the electrical field is defined as…

the direction of the electric force that would be exerted on a small positive test charge, q₀, at that point

The electric field produced by a negative charge is directed…

toward the charge

• negative source charge attracts positive test charges

The electric field produced by a positive charge is directed…

away from the charge

• positive source charge repels positive test charge

Electric Field Lines

• aid to visualize electric field patterns

• drawn pointing in the direction of the field vector at any point

• the electric field vector, E, is tangent to the electric field lines at each point

• the number of lines per unit area through a surface perpendicular to the lines is proportional to the strength of the electric field in a given region

Electric Field Line Patterns - Point Charge

• for a positive source charge

• for a negative source charge

• surround point charge, radiate equally in all directions

• for a positive source charge, the lines will radiate outward

• for a negative source charge, the lines will point inward

Electric Field Line Patterns - Dipole

• electric dipole = 2 equal and opposite charges

• the high density of lines between the charges indicates the strong electric field in this region

• no two field lines can cross each other

Electrostatic Equilibrium

when no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium

Isolated Conductor

• electric field is zero everywhere inside the conducting material

• any excess charge on an isolated conductor resides entirely on its surface

• the electric field just outside a charged conductor is perpendicular to the conductor’s surface

• on an irregularly shaped conductor, the charge accumulates at locations where the radius of curvature of the surface is smallest → at sharp points

Faraday Cage

A structure that shields its contents from external electric fields by redistributing charge, ensuring the electric field inside remains zero.

In an isolated conductor, if the electric field just outside a charged conductor is not perpendicular to the conductor’s surface, what would happen?

• the component along the surface would cause the charge to move

• it would not be in equilibrium

In an isolated conductor that is irregularly shaped, what is true?

• the charge accumulates at sharp points

• any excess carge moves to its surface

• the charges move apart until an equilibrium is achieved

• the amount of charge perunit area is greater at the sharp end(s)

• the forces from the charges at the sharp end(s) produce a larger resultant force away from the surface

• it’s why a lightning rod works (pointy)

Electric Flux

• Φ = E*A*cos(θ)

• Electric field through a surface

  • field lines penetrating an area A perpendicular to the field

  • the perpendicular to the area A is at an angle θ to the field

In electric flux, when the area is constructed such that a closed surface is formed, flux lines passing into the interior of the volume are…

negative

In electric flux, when the area is constructed such that a closed surface is formed, flux lines passing out of the interior of the volume are…

positive

The Area Vector

• a vector that points perpendicularly to a surface area, with a magnitude equal to the area of the surface. It is used in calculating electric flux through that surface.

• A(→) = An̂

  • vector in direction of n̂, perpendicular to the surface

  • vector with magnitude A equal to the area of the surface

• vector A(→) has units m²