PHYS 2210: Electricity

Potential difference between two points A and B in an electric field

(Line integral from A to B) ∆V = - ∫E ds

Electric force

Field force exerted between charged particles

Magnetic force

Field force exerted between moving charged particles

Direction of multiple charges

Multiple charges will create a net electric field at a location in space - Use superposition principle

What are the units of the Coulomb force?

N

If we know the electric field at a location in space and place a test charge at that spot, what do you need to have?

Need to get force on particle by multiplying charge by electric field

A charged spherical conductor acts like what at the center?

A point charge

Farady cage

Conducting sphere with a spherical cavity - If sphere is neutral, any field lines outside the sphere will terminate on sphere's surface which leaves the interior of the conductor at zero electric field - "Shields" electric field - When a charge hits insulator, it cannot move far

Grounding

Connecting an electroscope or other charged object to a larger conductor where the conductor allows a source or sink for electrons to move as they want to spread entirely on the outer surface of the new, big conductor - Effectively leaves charged object neutral

Is the Coulomb force conservative or nonconservative?

Conservative → maintains conservation of mechanical energy in an isolated system and work done by force is path independent

How can you find the electric field form the electric potential?

Potential is the integral of the field, then the field must be derivative of potential E = - DV / Dr (partial derivatives)

Dipole potential

Potential of an electric dipole along dipole axis can be found by adding the contribution of both charges V = kq/(x-a) - kq/(x+a) = 2kqa/(x²-a²)

What happens if we put a lot of charge on a 3D object?

Produces a field a distance away from the object and also produces an electric potential - Differential amount of potential a distance r away from one of the bits is dV = kdq/r V = ∫ kdq/r

Excess charge of an object

Q = ± Ne where N = number of particles

Quantized charge

Quantum of charge (indivisible unit of charge) - q proton = + e = 1.602 E-19 C - q electron = - e = -1.602 E-19 C - e = elementary charge

What is the direction of the field?

Radial outward direction which points away from the charge - If the source is positive, then the electric field points in the positive radial direction - Have removed test charge, but electric field still exists at P

If we place a dipole in a uniform external field E, the dipole naturally wants to what?

Rotate such that its dipole moment is aligned along the electric field - Net force on the dipole is zero because it has zero net charge - Net torque because of Coulomb force exerted by E on each of the charges - Dipole rotates such that its dipole moment is aligned along electric field

Glass rod experiment

Rubbing silk against a glass rod and then being able to pick up pieces of paper that sticks to rod - Energetically favorable is for a neutral atom where number of protons equals the number of electrons

Electric potential

Scalar quantity related to electric potential energy, but is NOT potential energy - Positive charge Q used to define electric field and place a positive test charge q₀ near "blob of charge" - Vector electric field located at position of test charge & electric potential (V) - Defined in terms of potential energy between source and test charge - Source of potential energy V = kQ/r - Source dependent U = V q → V = U/q

What causes polarization?

Separation of positive and negative charges - Charges are spatially separated by a finite distance, d - Dependent on the distance between charges and the amount of charge - p (vector) = d (vector) * Q - Only need polarization on atomic level - Can polarize a material regardless of conductivity

Negative test charge

Source and test charge attract each other - Coulomb force is opposite to the electric field - q₀ is negative

Electrostatics

Study of charges that do not move

Large charge distributions

Suppose we have a large object with many excess protons or electrons - Object will produce an electric field at a location in space - Can find electric field by splitting up object into differential point charges of charge dq and look at contribution of electric field at single point charge - Total field due all point charges that make up entire 3D object: E = ∫ kdq/r²

Equipotential surfaces

Surfaces that are two dimensional surfaces where the charge on the surface is constant (or equipotential)

Torque

T = p x E Magnitude of torque: T = pEsinθ

Gravitational force

The attractive field force that causes objects to fall to ground and keeps Earth in orbit around sun

Gauss's Law

The number of field lines determines the flux through the surface - The net flux through a closed surface is the total amount of charge inside the surface divided by the permittivity of free space - ∅E = ∫EdA = qenc/ε₀

Electric field

Vector field denoted by E and produced by a source charge - At the location of the test charge, E is defined by Coulomb force and test charge E = Fc/q₀ Fc = q₀E

Work done by Coulomb force is related to change in potential energy

W = - ∆U

What is the source of the Coulomb force?

When two charged particles are separated by a distance r, they exert equal and opposite Coulomb forces on one another - Not a contact force, but a field force - One charge produces an electric field which moves through space and pushes/pulls on the other charge ELECTRIC FIELD IS SOURCE OF COULOMB FORCE

What is the overall charge of a neutral atom?

Zero - As many protons as electrons AND as many electrons as protons

How does an electroscope work?

- Bring a positively charged rod to plate - Free electrons in electroscope (that was originally neutral) are attracted to top plate of glass - Leaves a net positive charge - Rigid beam and moveable beam are positively charged and repel each other

Uniform electric field

Consider potential energy in a uniform electric field created by a parallel plate capacitor - Positive point charge in plate capacitor is allowed to accelerate towards negative charge - Field is constant, so force is constant W = - ∆U = Fc * ∆r = (qE) * ∆r - ∆U = Ui - Uf = qE (xi-xf) Ui - Uf = qExi - qExf U = qEx + U₀ where U₀ determines here in space the electric potential is exactly zero

What happens if you place a charge in a uniform electric field?

Coulomb force accelerates proton to right along uniform electric field - Fc = qE = ma where a = qE/m where m is the mass of charged

Electric flux in an open surface and uniform electric field

Defined in terms of an imaginary "open surface" where that surface does not enclose a volume - ∅E = E * A = EAcosθ

Voltmeter

Device that can measure potential difference between its leads

Simple harmonic moment of dipoles

Dipole in external electric field will have a torque and will rotate clockwise, but will not stop rotating and continue to rotate counter-clockwise - If there are no non-conservative forces on system, system obeys conservation of mechanical energy

Potential of a spherical conductor

- Potential outside distribution is equal to V = kQ/r - At r = R (where R is the radius and r is the distance from the center of the sphere to a point P outside sphere), potential is V₀ = kQ/R V = V₀R/r

How do you get the electric field vector from field lines?

Draw lines tangent to the field line in the direction of the arrow → length of vector determined by bunching of lines

Smooth surfaces of a charged conductor

σ is lower because Q is smaller and A is large

Pointy surfaces of a charged conductor

σ is very high because Q is large and A is very small

Electric potential on the surface of a conductor

- Electric field on surface of conductor is perpendicular to surface - Differential displacement vectors (ds) are parallel to surface - In static equilibrium, ∆V = 0 and potential is constant on the surface of conductor - Field is zero inside, so electric field of integral is zero - Equipotential surface

Cylindrical symmetry

- Example is a very long wire (or line) carrying a uniform charge and constant linear charge density where l is very long - Field direction is along the radius of the cylinder ∅E = ∫E dA = E (2πrl) = q/ε₀ = λl/ε₀ E = λ/2πrε₀ = 2kλ/r

Charged conductor

- Field lines are perpendicular to surface of conductor because conductor is in static equilibrium - All charge lies on outer surface, but density is not constant - Gauss's law requires that most charge lies on "pointier" or "sharper" parts of conductor and less charge on smoother parts - Charge behaves this way to make electric field vectors add in the interior of conductor to zero

Describe charging by conduction

- First have a neutral conductor on an insulating pedestal - Bring a charged rod in direct contact to conducting sphere where electrons are transferred to conducting sphere - Take away charged rod - Electrons (or charge) spreads out quickly so it is uniformly distributed

Charged conducting sphere

- Has charge uniformly spread out on surface - Charge enclosed inside the Gaussian sphere is zero as all excess charge lies on the outer surface - Electric field inside conductor in static equilibrium is exactly zero

Field line potentials

- Lines terminate on the conductors along radii - Value of potentials on the surfaces get smaller the farther away from positive source - If equipotentials are "bunched up" the field lines are also bunched up (indicates stronger field) - Near the sphere, equipotentials are roughly spherical

Potential in a uniform field

A capacitor creates a uniform electric field & can find potential inside the capacitor V = U/q₀ = q₀Ex + U/q₀ = Ex + U₀/q₀ where V₀ = U₀/q₀ V = Ex + V₀

Electromagnetic force

A combination of electric and magnetic forces responsible for a wide range of phenomena including friction, phase transitions in matter, and light

Spherical distribution

A conducting sphere has a charge uniformly distributed on its surface - Charge uniformly spreads out - As long as charge is evenly spread out, it will behave like a point charge at the center of sphere

Field force of electrostatics

- Attractive and repulsive force - Depends on q1 and q2 - Inversely proportional to the distance squared

Describe charging by induction

- Begin with a charged sphere - Bring a negatively charged rod to sphere → polarizes sphere (positive charge is attracted to rod and negative charge is on the other side of sphere) - Connect a conducting wire which allows electrons to flow to ground - No electrons have been lost from rod - Conducting sphere is positive and electrons are absorbed by ground - Electrons redistribute and sphere is left with a surplus of positive charge

Spherical symmetry outside the sphere

- Use Gauss's law to find electric field everywhere in space due to a uniformly charged insulating sphere of radius R and charge Q of volume charge density p = Q/V where V is the volume of a sphere - Sphere is insulating which means charge is spread evenly through volume - Sphere is uniformly charged and volume charge density will be constant - Can use volume charge density to find charge of a certain volume of material ∅E = ∫E dA = E (4πr²) = Q / ε₀ E = Q / 4πε₀r² = kQ/r²

Electric flux in a closed surface

- dA vectors are perpendicular to surface and point away from volume (outward) - When all the flux is added together, the negative flux exactly cancels the positive flux and the net flux through the empty volume is exactly zero ∅E = ∫ EdA = 0 - Net flux through the surface is equal to the number of field lines "in" minus the number of field lines "out"

How to use Gauss's law

1. Determine the symmetry of the shape to be used - if it has spherical symmetry, use a sphere - if it has cylindrical symmetry, use a cylinder - if it has planar symmetry, use a cylinder 2. Draw shape such that the location that you want to find the field (point P) is on the surface of imaginary 3. Draw direction of field at point P - if spherical, field goes along radius - if cylinder, field goes along radius - if planar, field is perpendicular to top and bottom plates 4. Apply left integral - if sphere: ∅E = ∫E dA = E(4πr²) - if cylinder: ∅E = ∫E dA = E(2πrl) - if planar: ∅E = ∫E dA = EA (if one circle has a flux) ∅E = ∫E dA = 2 EA (if both circles have flux 5. Find right hand side of Gauss's Law qenc/ε₀ 6. Solve for field by setting right and left equal to each other

How to determine force of multiple point charges

1. Draw free body diagram of the charge you want to find force on. Must draw the force from each charge 2. Get magnitude of each force from Coulomb Law 3. Break each force into x and y components 4. Add components together

Electric field lines critera

1. Lines begin on positive charges and end on negative charges 2. Field line arrows represent field direction 3. Density or "bunching" of field lines is proportional to electric field strength 4. Field lines cannot cross because field lines represent field direction and electric field cannot have two directions at the same time

Electroscope

A special type of conductor used to measure charge - Fixed rod, moveable rod, and metal plate all electrically connected - Negative charge attracted to plate by positive charge → once rod removed, electrons distribute throughout scope & moving arm indicates amount of charge

Electron volt

An electron volt is the amount of work energy required to move one electron through one volt of potential difference - Work done by a Coulomb force is given by W = -∆U where electric potential ∆V = ∆U/q₀ where q₀ is test charge in the presence of a potential difference - W = ∆KE = -∆U = -q₀∆V - 1 eV = 1.602 E-19 J

Potential difference between two points A and B in a uniform electric field

Angle between the field and displacement is zero ∆V = - ∫E ds (from A to B) = - ∫ Edx (from 0 to l) = - E ∫ dx (from 0 to l) = - El

What is a point charge both a source of?

Both a source of electric field and electric potential → to find potential of a point charge, we use test charge and separate a distance r - To define potential: V = U/q₀ = kqq₀/rq₀ = kq/r

What happens if we have an electric dipole whose net charge is exactly zero?

By Gauss's law, the net flux through the sphere is zero (q enc = q - q = 0) so that the net electric field in space is nonzero - Flux is exactly zero because for every field line that leaves the sphere, one will come back into the sphere and terminate on the negative charge - Even though flux is zero, it does not mean that the field is zero

Which device accurately demonstrates the relationship between electric field and potential difference

Capacitor - E = ∆V/d - As long as you have an electric field in a region of space, there will be a potential difference between two points in that field - Work done by a Coulomb force in field is function of qEds, but work is related to potential difference by W = -q∆V

Charge separation

Charged objects near electroscopes cause charge separation because the electrons can move freely and the conductor has a positive and negative part

What happens when you place a point charge of - 3C in a spherical conductor with a spherical cavity?

Charges arrange themselves as to follow Gauss's law so that there is zero field and flux inside the conductor itself - -3C will repel exactly -3C of free electrons from the inner surface to the outer surface, leaving the outer surface with a charge of -3C - Inner surface will now have un-shielded protons which give the inner surface exactly +3C

What happens if we put excess charge on a conductor?

Charges rearrange fast and go into static equilibrium so that the charges no longer move - Excess charge is repulsive and will try to get away from each other

Negative source charge

Electric field behaves similarly, but vectors point towards a negative source and away from positive source

Point charge source

Electric field due to a point charge only depends on the source, so can use a test charge to find what the expression of field is - Source charge q straight line distance r away from test charge q₀ at point P - To find electric field at point P, we need to find Coulomb fore and then divide the value of the test charge q₀

What does electric field depend on?

Electric field only depends on the SOURCE charge - If you remove test charge, electric field still exists - As test charge gets closer to Q, the force and strength of electric field E increases

What is electric flux related to?

Electric flux is related to electric field lines and represents the number of field lines "piercing" through an imaginary surface - In a region of electric field, can imagine surfaces in any given orientation relative to field - Open and closed imaginary surfaces

Conductor

Electrons can freely move around

Field force of gravity

FG = G m1m2/r² - Gravity is always an attractive force - Depends on m1 and m2 (the masses ) - Inversely proportional to the distance squared

Infinite plane

Field is constant everywhere in space which is considered an infinite plane of constant surface density σ ∅E = ∫E dA = EA1 + EA2 = 2EA = qenc/ε₀ - To find the enclosed charge inside the Gaussian cylinder, the only real charged material inside the cylinder is a circle of the uniformly charged plane that has the same area as top and bottom circules of Gaussian cylinder of area A q enc = σA E = σ/2ε₀

Nonzero flux

Field lines of a point charge extend out to infinity (for positive point charges) or come from infinity (for negative point charges) - Net positive flux or net negative flux depending on line direction

Uniform disk

Field made up of a superposition of rings - Field along the axis will be confined to the z-axis E disk = σ/2ε₀ (1 - (z/√z²+R²)) where σ = surface charge density which is constant for a uniformly charged surface and is defined as charge per unit area on the surface σ = Q/A = Q/πr² for a disk since a disk is a circle - More charge we spread uniformly on surface of disk, larger σ - Think of σ as the thickness of charge

Coulomb force

Force between point charges - Like charges repel, opposites attract - Force obeys inverse square in distance law - Force is conservative - Direction of force is along line that connects charges - Force is proportional to the amount of charge - Force obeys Newton's Laws Fc = kq1q2/r² in the rhat direction

Newton's third law

Forces are equal in magnitude, but opposite in direction

Spherical symmetry inside the sphere

Gaussian sphere has a smaller radius than the real insulating material - Expect amount of charge inside blue sphere to be less than the total charge Q ∅E = ∫E dA = E (4πr²) = q/ε₀

Insulator

If electrons are bound and not flowing, material is an insulator - EX: rubber is an insulator → rub silk on one end of rubber and the other side of rod remains neutral where silk side has excess electrons that stay on the far side of the rubber rod - Build up of charge = -e where electrons are bound and not flowing - Electrons have a hard time moving around

Parallel plate capacitor

If we put a negative uniform plane next to positive uniform plane, we make a parallel plate capacitor - Direction of the fields causes a doubling effect - Positive and negative plate have same charge Q - Field lines point away from positive plate toward negative plate - Field is uniform because it is made up of planes, but only approximation because fields are not infinite - Assume that an ideal capacitor produces a uniform electric field E = σ/2ε₀ + σ/2ε₀ = Q/Aε₀ where A is the area of one of the plates

As the test charge gets closer to positive source, the potential energy of the system...

Increases - And so does electric potential

What is the general case for attraction and repulsion?

Like charges repel and unlike charges attract

In static equilibrium, where does all the excess charge lie?

On the outer surface of the conductor - For the case of conducting spheres, the only charge equally distributes itself on the outer surface leaving the interior neutral - Charge is more likely to congregate on pointy surfaces

What are the two types of electric charge?

Positive and negative

Electric field of a point charge from potential

Potential a distance r away from a source charge q is V = kq/r Er = - DV/Dr = - D(kq/r)/Dr = kq/r²

What happens when two point charges come near each other?

They have the potential for motion because they attract or repel each other - Potential energy can be found by getting work done by Coulomb force W = - ∆U = ∫Fc * dr = ∫ Fx * dx - ∆U = ∫ kq1q2/x² dx = -kq1q2 (1/xf - 1/xi) U = kq1q2/r + U₀

What are the two types of electrons?

Those bound to the nucleus (core electrons) and those that are valence and can move around freely

Why do you use Gauss's law?

To find the electric field of a point charge that has spherical symmetry - Suppose we have a point charge q and we want to find an expression for electric field at a point P a distance r from charge

Coulomb torsion pendulum

Torsion force from pendulum = electrostatic force in which charged particles move away from each other

Qualitative evaluation

Total amount of charge is conserved in the universe where ∆q = 0

Electric dippole

Two charges of equal magnitude, q, one positive and one negative, separated by a distance s - Along x axis, two charges have x components of field that cancel and the electric field on the axis perpendicular to dipole axis only points straight down - Dipole axis defined as axis that the dipole sits (y-axis)

Conservation of mechanical energy associated with torque

U = - p * E = -pEcosθ

What is the potential energy of a test charge?

U = kQqe/r where Q is the source and qe is the test charge

What is the potential due to a point charge?

V = kq/r

Electric dipole moment

Vector associated with the electric dipole - If a dipole moment exists, it will produce an electric field

Electric flux in an open surface and non-uniform electric field

d∅E = E * A Total flux on entire surface: ∅E = ∫EdA

What is the constant of proportionality

k = 8.99 E9 Nm²/C² - The Coulomb constant in terms of permittivity of free space ε₀ = 8.85 E-12 C²/Nm² k = 1/4πε₀

Charge density

p = Q / V

Dot product

u * v = |u| |v| cos θ