Chapter 20: Electric Potential and Electric Potential Energy

Electric Potential causes…

• voltage

• electric potential = height of hill, voltage = height difference between two points on hill

Total mechanical energy of a system is constant when…

• forces are conservative

• E = K + U

• K = ½ mv²

Potential Enegy

function of the position of particles

The change in potential energy of a particle moving between two points is __________ of the path taken.

independent

The gravitational potential energy of two masses separated by a distance r is:

F = GMm/r²

U = -Gm₁m₂/r

U_grav = mgy

Coulomb force is…

conservative

If electric force does negative work, potential energy…

increases

ΔU = -W_conservative = -FΔxcosθ

If electric force does positive work, potential energy…

decreases

ΔU = -W_conservative = -FΔxcosθ

a) the electric field does positive work on the electron

• The displacement and Force are pointing in the same direction, so work is positive

Electric field moves from…

positive to negative charge

Electric Potential, U_E equation derivation

E = F/q → F = qE

W = FΔxcosθ = qEΔxcosθ

ΔU = -W = -qEΔxcosθ

• if theta = 90, cos90=1, so can often omit

Potential Difference

• V (Voltage)

• ΔV = ΔU/q

  • units: unit/Volt (V) = J/C

  • Voltage = potential energy/charge, is like height for a charge

• ΔV = -EΔxcosθ = -EΔx

  • costheta = 1

  • q cancels out

• E = -ΔV/Δx = F/q

• ΔU = qΔV = -W

• when Voltage changes, it creates the Electric Field

Battery Example - Voltage/Potential Difference

• battery does not give charge to lightbulb, wire does

• if battery has a voltage of 12V, the positive terminal has a voltage of 12V and the negative has a voltage of 0

• ΔV = 120 V

• ΔV = -EΔx

• |E| = |-ΔV/Δx| = |-120V /0.013 m| = 9200 V/m = c

The center circle of an outlet is…

ground

Current

flow of charge, flows because of electric field

Conservation of Mechanical Energy: Mass

• E_A = E_B

  • E = K + U

  • K = 1/2mv²

  • U_g = mgh

• K_A + U_A = K_B + U_B

Conservation of Mechanical Energy: Charge

• E_A = E_B

  • k = 1/2mv²

  • U_E = qV = qΔV

• K_A + U_A = K_B + U_B

If m = 1kg and A = 5m height, what is velocity?

• use conservation of mechanical energy (mass) to find

• E_A = E_B

• K_A + U_A = K_B + U_B

  • K_A = 0, U_B = 0

• U_A = K_B

• mgh = ½mv²

• sqrt(2gh) = v = sqrt(2(9.8)(5)) = 9.9 m/s

b

In a problem, if there is a height difference or charge moving through voltage…

apply energy conservation

positive charges accelerate in the direction of…

• decreasing electric potential

  • because the force on a negative charge is opposite to the field direction

• moves to a region of lower energy

negative charges accelerate in the direction of…

• increasing electric potential

  • because the force on a negative charge is opposite to the field direction

• moves to a region of lower energy

The point of zero electric potential is taken to be at ______ distance from the charge

an infinite

T/F: A potential exists at some point in space whether or not there is a test charge at that point.

True

Electric Potential (V) for a point charge

• V = k(q/r)

  • k = 8.99×10⁹

  • q = charge

  • r = distance

Electric Potential Energy (U)

U = qV = k((q₁+q₂)/r)

• scalar

• V = V₁ + V₂

• V = k(q/r)

• q/r₁ = (5×10^-6C/+4)

• r₂ = sqrt((4²)+(3²)) = 5

• q/r₂ = (-2×10^-6C/5)

• V = (8.99×10^9)((5×10^-6C/+4)(-2×10^-6C/5)

• c: 7.6×10³

There is always _________ field, but _________ only exist if there is another charge nearby.

• always gravitational or voltage & electric field

• Force & U only exist if another nearby charge

d. zero

Equipotential Surface: Definition & Properties

• same voltage/potential all around surface, ΔV = 0

• all conductors have

• no work required to move across surface

• Properties:

  • W = qΔV = 0

  • ΔV = -EΔxcosθ

    • If ΔV=0, θ=90 degrees

    • E field is perpendicular

d. C & E

• V = scalar

  • V = k(q/r)

  • In this ex: V = k(q/r) - k(q/r) - k(q/r) + k(q/r) → cancel out

• E = vector

  • E = k(q/r²)

  • net E field goes towards left in this example

• d. E dne 0, V=0

  • electric field is not zero, potential is zero

An EKG plots…

the heart’s potential difference (V) vs. time

Capacitor

• 2 parallel plates, one positively charged and one negatively charged

• store charge and energy

• provide Electric Field

Capacitance (C)

• C = q/ΔV (Farads; F = C/V)

  • q = charge

  • deltaV = change in voltage

• depends only on geometry of conductors

Are batteries capacitors?

• No

• batteries release energy slowly, capacitor quickly

  • capacitor example: camera, touch screen (your finger = one plate, screen the other)

  • a battery can be used to charge the capacitor

Parallel Plate Capacitor

• C = ε₀ A/d

  • ε₀ = 8.85×10^-12 C²/nm²

• larger area, less space between plates = holds more charge

• C=Q/V, Q=CV

• d.

Energy Stored in Capacitor

• Energy = ½QΔV = ½CΔV² = Q²/2C

• extreme example: Z machine in New Mexico; used in experiments in controlled nuclear fusion

  • uses large number of capacitors in parallel to give tremendous equivalent capacitance

  • discharge energy into a target, heated to extremely high temp

• can be charged slowly and then release energy very quickly

  • defibrillator

  • fibrillation (heart muscles twitch randomly and cannot pump blood)

  • strong electric shock stops the heart, giving cells ability to restore proper heartbeat

• C = ε₀ A/d

• C = 8.85×10^-12(C²/Nm²)(0.0005m²)/(0.005m) = 8.85×10^-13 F

  • 5 cm² (0.01×0.01m²/1×1cm²) = 0.0005m²

• Energy = ½QΔV = ½CΔV²

  • Energy = ½ (8.85×10^-13F)(15.0V)² = 9.96×10^-11 = 10×10^-11

Capacitors with dielectrics

• dielectric: insulating material between capacitor plates that increases capacitance

• examples: rubber, plastic, or waxed paper

• when dielectric completely fills region between plates, multiply by factor κ:

  • C = κC₀ = κε₀(A/d)

If the potential difference between the plates of a capacitor is maintained, as by the presence of battery B, the effect of a dielectric is to…

• increase the charge on the plates

• wants to increase capacitance

• The battery keeps the voltage constant.

• Adding a dielectric increases the capacitance.

• Since Q=CV, more capacitance → more charge on the plates.

• The electric field and voltage stay the same, but the capacitor can store more energy.

Basically: dielectric = more “room” for charge when voltage is fixed.

If the charge on the capacitor plates is maintained, as in this case by isolating the capacitor, the effect of a dielectric is to…

• reduce the potential difference between the plates

• wants to increase capacitance

• Now the charge is fixed, so Q doesn’t change.

• Adding a dielectric increases capacitance (C=κC₀)

• Since V=Q/C, more capacitance → voltage decreases.

• Electric field E=V/d also decreases.

• Energy U=Q²/(2C) decreases too.

Atomic Description of Dielectrics

• presence of the positive charge on the dielectric effectively reduces some of the negative charge on the metal

• this allows more negative charge on the plates for a given applied voltage

• the capacitance increases

need to do out still xx.