Download Electric Potential around Point Charges

Electrostatics
Static electricity is the buildup and eventual release of charge in an object
due to the movement of electrons.
It is different from current electricity which is the constant flow of electrons through
a material.
Electrons can move from one atom to
another and from one material to another
because they are located in electron
clouds around the nucleus.
Protons CANNOT move from one material
to another because the protons are tightly
held in the nucleus.
Objects can obtain one of two types of
charge:
•A positively-charged object develops
if an object loses electrons
•A negatively-charged object
develops if an object gains electrons
Fundamental Rule for Electrostatic Charge
•Opposite charges attract
•Like charges repel
This is true for large-scale objects and for
atomic particles (protons, electrons, ions)
Electric charge is measured in coulombs (C)
A proton has a charge of +1.60 x 10-19 C (+e)
An electron has a charge of -1.60 x 10-19 C (-e)
Coulomb’s Law determines the amount of attraction or repulsion there is.
Fc = kq1q2/r2
Fc = electric force (N)
q1 = charge of one particle (C)
q2 = charge of second particle (C)
k = proportionality constant (8.99 x 109 Nm2/C2)
r = distance between charges (m)
Coulomb’s Law is also expressed this way.
Fc = (1/4πε0)q1q2/r2
ε0 = permittivity constant (8.85 x 10-12 C2/Nm2)
Referrring to the diagram to the
left, what is the net force acting
on Q3 by Q1 and Q2?
Electric Fields
Many forces can move things without
any type of contact.
Gravity moves objects without any
contact as long as the object is in the
Earth’s gravitational field. Gravity is a
“force field”
Force field – a region in space in which
an object can be placed and forces will
be exerted on the object without contact
Electrical charges can move other
electrical charges without making
contact; therefore, they have electric
fields.
Electric fields are represented with lines
that extend out from the center; the
closer the lines, the stronger the field.
Field lines are directional, depending on
the sign of the charge (Q).
To determine the direction of the field,
imagine the existence of a positive test
charge (P) close to the charge in
question (Q).
If Q is positive, P will be repelled.
The field lines move outward.
If Q is negative, P will be attracted.
The field lines move inward.
Rules for Field Lines
1. Pattern indicates the
direction of the field in a
given region (tangent to
point on line)
2. Concentration of lines
indicates the strength of the
field
3. Arrow indicates the direction
of the field (positive to
negative)
Figure (a) represents an electric
dipole – two equal charges of
opposite sign.
Electric Field Exerted on a Charge
The electric field exerted on a test charge is calculated as the force per unit of
charge. It is analogous to acceleration…
Test
Charge
(q)
a = F/m
E = F/q
a = acceleration (m/s2)
E = electric field (N/C)
F = gravitational force (N)
F = electric force (N)
m = mass (kg)
q = electric charge (C)
The larger the mass, the larger the
force exerted by gravity
Point
Charge
(Q)
The larger the test charge, the larger
the force exerted by point charge
What is the electric field strength at a point in space where a proton experiences
an acceleration of 1 million “g’s”? (mass of proton = 1.67 x 10-27 kg; charge of
proton = 1.60 x 10-19 C)
Reminder: Electric field lines always go from positive to negative regardless
of the force on the charge.
If the test charge in the field is
positive, the field is in the same
direction as the force (Figure “b”)
If the test charge in the field is
negative, the force is is in the
opposite direction (Figure “c”)
Electric Field Around a Charge
Think about your experience with gravity…the strength of the
gravitational field decreases as you move away from the Earth.
The strength of the electric field around a point charge decreases as
you move away from the charge
Test
Charge
(q)
We know that…
E = F/q
If F = kqQ/r2
Then E = (kqQ/r2)/q
E = kQ/r2
k = proportionality constant
Q = magnitude of point charge
r = distance to point charge
Point
Charge
(Q)
Therefore, the electric field at a given
distance r can be measured
independent of the magnitude of q
Two point charges are separated by a distance of 10.0 cm. One has a charge of 25 μC and the other +50 μC. Determine the magnitude and direction of the
electric field at a point P between the two charges that is 2.0 cm from the
negative charge.
Conductors and Insulators
Conductors – a material in which electrons move freely
from one atom to another and throughout the
material as a whole
Insulators –a material in which electrons are relatively
stationary and do NOT freely move throughout the
material
How Objects Obtain Charge
1. Charging by Friction (balloon on head)
Triboelectric
Series
Occurs when there is friction between two
different objects. They can be conductors or
insulators.
Electrons move to the material that “wants”
the electrons more.
How Objects Obtain Charge
2. Charging by Polarization (balloon with paper)
Occurs when a
charged object is
brought close to an
insulator.
Since electrons
cannot move freely,
atoms will turn
themselves so that
their similar side will
be turned away
from the object
Since the near side is oppositely charged, the object as a whole
attracts.
How Objects Obtain Charge
3. Charging by Conduction (electroscope, electrophorus)
When a charged
object makes
contact with a
neutral conductor,
electrons will move
accordingly until the
charge is equally
distributed.
Insulators can also be charged by contact, but since electrons
do not move freely, the charge remains localized.
How Objects Obtain Charge
4. Charging by Induction (electroscope, electrophorus)
When a charged
object is brought
close to a neutral
conductor,
electrons in the
conductor will
group up on one
side of the neutral
object “inducing” a
charge.
If the conductor is touched when a charge is induced, electrons
will move to neutralize the induced charge. This electron
movement actual gives the object a real charge.
Electric Potential
The comparison continues….
In figure “a”, the rock
has potential energy.
In figure “b”, assume
that the test charges
Q and 2Q are
positive. Why would
they have potential
energy?
The positive charges “want” to move off of the plate on the left and
move to the plate on the right.
With the rock…
With the charge…
PEtop = KE bot
PE pos = KE neg
∆KE = -∆PE
∆KE = -∆PE
In the same way that the larger rock has more potential energy, the larger
charge also has more potential energy.
Electric Potential – the electric potential
energy per unit charge
V = PE/q
V = Electric Potential (Volts or J/C)
PE = Potential Energy (J)
q = charge (C)
Potential Difference – the difference in
electric potentials between two
locations. Also known as the voltage
Vba = Vb - Va
= PEb / q - PEa / q
We know that
Vba = (PEb - PEa) / q
and
W = ∆KE = -∆PE
So,
Vba = -Wba / q
We also know that
W = Fd
and
E=F/q
So,
W = qEd
Therefore,
Vba = -(qEd) / q
Vba = -Ed
E=-V/d
E = Electric Field (N/C or V/m)
V = Potential Difference (V)
d = distance (m)
The electric field increases as the
voltage across the plates increase,
but it decreases as the distance
between the plates decreases.
When an electron in a television tube is accelerated from rest through a potential
difference of 5000 V…
(a) What is the change in potential energy of the electron?
(b) What is the speed of the electron at the other end of the tube?
(c) What is the electric field across the tube if the distance across the tube is 50
cm?
Equipotential Lines
What do the lines on this map represent?
Lines of equal elevation. No work needed to walk along line.
Close lines mean the elevation changes dramatically.
Equipotential lines show were the electric potential
around charged objects is equal.
They are always perpendicular to electric field lines.
No work needed to move charge along equipotential
line.
Lines that are close together that potential
difference is great in short distance.
Electric Potential around Point Charges
The electric field around a point charge is:
E = kQ/r2
The electric field is determined by the
strength of the charge but also the
distance away from it.
The electric potential around a point
charge is:
V = kQ/r
If you double the distance from the point
charge,
The electric potential is ½ the original
amount, but the electric field is ¼ the
original amount.
Because electric potential is a scalar (like energy), direction does not matter.
However, positive charges have positive potentials and negative charges have
negative potentials (like energy). The total electric potential is the sum of all of
the potentials.
V = Σ kQ / r
What is the electric potential at point A and the electric potential at point B due
to charges Q1 and Q2?
Capacitance
Capacitors are devices that can
store electric charge and then
release it in a short burst of
charge.
They consist of two conductive
materials that are placed near
each other but not touching.
Figure (a) shows a
capacitor made up of
parallel plates that are
separated by a layer
of air.
Figure (b) shows two
conductive layers
separated by an
insulating material
A capacitor loads up with a
potential difference (voltage)
when opposite charges get
established on the different
plates.
Since the plates are separate
but not touching, there is no
way initially for the charge to
move from one plate to the
other.
The voltage will eventually become large enough to overcome the gap
between the plates resulting in a sudden movement of charge (discharge).
Capacitance – the ability of an object to
store electrical charge. Measures the
amount of charge that can be held for a
given voltage
Capacitance is determined by the gap
between the plates and the area of the
plates.
Some capacitors have dielectric
materials which are inserted in the air
gap to improve the insulating property
between the plates
C=Q/V
Q = charge (C)
C = capacitance (farad)
V = voltage (V)
C = Kε0A / d
C = capacitance (F)
K = dielectric constant
ε0 = permittivity constant
(8.85 x 10-12 C2/Nm2)
A = area of one plate (m2)
d = distance between plates (m)
A capacitor has plates that are 20 cm x 3.0 cm and are separated by a gap of
1.0 mm of air.
a. What is the capacitance?
b. What will be the charge on the plates if a 12-V battery is connected
across the plates?
c. What is the magnitude of the electric field between the plates?
d. How big would the plates have to be to get a capacitance of 1.0 F?
Energy Storage in Capacitors
The potential energy stored in a
capacitor is equal to the work that is
done to move charge to the plates of
the capacitor.
PE = W = VQ
As more charge gets built up on the
capacitor, more work is needed to
move charge onto the plates.
The total work needed to move all of
the charge is equal to the average
voltage across the capacitor during the
process
Average V = (Vf + 0) / 2 = ½Vf
PE = ½VfQ
Q = CV
PE = ½CV2
PE = Energy (J)
C = Capacitance (F)
V = Voltage (V)
A camera flash unit stores energy in a 150-μF capacitor at 200 V. How much
electric energy can be stored?
Charge in Batteries
Batteries consist of two different types of metals and a medium that allows
electrons to move.
Oxidation occurs at the carbon electrode when
it to lose electrons making it positive.
Reduction occurs at the zinc electrode when it
gains electrons becoming negative.
Voltage is created between terminals based on
the amount of charge on each of the terminals.
What makes batteries have different
voltages?
When is a battery dead?
Since the electrons have the potential to move, all they need is a path to
return.
Circuit – a complete path from one battery terminal to another through a
conductive material
The flow of electrons can be used to make electrical devices function
provided they become part of the circuit.
Which of the following would cause the bulb to light up?
Light bulbs are
designed to allow the
flow electrons in one
way, passing through
the filament, and out a
different way.
Circuits are represented by
schematic designs.
The actual flow of electrons
is from the negative terminal
to the positive terminal.
The historical convention for
current flow is as positive
charge flowing from the
positive terminal to negative
terminal.
Current – the amount of charge that passes a given point in a conductor
I = Q / ∆T
I = Current (Amperes)
∆Q = change in charge (C)
∆T = elapsed time (s)
Current, Voltage, and Resistance
If voltage is analogous to a change in elevation, current is analogous to a
flowing river.
As elevation increases, the flow of water increases.
As voltage increases, the flow of charge (current) increases.
Therefore, current is directly proportional to voltage.
Resistance impedes flow.
Obstacles in a river slow down the flow of water
Conductors can have electrical resistance that slows down the flow of
charge.
As resistance increases, current decreases.
Therefore, current is inversely proportional to resistance.
Ohm’s Law
I=V/R
Or
V = IR
V = voltage (V)
I = current (A)
R = resistance (Ohms)
Clarifications about Voltage, Current and Resistance
1. Batteries are sources of constant voltage, not constant current.
The current out of a batteries changes depending on the resistance
of the circuit.
2. Resistance is a property of a device or conducting material.
Voltage, however, is applied “across” a device. Current is a
“response” to the voltage across a device depending on the
resistance.
3. Current is directional, but it is not a vector, It always moves parallel
to a wire.
4. Current and charge does not get used up by a device in a circuit.
The charge that leaves one terminal of a battery travels through the
entire circuit and returns to the other terminal.
5. Electric potential energy gets used up by a specific device.
Resistances have voltage “drops” where electric potential energy
is converted to other forms of energy (light, heat, etc.)
Electric Power
Devices in a circuit that are “voltage
drops“ convert the potential energy in the charge
to other types of energy.
Motors convert electric energy to mechanical
energy (movement)
Light bulbs convert electrical energy to radiant
energy (light)
Heaters convert electrical energy to thermal
energy (heat)
All of these devices use current at different rates
which is their power
We know power is amount of work done per unit of
time
P=W/t
In an electrical device, work comes from the
potential energy in the charge, so
P = PE / t
Recall the relationship between potential energy, charge, and voltage
V = PE / Q
PE = QV
So,
P = QV / t
Since Q / t = I
P = IV
Since V = IR
And since I = V/R
P = power (Watts)
P = I(IR)
P = (V/R)V
I = current (A)
P = I2R
P = V2/R
V = voltage (V)
Since resistance is a property of a conducting material, it is determined by
physical characteristics of the material.
R = ρL / A
R = resistance (Ω)
ρ = resistivity coefficient (Ωm)
L = length of wire (m)
A = cross-sectional area (m)
A hair dryer draws 7.5 A when plugged into a 120-V line.
a. What is its resistance?
b. How much charge passes through it in 15 minutes?
c. What is the length of copper wire running through the hair dryer assuming it is
14-gauge wire with a diameter of 1.6 mm?
Batteries and other devices provide
electromotive force (emf) to circuits.
Electromotive force (E) – the process
of transforming chemical energy to
electrical energy. Think of it as the
peak voltage of a battery (measured
in volts)
Batteries also have internal resistance
(r) that develops when current is
drawn from the battery. This results in
a slighty lower voltage provided by the
battery.
Terminal voltage – the actual voltage
provided by a battery when in a circuit
V = E - Ir
Series Circuit
A series circuit is one that has only one path through which current can travel
Current flows equally through all three resistors even though they are not
necessarily the same. There is no buildup of electrons at higher resistors.
I = constant
Since charge loses its all of its energy by the time it gets back to the
battery, the voltage drops across all three resistors is equal to the voltage
gain in the battery.
V = V1 + V2 + V3
Equivalent resistance – the sum of the individual resistances in the circuit
V = V1 + V2 + V3
In a series circuit:
IR = IR1 + IR2 + IR3
Current constant
IR = I (R1 + R2 + R3)
Voltages add
Req = R1 + R2 + R3
Resistances add
Parallel Circuit
A parallel circuit is one that has many paths through which current can travel
Current divides
whenever it gets
to a place where
one wire
branches into
two or more
wires. Current
recombines
when wires
merge
I = I1 + I2 + I3
Each charge loses energy across a different resistor, but they each lose
the same amount. This is because they leave the battery with the same
energy and return to the battery with zero energy.
V = V1 = V2 = V3
V = constant
To find the equivalent resistance…
I = I1 + I2 + I3
V/R = V/R1 + V/R2 + V/R3
V/R = V (1/R1 + 1/R2 + 1/R3)
1/R = 1/R +1/R +1/R
eq
1
2
3
In a parallel circuit:
Voltage constant
Currents add
Reciprocal resistances add
What does that mean in real life?
Two light bulbs, each with a resistance of 10 Ω, are connected across a
battery. If the terminal voltage of the battery is 24 V,
a) What is the equivalent resistance and current if the bulbs are connected in
series?
b) What is the equivalent resistance and current if the bulbs are connected in
parallel?
Combination circuits have
resistors in series and
parallel.
To find the equivalent
resistance, work from the
inside out.
Find the equivalent resistance of the circuit.
Find the individual currents and voltage drops across each resistor.
1. Find equivalent resistance
2. Find total current
3. Find individual voltage drops
4. Find individual currents
Capacitors in Circuits
Capacitors in circuits are used to store up charge and to control the flow
of current.
When a capacitor is put in a circuit
with a battery, electrons flow
building up positive and negative
charge on the plates
As charge builds up, voltage
approaches voltage of battery and
current slows down.
When voltage of capacitor equals
voltage of battery, current stops
flowing because there is no
potential difference between
capacitor and battery.
If R is a light bulb, it lights up initially
and slowly gets dimmer.
That same charged capacitor
can be used like a battery. It
acts like a short-term power
source.
When switch is closed,
charge starts to flow from the
capacitor through the
resistor.
Once charge moves around
the loop, capacitor is
discharged.
If R is a light bulb, how would
it respond to the capacitor
being discharged?
Capacitors in Parallel
When capacitors are in parallel, the plates
all have the same voltage across them.
V = V1 = V2 = V3 V = constant
Also, total charge gets divided in each path
Q = Q1 + Q2 + Q3
We know Q
= CV,
CV = C1V + C2V + C3V
CV = (C1 + C2 + C3)V
Ceq = C1 + C2 + C3
Capacitors in Series
When capacitors are connected in
series, the individual voltages
across each capacitor have to add
up to the total voltage from the
battery.
V = V1 + V2 + V3
The magnitudes of all Q’s on A
and B are equal and electrically
neutral (no net charge)
Q = constant
We know V
= Q/C
Q/C = Q/C1 + Q/C2 + Q/C3
Q/C = Q (1/C1 + 1/C2 + 1/C3)
1/Ceq = 1/C1 + 1/C2 + 1/C3