Lesson 3
• Electric Potential
 Potential Differences in a Uniform Electric Field
 Electric Potential and Potential Energy
 The Millikan Oil-Drop Experiment
• Capacitors
• Current Electricity
 Ohm’s Laws
 Resistance
Potential Difference and Electric
Potential
• When analyzing electric and magnetic fields, it is common
practice to use the notation ds to represent an infinitesimal
displacement vector that is oriented tangent to a path through
space. This path may be straight or curved, and an integral
performed along this path is called either a path integral or a line
integral (the two terms are synonymous).
• For an infinitesimal displacement ds of a charge, the work
done by the electric field on the charge is F . ds = q0E . ds.
• As this amount of work is done by the field, the potential energy of
the charge–field system is changed by an amount dU=-q0E . ds.
• For a finite displacement of the charge from point A to point B, the
change in potential energy of the system ΔU = UB - UA is
Potential Difference and Electric
Potential
Potential Difference and Electric
Potential
Potential Difference and Electric
Potential
Potential Differences in a Uniform
Electric Field
• Let us calculate the potential difference between two points A
and B separated by a distance IsI = d, where s is parallel to the
field lines. Then
Two points in an electric field.
Potential Differences in a Uniform
Electric Field
• The negative sign indicates that the electric potential at point
B is lower than at point A; that is, VB < VA. Electric field lines
always point in the direction of decreasing electric potential,
as shown in the Figure below.
(a) When the electric field E is directed downward, point B is at a lower electric potential than point A. When a positive test charge
moves from point A to point B, the charge–field system loses electric potential energy.
(b) When an object of mass m moves downward in the direction of the gravitational field g, the object–field system loses gravitational
potential energy.
Potential Differences in a Uniform
Electric Field
A uniform electric field directed along the
positive x axis. Point B is at a lower electric
potential than point A. Points B and C are at
the same electric potential.
• A system consisting of a positive charge and an electric field loses electric potential energy
when the charge moves in the direction of the field.
• As the charged particle gains kinetic energy, the charge–field system loses an
equal amount of potential energy.
• A system consisting of a negative charge and an electric field gains electric potential
energy when the charge moves in the direction of the field.
• The more general case of a charged particle that moves between A and B in a
uniform electric field such that the vector s is not parallel to the field lines, gives
thus
Electric Potential Due to Various
Charge Distributions
The Electric Potential Due to Two Point
Charges
• Example 1: The Electric Potential Due to Two Point Charges
A charge q 1= 2.00 μ C is located at the origin, and a charge q 2 =- 6.00 μC is
located at (0, 3.00) m,
(i) Find the total electric potential due to these charges at the point P,
whose coordinates are (4.00, 0) m
(ii) Find the change in potential energy of the system of two
charges plus a charge q3 = 3.00 μC as the latter charge moves from
infinity to point P
Solution (i)
For two charges,
The Electric Potential Due to Two Point
Charges
(ii)
The Millikan Oil-Drop Experiment
• During the period from 1909 to 1913, Robert Millikan performed a
brilliant set of experiments in which he measured e, the magnitude
of the elementary charge on an electron, and demonstrated the
quantized nature of this charge. His apparatus, diagrammed in the
Figure , contains two parallel metallic plates.
• Oil droplets from an atomizer are allowed to pass through a small hole in
the upper plate. Millikan used x-rays to ionize the air in the chamber, so
that freed electrons would adhere to the oil drops, giving them a
negative charge.
• A horizontally directed light beam is used to illuminate the oil
droplets, which are viewed through a telescope whose long axis is
perpendicular to the light beam. When the droplets are viewed in
this manner, they appear as shining stars against a dark background,
and the rate at which individual drops fall can be determined.
• Let us assume that a single drop having a mass m and carrying a charge q
is being viewed and that its charge is negative. If no electric field is present
between the plates, the two forces acting on the charge are the
gravitational force mg acting downward and a viscous drag force FD
acting upward.
The Millikan Oil-Drop Experiment
Schematic drawing of the Millikan oil-drop apparatus.
The forces acting on a
negatively charged oil droplet
in the Millikan experiment
The Millikan Oil-Drop Experiment
• Now suppose that a battery connected to the plates sets up an electric
field between the plates such that the upper plate is at the higher electric
potential. In this case, a third force qE acts on the charged drop.
Because q is negative and E is directed downward, this electric force
is directed upward, as shown in Figure b.
• If this force is sufficiently great, the drop moves upward and the
drag force FD’ acts downward. When the upward electric force qE
balances the sum of the gravitational force and the downward drag force
FD’ , the drop reaches a new terminal speed ν’ in the upward direction.
• With the field turned on, a drop moves slowly upward, typically at
rates of hundredths of a centimeter per second. The rate of fall in the
absence of a field is comparable. Hence, one can follow a single droplet
for hours, alternately rising and falling, by simply turning the electric field
on and off.
• After recording measurements on thousands of droplets, Millikan and his
co-workers found that all droplets, to within about 1% precision, had a
charge equal to some integer multiple of the elementary charge e :
Applications of Electrostatics
• The practical application of electrostatics is
represented by such devices as lightning rods and
electrostatic precipitators and by such processes as
xerography and the painting of automobiles.
• Scientific devices based on the principles of
electrostatics include electrostatic generators, the
field-ion microscope, and ion-drive rocket engines.
Assignment 1
With aid a digram describe the Van de Graaff Generator
Capacitors
Capacitors
• Capacitors are devices that store electric charge.
• Capacitors are commonly used in a variety of electric
circuits. For instance, they are used to tune the
frequency of radio receivers, as filters in power
supplies, to eliminate sparking in automobile ignition
systems, and as energy-storing devices in electronic
flash units.
• A capacitor consists of two conductors separated
by an insulator. The capacitance of a given capacitor
depends on its geometry and on the material—called a
dielectric- that separates the conductors.
Capacitors
Definition of Capacitance
Capacitors
Capacitance of a parallel-plate capacitor
• The capacitance of a parallel-plate capacitor is
proportional to the area of its plates and
inversely proportional to the plate separation,
Capacitors
• Example: Parallel-Plate Capacitor
Capacitors
• Example: The Cylindrical Capacitor
A solid cylindrical conductor of radius a and charge Q is coaxial with a cylindrical shell
of negligible thickness, radius b > a, and charge -Q. Find the capacitance of
this cylindrical capacitor if its length is l
Solution
Capacitors
• Example: The Spherical Capacitor
A spherical capacitor consists of a spherical conducting shell of radius b and
charge - Q concentric with a smaller con-ducting sphere of radius a and charge
Q . Find the capacitance of this device.
Solution
Capacitors
• Combinations of Capacitors:
• Parallel Combination
Capacitors
• Series Combination
Capacitors
Example:
Find the charges on the capacitors in the figure and the
potential difference across them
Capacitors
Example: A Paper-Filled Capacitor
i. Find its capacitance
ii. What is the maximum charge that can be placed on the
capacitor?
Solution (i)
Capacitors
(ii) The dielectric strength of paper is 16 x 106 V/m. Because the thickness
of the paper is 1.0 mm, the maximum voltage that can be applied
before breakdown is
Current Electricity
• Electric Current
The current is the rate at which charge flows through this surface. If ΔQ is the
amount of charge that passes through this area in a time interval Δt, the average
current Iav is equal to the charge that passes through A per unit time:
•The instantaneous current is defined as
•The steady current is defined as
I
I
dQ
dt
Q
t
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Current Electricity
• Microscopic Model of Current
A section of a uniform conductor of cross-sectional area A.
Current Electricity
• Microscopic Model of Current
Current Electricity
Example : Drift Speed in a Copper Wire
Current Electricity
• Resistance
Current Electricity
Ohm’s law
Materials that obey the above equation are said to follow Ohm’s law,
Ohm’s law states that
Current Electricity
Ohm’s law
A uniform conductor of length l and cross-sectional area A. A
potential difference ΔV=Vb-Va maintained across the
conductor sets up an electric field E, and this field produces
a current I that is proportional to the potential difference.
A potential difference ΔV=Vb-Va is maintained across the wire, creating in the wire an
electric field and a current. If the field is assumed to be uniform, the potential difference is
related to the field through the relationship
Current Electricity
Resistance
Resistance has SI units of volts per ampere. One volt per ampere is
defined to be one ohm (Ω):
The inverse of conductivity is resistivity
Current Electricity
Example: The Resistance of a Conductor
Calculate the resistance of an aluminum cylinder that has a length of 10.0 cm and a
cross-sectional area of 2.00 x 10-4 m2. Repeat the calculation for a cylinder of the
same dimensions and made of glass having a resistivity of 3.0 x 1010 Ωm.
Solution
Current Electricity
Example : The Resistance of Nichrome Wire
(i) Calculate the resistance per unit length of a 22-gauge Nichrome wire, which has
a radius of 0.321 mm.
(ii) If a potential difference of 10 V is maintained across a 1.0-m length of the
Nichrome wire, what is the current in the wire?
Solution
(i) The cross-sectional area of this wire is
(ii)
Current Electricity
Example: The Radial Resistance of a Coaxial Cable
Coaxial cables are used extensively for cable television and other electronic applications.
A coaxial cable consists of two concentric cylindrical conductors. The region between
the conductors is completely filled with silicon, as shown in Figure, and current leakage
through the silicon, in the radial direction, is unwanted. The radius of the inner
conductor is a " 0.500 cm, the radius of the outer one is b " 1.75 cm, and the length is
L " 15.0 cm. Calculate the resistance of the silicon between the two conductors.
Solution
Current Electricity
• Resistance and Temperature
Over a limited temperature range, the resistivity of a conductor varies approximately
linearly with temperature according to the expression
Where ρ is the resistivity at some temperature T (in degrees Celsius), ρ0 is the resistivity
at some reference temperature T0 (usually taken to be 20°C), and α is the
temperature coefficient of resistivity. The temperature coefficient of resistivity can be
expressed as
Current Electricity
Example: A Platinum Resistance Thermometer
A resistance thermometer, which measures temperature by measuring the change in
resistance of a conductor, is made from platinum and has a resistance of 50.0 Ω at
20.0°C. When immersed in a vessel containing melting indium, its resistance
increases to 76.8 Ω. Calculate the melting point of the indium.
Solution