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

• 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 :

The Millikan Oil-Drop Experiment

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

• The capacitance of a parallel-plate capacitor is proportional to the area of its plates and inversely proportional to the plate separation,

Capacitance of a parallel-plate capacitor

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

Find the charges on the capacitors in the figure and the potential difference across them Example:

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 dQI

dt

•The steady current is defined as QI

t

28

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

Materials that obey the above equation are said to follow Ohm’s law,

Ohm’s law states that

Ohm’s law

Current Electricity

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

Ohm’s law

Current Electricity

Resistance has SI units of volts per ampere. One volt per ampere is defined to be one ohm (Ω):

Resistance

The inverse of conductivity is resistivity

Current Electricity

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.

Example: The Resistance of a Conductor

Solution

Current Electricity

(i) Calculate the resistance per unit length of a 22-gauge Nichrome wire, which has a radius of 0.321 mm.

Example : The Resistance of Nichrome Wire

(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 The cross-sectional area of this wire is (i)

(ii)

Current Electricity

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.

Example: The Radial Resistance of a Coaxial Cable

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

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

Example: A Platinum Resistance Thermometer