Physics 1201

More Properties of Charge

* Nature's basic carrier of negative charge is the electron. - Gaining or losing electrons is how an object becomes charged. • Electric charge is always conserved. - Charge is not created, only exchanged - Objects become charged because negative charge is transferred from one object to another.

Potential Energy

* The concept of poten:tial energy is useful in the study of electricity. • A poten:al energy function can be defined corresponding to the electric force. • Electric potential can also be defined. • The concept of potential relates to circuits.

Charles Coulomb

• 1736 - 1806 • Studied electrostatics and magnetism • Investigated strengths of materials - Identified forces acting on beams

Georg Simon Ohm

• 1787 - 1854 • Formulated the concept of resistance • Discovered the proportionality between current and voltages

Benjamin Franklin

* 1706 - 1790 * Printer, author, founding father, inventor, diplomat * Physical ScienAst - 1740's work on electricity changed unrelated observations into coherent science

Problem Solving Strategies

* Check Them Out

Faraday's Ice-Pail Experiment

* Concluded a charged object suspended inside a metal container causes a rearrangement of charge on the container in such a manner that the sign of the charge on the inside surface of the container is opposite the sign of the charge on the suspended object

Polarization

* In most neutral atoms or molecules, the center of positive charge coincides with the center of negative charge. • In the presence of a charged object, these centers may separate slightly. - This results in more positive charge on one side of the molecule than on the other side • This realignment of charge on the surface of an insulator is known as polarization.

Properties of Electric Charges

*Two types of charges exist - They are called posiAve and negaAve - Named by Benjamin Franklin * Like charges repel and unlike charges aIract one another. * Nature's basic carrier of posiAve charge is the proton. - Protons do not move from one material to another because they are held firmly in the nucleus.

Capacitance (

• A capacitor is a device used in a variety of electric circuits. • The capacitance, C, of a capacitor is defined as the ra:o of the magnitude of the charge on either conductor (plate) to the magnitude of the potential difference between the conductors (plates). * C= Q/ΔV • Units: Farad (F) - 1 F = 1 C / V - A Farad is very large • Often will see μF or pF • ΔV is the poten:al difference across a circuit element or device. • V represents the actual potential due to a given charge at a given location.

Circuits

• A circuit is a closed path of some sort around which current circulates. • A circuit diagram can be used to represent the circuit. • Quantities of interest are generally current and potential difference.

Capacitors in Circuits

• A circuit is a collection of objects usually containing a source of electrical energy (such as a battery) connected to elements that convert electrical energy to other forms. • A circuit diagram can be used to show the path of the real circuit.

Electric Field Lines (E->)

• A convenient aid for visualizing electric field paIerns is to draw lines pointing in the direction of the field vector at any point. • These lines are called electric field lines and were introduced by Michael Faraday. • The field lines are related to the field in the following manners: - The electric field vector, , is tangent to the electric field lines at each point. - The number of lines per unit area through a surface perpendicular to the lines is proportional to the strength of the electric field in a given region.

Charging by Induction

• A negatively charged rubber rod is brought near an uncharged sphere. • The charges in the sphere are redistributed. - Some of the electrons in the sphere are repelled from the electrons in the rod. • The region of the sphere nearest the negatively charged rod has an excess of positive charge because of the migraAon of electrons away from this location. • A grounded conducting wire is connected to the sphere. - Allows some of the electrons to move from the sphere to the ground • The wire to ground is removed, the sphere is lea with an excess of induced positive charge • Initially, the positive charge on the sphere is nonuniformly distributed. • Eventually, the excess positive charge becomes evenly distributed due to the repulsion between the positive charges. • Charging by induction requires no contact with the object inducing the charge.

Meters in a Circuit - Voltmeter

• A voltmeter is used to measure voltage (potential difference). - Connects to the two contacts of the bulb

Meters in a Circuit - Ammeter

• An ammeter is used to measure current. - In line with the bulb, all the charge passing through the bulb also must pass through the meter.

Electric Field Line Patterns (Part 2)

• An electric dipole consists of two equal and opposite charges. • The high density of lines between the charges indicates the strong electric field in this region. • Two equal but like point charges • At a great distance from the charges, the field would be approximately that of a single charge of 2q • The bulging out of the field lines between the charges indicates the repulsion between the charges. • The low field lines between the charges indicates a weak field in this region. • Unequal and unlike charges • Note that two lines leave the +2q charge for each line that terminates on -q • At a great distance from the charges, the field would be equivalent to that of a single charge +q

Van de Graaff Generator

• An electrostatic generator designed and built by Robert J. Van de Graaff in 1929 • Charge is transferred to the dome by means of a rotating belt. • Eventually an electrostatic discharge takes place.

Equipotential Surfaces

• An equipotential surface is a surface on which all points are at the same potential. - No work is required to move a charge at a constant speed on an equipotential surface. - The electric field at every point on an equipotential surface is perpendicular to the surface.

Property 2

• Any excess charge on an isolated conductor resides enArely on its surface. - A direct result of the 1/r2 repulsion between like charges in Coulomb's Law - If some excess of charge could be placed inside the conductor, the repulsive forces would push them as far apart as possible, causing them to migrate to the surface.

Electrical Force Compared to GravitaAonal Force

• Both are inverse square laws. • The mathemaAcal form of both laws is the same. - Masses replaced by charges - G replaced by ke • Electrical forces can be either aIracAve or repulsive. • Gravitational forces are always attractive. • Electrostatic force is stronger than the gravitational force.

Properties of Charge, Final

• Charge is quantized. - All charge is a mulAple of a fundamental unit of charge, symbolized by e • Quarks are the excepAon - Electrons have a charge of -e - Protons have a charge of +e - The SI unit of charge is the Coulomb (C) • e=1.6x10-19 C

Current and Drift Speed

• Charged particles move through a conductor of cross-sectional area A. • n is the number of charge carriers per unit volume. • n A Δx is the total number of charge carriers. • The total charge is the number of carriers times the charge per carrier, q - ΔQ = (n A Δx) q • The drift speed, vd, is the speed at which the carriers move. - vd =Δx/Δt • Rewritten: ΔQ = (n A v d Δt)q • Finally, current, I = ΔQ/Δt = n q v d A • If the conductor is isolated, the electrons undergo random motion. • When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current.

Conductors

• Conductors are materials in which the electric charges move freely in response to an electric force. - Copper, aluminum and silver are good conductors. - When a conductor is charged in a small region, the charge readily distributes itself over the entire surface of the material.

Coulomb's Law

• Coulomb shows that an electric force has the following properties: - It is directed along the line joining the two particles and inversely proportional to the square of the separation distance, r, between them - It is proportional to the product of the magnitudes of the charges, |q1|and |q2|on the two particles - It is attractive if the charges are of opposite signs and repulsive if the charges have the same signs • Mathematically, • ke is called the Coulomb Constant - ke =8.9875x109 Nm2/C2 • Typical charges can be in the μC range - Remember, Coulombs must be used in the equation • Remember that force is a vector quanAty • Applies only to point charges and spherical distributions of charges - r is the distance between the two centers of charge

Problem Solving with Electric Potential (Point Charges)

• Draw a diagram of all charges. - Note the point of interest. • Calculate the distance from each charge to the point of interest. • Use the basic equation V = keq/r - Include the sign - The potential is positive if the charge is positive and negative if the charge is negative. • Use the superposition principle when you have multiple charges. - Take the algebraic sum • Remember that potential is a scalar quantity. - So no components to worry about

Electric Flux

• Electric flux is a measure of how much the electric field vectors penetrate through a given surface. • Field lines penetraAng an area A perpendicular to the field • The product of EA is the flux, Φ • In general: - Φ E = E A cos θ • ΦE =EAcosθ - The perpendicular to the area A is at an angle θ to the field - When the area is constructed such that a closed surface is formed, use the convenAon that flux lines passing into the interior of the volume are negaAve and those passing out of the interior of the volume are positive. - SI unit : N * m2 /C

Potential Energy Compared to Potential

• Electric poten:al is characteristic of the field only. - Independent of any test charge that may be placed in the field • Electric potential energy is characteris:c of the charge-field system. - Due to an interaction between the field and the charge placed in the field

Energy Stored in a Capacitor

• Energy stored = 1⁄2QΔV • From the definition of capacitance, this can be rewritten in different forms. * Energy = 1/2QΔV = 1/2CΔV^2 = Q^2/2C

Equipotentials and Electric Fields Lines - Dipole

• Equipotential lines are shown in blue. • Electric field lines are shown in orange. • The field lines are perpendicular to the equipotential lines at all points.

Ohm's Law

• Experiments show that for many materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents. • This statement has become known as Ohm's Law. - ΔV = I R • Ohm's Law is an empirical rela<onship that is valid only for certain materials. - Materials that obey Ohm's Law are said to be ohmic. • An ohmic device • The resistance is constant over a wide range of voltages. • The relationship between current and voltage is linear. • The slope is related to the resistance. • Non-ohmic materials are those whose resistance changes with voltage or current. • The current-voltage relationship is nonlinear. • A diode is a common example of a non-ohmic device.

Electrical Field

• Faraday developed an approach to discussing fields. • An electric field is said to exist in the region of space around a charged object. - When another charged object enters this electric field, the field exerts a force on the second charged object. • A charged parAcle, with charge Q, produces an electric field in the region of space around it. • A small test charge, qo, placed in the field, will experience a force. • Mathematically, • SI unit: N/C • Use this for the magnitude of the field • The electric field is a vector quantity • The direction of the field is defined to be the direction of the electric force that would be exerted on a small positive test charge placed at that point.

Temperature Varia<on of Resistivity

• For most metals, resistivity increases with increasing temperature. - With a higher temperature, the metal's constituent atoms vibrate with increasing amplitude. - The electrons find it more difficult to pass through the atoms. • For most metals, resistivity increases approximately linearly with temperature over a limited temperature range. ρ = ρ0[1+α(T-T0)] - ρ is the resistivity at some temperature T - ρo is the resistivity at some reference temperature To • To is usually taken to be 20° C - α is the temperature coefficient of resistivity • Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, you can find the effect of temperature on resistance. R = R0[1+α(T-T0)]

Gauss' Law (Theres an equation)

• Gauss' Law states that the electric flux through any closed surface is equal to the net charge Q inside the surface divided by εo - εo is the permittivity of free space and equals 8.85 x 10-12 C2/Nm2 - The area in Φ is an imaginary surface, a Gaussian surface, it does not have to coincide with the surface of a physical object.

Notes About Electric Potential Energy of Two Charges

• If the charges have the same sign, PE is positive. - Positive work must be done to force the two charges near one another. - The like charges would repel. • If the charges have opposite signs, PE is negative. - The force would be attractive. - Work must be done to hold back the unlike charges from accelerating as they are brought close together.

Power

• In a conductor carrying a current, the electric potential of the charges is continually decreasing. • Positive charges move from regions of high potential to regions of low potential. • ΔUcharges = q ΔV is negative - Often only the magnitude is desired • The power delivered to the circuit element is the energy divided by the elapsed time.

Resistance

• In a conductor, the voltage applied across the ends of the conductor is proportional to the current through the conductor. • The constant of proportionality is the resistance of the conductor. R = ΔV/ I • Units of resistance are ohms (Ω) - 1 Ω = 1 V / A • Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor.

Insulators

• Insulators are materials in which electric charges do not move freely. - Glass and rubber are examples of insulators. - When insulators are charged by rubbing, only the rubbed area becomes charged. • There is no tendency for the charge to move into other regions of the material.

Millikan Oil-Drop Experiment

• Measured the elementary charge, e • Found every charge had an integral multiple of e - q = n e

More Properties of Charge

• Nature's basic carrier of negaAve charge is the electron. - Gaining or losing electrons is how an object becomes charged. • Electric charge is always conserved. - Charge is not created, only exchanged - Objects become charged because negaAve charge is transferred from one object to another.

Property 4

• On an irregularly shaped conductor, the charge accumulates at locations where the radius of curvature of the surface is smallest (that is, at sharp points). • Any excess charge moves to its surface. • The charges move apart until an equilibrium is achieved. • The amount of charge per unit area is greater at the flat end. • The forces from the charges at the sharp end produce a larger resultant force away from the surface.

Electric Field Line Patterns

• Point charge • The lines radiate equally in all directions. • For a positive source charge, the lines will radiate outward. • For a negative source charge, the lines will point inward.

Dipole Example

• Potential is plotted on the vertical axis. - In arbitrary units • Two charges have equal magnitudes and opposite charges. • Example of superposition

Current

• Practical applica<ons were based on static electricity. • A steady source of electric current allowed scientists to learn how to control the flow of electric charges in circuits.

Potentials and Charged Conductors

• Since W = -q(VB - VA), no net work is required to move a charge between two points that are at the same electric poten:al. - W= 0 when VA =VB • All points on the surface of a charged conductor in electrostatic equilibrium are at the same potential. • Therefore, the electric poten:al is constant everywhere on the surface of a charged conductor in electrostatic equilibrium.

Electric Potential of Multiple Point Charges

• Superposition principle applies • The total electric potential at some point P due to several point charges is the algebraic sum of the electric potentials due to the individual charges. - The algebraic sum is used because potentials are scalar quantities.

Electric Potential Energy

• The Coulomb force is a conserva:ve force. • It is possible to define an electrical poten:al energy func:on with this force. • Work done by a conserva:ve force is equal to the nega:ve of the change in poten:al energy.

Electric Field of a Charged Thin Spherical Shell (Theres an equation)

• The calculaAon of the field outside the shell is idenAcal to that of a point charge. • The electric field inside the shell is zero.

Parallel-Plate Capacitor, Example (Theres an equation here)

• The capacitor consists of two parallel plates. • Each has area A. • They are separated by a distance d. • The plates carry equal and opposite charges. • When connected to the ba`ery, charge is pulled off one plate and transferred to the other plate. • The transfer stops when ΔVcap = ΔVbattery • The capacitance of a device depends on the geometric arrangement of the conductors. • For a parallel-plate capacitor whose plates are separated by air: C = squigglye0 A/d

Examples of Polarization

• The charged object (on the lea) induces charge on the surface of the insulator. • A charged comb aIracts bits of paper due to polarization of the paper.

Conductors in Equilibrium

• The conductor has an excess of posi:ve charge. • All of the charge resides at the surface. • E = 0 inside the conductor. • The electric field just outside the conductor is perpendicular to the surface. • The potential is a constant everywhere on the surface of the conductor. • The potential everywhere inside the conductor is constant and equal to its value at the surface.

Electric Current ( Theres an equation)

• The current is the rate at which the charge flows through a surface. - Look at the charges flowing perpendicularly through a surface of area A. Iav = ΔQ/Δt • The SI unit of current is Ampere (A) - 1 A = 1 C/s • The direction of the current is the direction positive charge would flow. - This is known as conventional current direction. • In a common conductor, such as copper, the current is due to the motion of the negatively charged electrons. • It is common to refer to a moving charge as a mobile charge carrier. - A charge carrier can be positive or negative.

Parallel Plate Capacitor (Theres an equation)

• The device consists of plates of posiAve and negaAve charge. • The total electric field between the plates is given by • The field outside the plates is zero.

Electric Field in a Parallel-Plate Capacitor

• The electric field between the plates is uniform. - Near the center - Nonuniform near the edges • The field may be taken as constant throughout the region between the plates.

Electric Field and Electric Potential Depend on Distance

• The electric field is proportional to 1/r2 • The electric potential is proportional to 1/r

Property 1

• The electric field is zero everywhere inside the conducting material. - Consider if this were not true • If there were an electric field inside the conductor, the free charge there would move and there would be a flow of charge. • If there were a movement of charge, the conductor would not be in equilibrium.

Properties, Cont.

• The electric field just outside a charged conductor is perpendicular to the conductor's surface. • On an irregularly shaped conductor, the charge accumulates at locaAons where the radius of curvature of the surface is smallest (that is, at sharp points).

Electric Field Lines, Final

• The electric field lines are not material objects. • They are used only as a pictorial representation of the electric field at various locations. • They generally do not represent the path of a charged particle released in the electric field.

Direction of Electric Field

• The electric field produced by a negative charge is directed toward the charge. - A positive test charge would be attracted to the negative source charge. • The electric field produced by a posiAve charge is directed away from the charge. - A positive test charge would be repelled from the positive source charge.

Potential Difference

• The electric potential difference ΔV between points A and B is defined as the change in the potential energy (final value minus initial value) of a charge q moved from A to B divided by the size of the charge. - ΔV=VB -VA =ΔPE/q • Potential difference is not the same as potential energy. • Another way to relate the energy and the potential difference: ΔPE = q ΔV • Both electric poten:al energy and poten:al difference are scalar quantities. • Units of potential difference - V = J/C • A special case occurs when there is a uniform electric field. - ΔV=-Ex Δx • Gives more information about units: N/C = V/m

The Electron Volt

• The electron volt (eV) is defined as the kinetic energy that an electron gains when accelerated through a potential difference of 1 V. - Electrons in normal atoms have energies of 10's of eV. - Excited electrons have energies of 1000's of eV. - High energy gamma rays have energies of millions of eV. • 1eV=1.6x10-19 J

Equipotentials and Electric Fields Lines - Positive Charge

• The equipotentials for a point charge are a family of spheres centered on the point charge. - In blue • The field lines are perpendicular to the electric potential at all points. - In orange

Instantaneous Current (Theres an equation)

• The instantaneous current is the limit of the average current as the <me interval goes to zero: • If there is a steady current, the average and instantaneous currents will be the same. • SI unit: A

Rules for Drawing Electric Field Lines

• The lines for a group of charges must begin on positive charges and end on negative charges. - In the case of an excess of charge, some lines will begin or end infinitely far away. • The number of lines drawn leaving a positive charge or ending on a negative charge is proportional to the magnitude of the charge. • No two field lines can cross each other.

Electric Potential of a Point Charge

• The point of zero electric potential is taken to be at an infinite distance from the charge. • The potential created by a point charge q at any distance r from the charge is V = Ke * q/r

Resistivity

• The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A. R = ρ L/A - ρ is the constant of proportionality and is called the resistivity of the material. - See table 17.1

The Superposition Principle

• The resultant force on any one charge equals the vector sum of the forces exerted by the other individual charges that are present. - Find the electrical forces between pairs of charges separately - Then add the vectors • Remember to add the forces as vectors

Electric Fields and Superposition Principle

• The superposition principle holds when calculating the electric field due to a group of charges. - Find the fields due to the individual charges. - Add them as vectors. - Use symmetry whenever possible to simplify the problem.

More About a Test Charge and The Electric Field

• The test charge is required to be a small charge. - It can cause no rearrangement of the charges on the source charge. - Mathematically, the size of the test charge makes no difference. • Using qo = 1 C is convenient • The electric field exists whether or not there is a test charge present.

Charge Carrier Motion in a Conductor

• The zig-zag black line represents the motion of a charge carrier in a conductor. - The net drift speed is small. • The sharp changes in direction are due to collisions. • The net motion of electrons is opposite the direction of the electric field.

Work and Potential Energy

• There is a uniform field between the two plates. • As the charge moves from A to B, work is done on it. • WAB =Fx Δx= qEx (xf -xi) • ΔPE=-WAB = -qEx Δx - Only for a uniform field for a particle that undergoes a displacement along a given axis • SI unit of energy: J

Electrical Forces are Field Forces

• This is the second example of a field force. - Gravity was the first • Remember, with a field force, the force is exerted by one object on another object even though there is no physical contact between them. • There are some important similarities and differences between electrical and gravitational forces.

Vector Nature of Electric Forces

• Two point charges are separated by a distance r • The like charges produce a repulsive force between them • The force on q1 is equal in magnitude and opposite in direction to the force on q2 • Two point charges are separated by a distance r • The unlike charges produce an aIracAve force between them • The force on q1 is equal in magnitude and opposite in direcAon to the force on q2

Electric Field of a Nonconducting Plane Sheet of Charge

• Use a cylindrical Gaussian surface • The flux through the ends is EA, there is no field through the curved part of the surface. • The total charge is Q = σA • Note, the field is uniform. • The field must be perpendicular to the sheet. • The field is directed either toward or away from the sheet.

Electrical Potential Energy of Two Charges PE = q2 * V1 = q1q2/r

• V1 is the electric potential due to q1 at some point P • The work required to bring q2 from infinity to P without acceleration is q2V1 • This work is equal to the potential energy of the two particle system

Capacitors in Parallel

• When connected in parallel, both have the same potential difference, ΔV, across them. • When capacitors are first connected in the circuit, electrons are transferred from the lej plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged. • The flow of charges ceases when the voltage across the capacitors equals that of the battery. • The capacitors reach their maximum charge when the flow of charge ceases. • The potential difference across the capacitors is the same. - And each is equal to the voltage of the battery • The total charge, Q, is equal to the sum of the charges on the capacitors. - Q=Q1 +Q2 • The capacitors can be replaced with one capacitor with a capacitance of Ceq - The equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors. • Ceq =C1 +C2+... • The equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors.

Capacitors in Series (Theres an equation)

• When in series, the capacitors are connected end-to-end. • The magnitude of the charge must be the same on all the plates. • When a battery is connected to the circuit, electrons are transferred from the left plate of C1 to the right plate of C2 through the battery. • As this negative charge accumulates on the right plate of C2, an equivalent amount of negative charge is removed from the left plate of C2, leaving it with an excess positive charge. • All of the right plates gain charges of -Q and all the left plates have charges of +Q. • An equivalent capacitor can be found that performs the same function as the series combination. • The potential differences add up to the battery voltage. • ΔV = V1 + V2 1/Ceq = 1/C1 + 1/C2 +... • The equivalent capacitance of a series combina:on is always less than any individual capacitor in the combina:on.

Conductors in Electrostatic Equilibrium

• When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium. • An isolated conductor has the following properties: - The electric field is zero everywhere inside the conducting material. - Any excess charge on an isolated conductor resides entirely on its surface.

Electric Potential and Charge Movements

• When released from rest, posi:ve charges accelerate spontaneously from regions of high potential to low potential. • When released from rest, negative charges will accelerated from regions of low poten:al toward region of high potential. • Work must be done on a negative charges to make them go in the direction of lower electric potential.