| Welcome to Physics 112N : Welcome to Physics 112N Professor Charles E. Hyde-Wright
Spring 2005
Navigate from http:www.physics.odu.edu, or
http:www.physics.odu.edu/hyde/Teaching/Spring05/Phys112_2005.htm
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| Topics to be covered : Topics to be covered Electricity and Magnetism (Chapters 19-24)
Light and Optics (Chapters 25, 26, 28)
Modern Physics (Chapter 30) |
| Phys 111: Chapters 1-18 : Phys 111: Chapters 1-18 Description of motion: Kinematics
Position (in 1-, 2-, 3- dimensions)
Velocity (rate of change of position)
Acceleration (rate of change of velocity)
Relationship between Force and Motion
Net Force equals mass times acceleration
Description of motion in terms of Energy
Kinetic Energy
Potential Energy (Gravity, Springs…)
Thermal Energy (non conservative forces)
Examples of forces:
Contact forces (friction, “normal” force = force perpendicular to surface)
Spring Force F = - kx
Gravity |
| Gravity : Gravity |F| = G M m / r2
Near surface of earth ( h << R )
|F| = G M m /(R+h)2 m [ GM/R2 ] = mg
Circular, Elliptical, Parabolic, & Hyperbolic orbits of moons, planets, asteroids, comets, possible visitors from outer space.
Potential Energy:
U = - G M m /r
Note minus sign, Potential energy decreases as two masses approach: Conservation of energy means Kinetic Energy increases as Potential energy decreaces. |
| Chapter 19 Electric Charges, Forces, and Fields : Chapter 19 Electric Charges, Forces, and Fields
Fundamental Forces in Physics
Gravity (gravitons)
Electromagnetism (photons)
Weak Interaction (W and Z bosons)
Strong Interaction (gluons)
All of physics is based on these four forces
All four forces have similar equations. |
| Energy in our World : Energy in our World Nuclear Fusion in sun E=mc2
H H H H He + n + n + Energy
Thermal Energy at surface converted to visible light energy
Light Energy Chemical Energy (photosynthesis)
Plants Fossil Fuels
Fuel for cars (motion)
Fuel for power plants
(electrical energy lighting for your Physics HW).
Plants Food –> Krebs cycle ADP/ATP
Energy for thought, motion of muscles HW Fusion Radiation |
| Electromagnetism in our World : Electromagnetism in our World Gravity holds us to the earth.
Electromagnetism dominates every other aspect of our physical world
Atoms, Molecules, Solids, & Liquids held together by electrostatics
Chemistry
Light
Virtually all technology:
Electronics, Electric motors, Electric Lighting
Even fire is fundamentally an electromagnetic phenomenon
Profound insights into the physical nature of life.
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| Electrostatic Phenomena : Electrostatic Phenomena Rubbing things makes an electrostatic charge
Spark, Hair standing on end
Thunderclouds rub rising microscopic ice-crystals against falling hail Clouds charge up
Electrostatic phenomena do not require any obvious macroscopic change (mass, material change) |
| Basic Model of Electrostatics Franklin, Coulomb, 18th Century : Basic Model of Electrostatics Franklin, Coulomb, 18th Century When two dissimilar materials are in contact, microscopic particles can be transferred from one to the other
Modern view: electrons e-, or ions Ca++…
These particles carry a property (like mass) called electric-charge.
Electric charge can be positive or negative
Electric charge is a scalar: It is a quantity independent of any direction in space (compare temperature vs. velocity)
Positive attracts negative
Positive repels positive, Negative repels negative
Charge adds linearly: put together charge q1 and charge q2, they act like q3 = q1 + q2.
Charge is conserved: 0 = +q + (-q) |
| Insulators & Conductors : Insulators & Conductors Charges placed on an insulator (plastic, wood, ceramic) stay put—in spite of the electric forces on them:
Fnet=0
Binding force acts like a microscopic spring. As external electric force pulls on charge on insulator, the binding force pulls back (up to some limit: spring breaks)
Charges placed on a metal are free to move in response to electro-magnetic (or other forces): ma = F |
| Electrical Charge : Electrical Charge All physical quantities must be measured as multiples of a standard
Time is measured in multiples of the standard second (now defined by atomic physics phenomena)
Distance is measured in multiples of the standard meter (now defined in reference to the speed of light times the second).
Mass is measured in multiples of the standard kilogram, housed near Paris, France.
The SI unit of electrical charge is the Coulomb
The Coulomb (C ) is defined from magnetic phenomena (skip till later).
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| The Structure of an Atom : The Structure of an Atom The atom consists of a positively charged nucleus, orbited by negatively charged electrons. The nucleus contains protons (positive) and neutrons (neutral).
The orbital lines are an accurate description of the orbits of electrons in a highly excited atom
The fuzzy red blob is a better representation of the electron wave in the atomic ground state (see Chap. 30). |
| The Electron : The Electron One of the fundamental particles found in nature is the electron.
The electron mass is 9.11 10-31 kg.
The electron charge (-e) is -1.6 10-19 C.
The symbol e is the magnitude of the electron’s charge
The electron is part of a family of fundamental particles known as leptons.
Electron lifetime > 1023 years (test of charge conservation). Eur. Phys. J. C 3, 1 (1998) |
| The Proton : The Proton The proton is not a fundamental particle. It has a finite size (10-15 m) and a spectrum of excited states. It is understood to consist of three quarks bound together by a cloud of gluons and quark—anti-quark pairs.
The proton mass is 1.67 10-27 kg.
The proton is 2000 times heavier than the electron, so the vast majority of an atom’s mass resides in the nucleus.
The proton charge (+e) is +1.6 10-19 C.
The proton charge and electron charge are known to be equal and opposite to very high precision.
|qp + qe|/e < 10-21 Eur. Phys. J. C 3, 1 (1998) |
| Slide15 : An object may contain both positive and negative charges. If the object possesses a net charge it is said to be charged. If the object possesses no net charge it is said to be neutral.
An atom is normally neutral, because it possesses an equal number of electrons and protons. However, if one or more electrons are removed from or added to an atom, an ion is formed, which is charged.
Charge is always conserved: charge may be transferred but it is never created or destroyed.
However, charges can be created and destroyed in positive and negative pairs, so that the net charge in the universe does not change. |
| Electrical Forces : Two charged objects will exert forces on one another.
Unlike charges attract one another.
Like charges repel one another.
The force decreases with the square of the distance between the charges
Electrical Forces |
| Polarization : Polarization An object is polarized when its charges are rearranged so that there is a net charge separation. Charged objects can be attracted to neutral objects because of polarization. neutral & polarized charged |
| Insulators and Conductors : Insulators and Conductors Materials are classified by how easily charged particles can “flow” through them.
If charges flow freely, the material is a conductor (metals, for example)
If charges are unable to move freely, the material is an insulator (glass, for example)
Some materials have properties in between insulators and conductors, these are called semiconductors. |
| Charge Transfer : Charge Transfer Charge is usually transferred because electrons move from one place to another.
But sometimes the flow of both positively or negatively charged ions (atoms or molecules) is important (cells, batteries…).
The earth can be viewed as an infinite (conducting) reservoir of electrons. An object in electrical contact with the earth is said to be grounded.
What happens when I ground the Van de Graaff generator? [And why do I do this before touching the generator?] |
| Properties of the Mutual Electrical Forces Acting on Two Charges : Properties of the Mutual Electrical Forces Acting on Two Charges Each of the two charged object experiences a force that is equal and opposite to the force experienced by the other charge (Newton’s Third Law).
The force is attractive if the charges are unlike and repulsive if the charges are like.
The force is inversely proportional to the square of the separation of the two charges, and is directed along the line joining them (attractive or repulsive).
The force is proportional to the product of the magnitudes of the 2 charges.
Remember force is a vector! |
| Coulomb’s Law : Coulomb’s Law The magnitude of the force between two point* objects separated by a distance r with charges q1 and q2 is given by Coulomb’s Law:
where k = 8.99… 109 Nm2/C2 , precision of 10-7 is linked to measurement of electron charge.
*or spherical charge distributions, or any objects whose size is much less than the separation distance r
The direction of the force on one charge is either toward (negative) or away (positive) from the other charge. q1 and q2 are the values (+ or -) of the two charges |
| Force: vector, magnitude, component : Force: vector, magnitude, component Magnitude (strictly positive) Component along direction from q1 to q2 of Force from q1 acting on q2.
If q1q2> 0, force is repulsive (pushes q2 away from q1)
If q1q2< 0, force is attractive (pulls q2 towards q1)
r q1 q2 |
| Comments on Coulomb’s Law : Comments on Coulomb’s Law 1/r2 Charge is conserved, Gauss’ Law
Deviations from 1/r2 are measured to be less than 1 part in 1010 over distance scales from (10-10 m to 1.0 m)
The force is linear in the value of each charge.
If an amount of charge 0.2q1 is brought from far away and added to q1, the force on q2 is increased to
Why k? (Why not k=1?)
In Gaussian (or cgs) units, 1.00 esu is defined such that
Two charges of 1.00 esu each separated by 1cm exert mutual forces on each other of 1 dyne = 1 gm cm2/sec2
k = 8.99 109 Nm2/C2 = 1.00 dyne (cm)2/esu2.
The value of k depends upon our choice of units for Force, Distance and Charge.
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| Subscript labels on Force : Subscript labels on Force |
| Vectors and Scalars : Vectors and Scalars A scalar is a physical quantity with magnitude, but without direction in space.
Temperature
Mass
Energy, Time, Charge …
A Vector is a physical quantity with magnitude and direction in space.
Displacement
Momentum
Velocity, Force |
| Vector Components & Unit Vectors : Vector Components & Unit Vectors A vector can be expressed in terms of a coordinate system.
Force vector
F = 1.6 N oriented 110° counter-clockwise from x-axis.
Force Vector
F = (1.6 N)(cos110°) along +x-axis plus (1.6N)(sin110°) along +y-axis x q=110 |
| Walker Problem 13, pg. 641 : Walker Problem 13, pg. 641 Given that q = +12 mC and d = 16 cm, (a) find the direction and magnitude of the net electrostatic force exerted on the point charge q2 in Figure 19-30. (b) How would your answers to part (a) change if the distance d were tripled? |
| Solution : Solution Draw free body for JUST q2 |
| Problem 13, Solution, cont’d : Problem 13, Solution, cont’d B) Tripling the separations decreases all forces by a factor of 32=9
F2 = 22.4 N, +x direction |
| Relative Strength of Gravity and Electrostatics : Relative Strength of Gravity and Electrostatics In the hydrogen atom, the electron and proton are separated by 0.5·10-10 m
The ratio of gravitational attraction between the electron and proton divided by the electrostatic attraction is
FG/FQ = 10-39 (see text)
This ratio is independent of the separation
Both forces are 1/r2.
Why is gravity so much more important in the solar system? |
| Multiple Charges : Multiple Charges If there are more than two charges present, the net force on any one charge is given by the vector sum of the forces on that charge from all surrounding charges. This is an example of the Principle of Superposition. + + F- F+ What is the direction of the net force on each charge (roughly)? |
| Walker (1st edition) Problem 19, pg. 641 : Walker (1st edition) Problem 19, pg. 641 (a) Find the direction and magnitude of the net electrostatic force exerted on the point charge q3 in the Figure. Let q = +1.8 mC and d = 22 cm. (b) How would your answers to part (a) change if the distance d were doubled? |
| Solution : Solution Force F3,2 on q3 from q2 is repulsive
Force F3,1 on q3 from q1 is attractive
Force F3,4 on q3 from q4 is repulsive
Distance from q2 to q3 is d
Distance from q4 to q3 is d
Distance from q1 to q3 is (2)d
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| Slide34 : Add the force vectors graphically F3,4 F3,2 F3,1 FNet Solution, cont’d |
| Solution, four charges : Solution, four charges Find angle q from x-axis
Cosq = [FNet,x]/ |FNet|
Cosq = (2.97N)/(7.22N)
q = 65.7 FNet x y q |
| Spherical Charge Distributions : Spherical Charge Distributions In general a spherical charge distribution behaves as if all of its charge were at the center of the sphere. Use the distance to the center of the sphere to calculate the electrostatic force. q1 q2 r |
| Newton: Action at a Distance Faraday: Force Fields : Newton: Action at a Distance Faraday: Force Fields A mass m exerts a gravitational force GmM/r2 on a second mass M separated by a distance r, and vice versa.
Coulomb gave us the same picture for electrostatic forces
Faraday offered a new insight, introducing the Electric Field, which can be thought of as carrying the force from charge q to charge Q.
In physics, a field means a physical variable that has a defined value at every point in space. Examples:
Temperature map, Barometric Pressure map (a scalar field)
Wind velocity map (a vector field)
Initially just a mathematical trick, with our understanding of electromagnetic waves and the quantum nature of light, Electric and Magnetic fields are as real as charge and mass. |
| Electric Field : Electric Field If a test charge q0 experiences a force F at a given location r, the magnitude of the electric field at that location is defined by
The electric field is a “what if” concept. What would be the electrostatic force acting on a charge q0 if it were placed at position r?
The electric field can also be thought of as a disturbance in space caused by nearby charges.
The electrostatic force experienced by a charge is the interaction between the charge and the electric field at that position.
The SI units of electric field are Newtons/Coulomb = N/C |
| Electric Force F(r) from charge Q acting on a test charge q0 at various locations r =(x,y,z): F=kQq0/r2 Electric Field E(r)= F/ q0 : Electric Force F(r) from charge Q acting on a test charge q0 at various locations r =(x,y,z): F=kQq0/r2 Electric Field E(r)= F/ q0 Q q0 |
| Electric Field E(r) from charge Q at various locations r: E=kQ/r2 : Electric Field E(r) from charge Q at various locations r: E=kQ/r2 Q r |
| Electric Field : Electric Field A vector at every point in space that tells us the magnitude and direction of the force a charge q will experience if the charge q is placed at the position (x,y,z).
If q<0, then the force F on q is opposite E.
To measure E = F/q, q must be small enough that it doesn’t change the distribution of charges that created the electric field in the first place. |
| Electric Field Direction : Electric Field Direction The direction of the electric field is defined to be the direction of the force that would be experienced if the test charge is positive. Because the field has a direction, it must be a vector.
+ q0 q0 E E |
| Electric Field (cont.) : Electric Field (cont.) The electric field is the force per charge at a given location. If you know the electric field, then the force on a charge can easily be found using
F = qE Example: A charge q of 8 mC experiences a uniform electric field of 1000 N/C to the right. (a) What is the force on the charge? (b) What would the force be if the charge were –8 mC? Note: In problems like this we do not need to know what charges created the electric field. |
| Electric Field of a Point Charge : Electric Field of a Point Charge From Coulomb’s Law, the magnitude of the force experienced by a test charge q0 a distance r from a charge q is Since the definition of the electric field is the magnitude of the electric field from a point charge is given by |
| Walker Problem 28, pg. 642 : Walker Problem 28, pg. 642 What is the magnitude of the electric field produced by a charge of magnitude 10.0 mC at a distance of (a) 1.00 m and (b) 2.00 m? k = 8.99 ·109 N m2/C2 |
| Electric Field & Polarization : Electric Field & Polarization What is the magnitude of an electric field strong enough to polarize the molecules in the air to the point that electrons are pulled out of the air (ionization produces a spark)?
Several Million Newton/Coulomb.
Several Million Volt/meter |
| Electric Fields in Nuclear Physics : Electric Fields in Nuclear Physics What is the electric field at the surface of a proton?
(radius 10-15 m, charge 1.6·10-19 C)
E = (8.99 ·109 N m2/C2)(1.6·10-19 C)/(10-15 m) 2
E=(14.4) 109-19+30 N/C
E=1.44 · 1019 N/C
That’s big! |
| Electric Fields in Atomic/Molecular physics : Electric Fields in Atomic/Molecular physics What is the electric field from the hydrogen nucleus (proton) at a distance of one atomic radius (r=0.5Å=0.5·10-10m)
E = k q / r2
E = (8.99 ·109 N m2/C2)(1.6·10-19 C)/(0.5·10-10 m) 2
E = (58)(109-19+20) ( N/C)
E = 5.8 ·1011 N/C
Smaller, but still very large. |
| Superposition : Superposition Just like with forces, electric fields must be added as vectors. The electric field from several charges is the vector sum of the electric field from each charge. Example: Consider two identical negative charges as shown. At which lettered point is the magnitude of the electric field greatest? Least? a d c b |
| Superposition : Superposition - - Q1<0 Q2 = Q1 E1 E2 E E2 E1 E |
| Walker Problem 66, pg. 644 : Walker Problem 66, pg. 644 An object of mass m = 3.7 g and charge q = +44 mC is attached to a string and placed in a uniform electric field that is inclined at an angle of 30.0° with the horizontal. The object is in static equilibrium when the string is horizontal. Find (a) the magnitude of the electric field and (b) the tension in the string. |
| Walker Problem 66, pg. 644 : Walker Problem 66, pg. 644 Free Body Diagram
Net force = 0
S Fx=0: qEsin30o – mg = 0
m = 3.7E-3 kg, q = 44.E-6 C
E = mg/(q sin30o) = (3.7E-3 kg)(9.8m/s2)/(0.5*44.E-6 C)
E = 1.65E+3 (kg·m/s2)/C = 1.65E+3 N/C
S Fy=0: qEcos30o – T = 0
T = (44.E-6 C) (1.65E+3 N/C)0.866 = 6.3E-2 N qE T mg |
| Electric Field Lines : Electric Field Lines In order to visualize the electric field in space it is convenient to draw Electric field-lines (see Fig. 19-13). The field lines are directional [curved] lines that everywhere point in the direction of the electric field at that point. + + Dipole |
| Field Line Properties : Field Line Properties The electric field is tangent to the field line at any point in space.
The strength of the electric field is proportional to the density of field lines (areal density measured perpendicular to field line).
The field lines always begin on positive charges or at infinity and end on negative charges or at infinity.
No two field lines can ever cross.
The number of field lines leaving a positive charge or approaching a negative charge is proportional to the magnitude of the charge. |
| Electric Field Lines : Electric Field Lines Note that twice as many field lines originate from the +2q charge than the +q or –q charges. |
| Lecture 2, Quiz 1 : Lecture 2, Quiz 1 1. The net charge inside the green blob is:
Positive
Zero
Negative
Hint: Are there more Electric Field lines entering, or leaving the blob, or is it equal? |
| Lecture 2, Quiz 2 : Lecture 2, Quiz 2 2. The net charge inside the green blob is:
Positive
Zero
Negative
Hint: Are there more Electric Field lines entering, or leaving the blob, or is it equal?
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| Lecture 2, Quiz 3 : Lecture 2, Quiz 3 3. The net charge inside the green blob is:
Positive
Zero
Negative
Hint: Are there more Electric Field lines entering, or leaving the blob, or is it equal?
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| Walker Problem 37, pg. 642 : Walker Problem 37, pg. 642 The electric field lines surrounding three charges are shown in the Figure. The center charge is q2 = -10.0 mC. (a) What are the signs of q1 and q3? (b) Find q1. (c) Find q3. |
| Parallel-Plate Capacitor : Parallel-Plate Capacitor Two parallel conducting plates with opposite charge, separated by a distance d, is known as a parallel-plate capacitor. The electric field is uniform between the plates (except near the edges, not shown).
Uniform means the electric field magnitude and direction are the same everywhere (in gap). This is because of, not in spite of Coulombs 1/r2 law!! |
| Electrostatic Equilibrium : Electrostatic Equilibrium Recall that charges within a conductor are free to move around easily.
If the charges within a conductor are not in motion, then the system is said to be in electrostatic equilibrium. |
| Properties of Electrostatic Equilibrium : Properties of Electrostatic Equilibrium In the presence of electrostatic forces, the charges on the conductor move around until the following static conditions are achieved:
The electric field is zero everywhere inside a conductor.
The excess charge on a conductor resides entirely on its surfaces.
The electric field just outside a charged conductor is perpendicular to its surface.
On irregularly shaped objects, the charge accumulates at sharp points, and the electric field is most intense at sharp points. |
| Electric Flux : Electric Flux We define electric flux F as the product of the surface area A times the component Ecosq of the electric field perpendicular to the surface.
In general, F = EAcosq.
(a) F = EA
(b) F = 0
(c) F = EAcosq q is the angle between the electric field and the line perpendicular to the surface. |
| Gauss’s Law : Gauss’s Law Consider an arbitrary (imaginary) closed surface (called a Gaussian surface) enclosing a total charge q. The electric flux through the surface is This integral property is a consequence of the 1/r2 Coulomb Law, and is valid for any irregular surface, no matter how complicated the electric field produced by internal or external charges. |
| Example : Example Three point charges are arranged as shown. q1 = +4 mC, q2 = -6 mC and q3 = -4 mC. Find the electric flux through the three Gaussian surfaces labeled a, b and c. q1 q3 q2 a c b |
| Walker Problem 49, pg. 643 : Walker Problem 49, pg. 643 A thin wire of infinite extent has a charge per unit length of l. Using the cylindrical Gaussian surface shown in the Figure, show that the electric field produced by this wire at a radial distance r has a magnitude given by |
| Walker Problem 49, pg. 643 solution : Walker Problem 49, pg. 643 solution By symmetry, Electric force on a test charge is directed radially outward (if l>0).
Closed Gaussian surface consists of the cylinder and its two end caps.
Electric flux through end caps is zero because E is parallel to surface.
Electric flux through cylinder wall: F=Area · E(r )
F = 2p r L E(r )
Net Flux = 0 + 0 + 2p r L E(r )
= (charge enclosed)/e0 = L l /e0 |
| Charges on (and in) a conductor : Charges on (and in) a conductor Charge on a conductor is free to move under the influence of its mutual repulsion.
Are the charges in a) or b) “farther apart”?
The quantitative meaning to this question is “Which configuration gives the lowest value for the electrostatic energy?” (See Chap 20.)
It is a property of the 1/r2 law (not just repulsion) that all the excess charge on a conductor ends up on the SURFACE.
This can be an inside, as well as outside surface!! |
| Quiz 1 Jan 10, 2005 : Quiz 1 Jan 10, 2005 Two charges Q1 and Q2 are separated by a distance of 0.010 m. The Electrostatic force of Q1 on Q2 is 2.0e-5 N.
At what distance of separation between Q1 and Q2 would the force be 1.0e-5N?
a) 0.02 m b) 0.014 m c) 0.01 m
d) 0.007m e) 0.005 m |
| Quiz 1 12 January 2004 : Quiz 1 12 January 2004 In the diagram at right, F1 is the electrostatic force of Q1 acting on charge q=1.0E-9C .
Draw a vector with its tail at q to represent the magnitude and direction of the electrostatic force F2 of Q2 acting on charge q (the length of your vector should roughly describe the relative magnitudes of F2 and F1.
Draw a vector with its tail at q to represent the magnitude and direction of the net force FNet acting on q from both Q1 and Q2
Label your vectors F1 and FNet
Note: Name…………………… Q1 = +1.0E-6 C Q2 = -1.0E-6 C F1 q |
| Quiz 2 2 February 2004 : Quiz 2 2 February 2004 Name…………………… +4mC -2mC Sketch the electric field lines generated by these two charges.
Hint: Consider the electric flux through the three gaussian surfaces defined by the three dashed lines. |
| Preparation for Lab 2 (Chapter 21) : Preparation for Lab 2 (Chapter 21) Electric Current in wire equals steady flow of charge (not equilibrium!).
Unit of measure is Coulomb per second = Amp
1.00 C/s = 1.0 A
Think of electric current like flow of water in pipe.
Voltage = Electrostatic Potential difference of power supply or battery (e.g. AA=1.5 V)
How hard the current is being forced around circuit.
Think of difference in height of two ends of a water pipe. Water flows with greater force when the height difference is greater.
Resistance R = measure of how hard you have to push to obtain current (flow). R = V/I
Think of long thin pipe (high resistance to flow) versus short broad pipe (low resistance to flow). Pump |
| Equivalence of Gauss’ Law and Coulomb’s Law : Equivalence of Gauss’ Law and Coulomb’s Law Coulomb: Electric field at a distance r from a point charge Q:
E(r) = k Q / r2 = Q / (4p e0 r2)
For Q>0, E>0: E points away from Q
For Q<0, E<0: E points towards Q.
Gauss: Electric flux through an imaginary closed spherical shell a distance r from Q
Flux = E(r)•(Surface area of shell)= E(r) 4p r2
Outward flux is positive
Inward flux is negative
Gauss: Flux = Q/ e0.
E(r) = Q / (4p e0 r2)
r Q E |
| Gauss’ Law and the Parallel Plate Capacitor : Gauss’ Law and the Parallel Plate Capacitor Consider a rectangular Gaussian surface penetrating into the metal of a parallel plate capacitor:
Total Charge on left plate = Q, right plate = -Q
Total area or each plate = A
Surface charge density = s = Q/A
Surface area of face of Gaussian surface parallel to plate = a.
Charge enclosed by Gaussian surface: sa
Flux through portion of Gaussian surface inside metal = 0 (E=0).
Flux through top and bottom surfaces outside metal = 0 (Electric field parallel to surface).
Flux through face of Gaussian surface parallel to plate (outside) = Ea.
Gauss’ Law: sa/e0 = Ea
Uniform Electric Field in gap: E = s/e0 +
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