Physics II: Electricity & Magnetism

Physics II: Electricity & Magnetism
Chapter 22

Friday (Day 1) Add 21.9 Figure: E-field in a conductor.
Fix derivations for uniform charge density (dr is wrong, etc) Add EM Field Activities

Warm-Up Fri, Feb 20 Write down the steps you give a visually-disabled individual to explain to how to fill a coffee cup with water. Place your homework on my desk: “Foundational Mathematics’ Skills of Physics” Packet (Page 9) For future assignments - check online at

Essential Question(s)
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II? HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS AND APPLY IT TO VARIOUS SITUATIONS? How do we describe and apply the concept of electric field? How do we describe and apply the electric field created by uniformly charged objects? How do we describe and apply the relationship between the electric field and electric flux? How do we describe and apply Gauss’s Law? How do we describe and apply the nature of electric fields in and around conductors? How do we describe and apply the concept of induced charge and electrostatic shielding?

Vocabulary Flux Electric Flux Flux Lines Surface Area Perpendicular
Orthogonal Normal Dot Product Gauss’s Law Charge Density

Foundational Mathematics Skills in Physics Timeline
Day Pg(s) 1 2 6 3 11 16 21 13 14 7 4 12 17 8 22 23 5 18 9 24 †12 19 10 15 20 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?

Agenda Review “Foundational Mathematics’ Skills of Physics” Packet (Page 9) with answer guide. Review electric fields, electrons, and conductors Discuss the following: Electric Flux Work Day: Chapter 21 Web Assign Problems Uniform Charge Distribution Derivations Chapter 22 Web Assign Problems

A Review of Electric Fields and Conductors
The electric field is perpendicular to the surface of a conductor – again, if it were not, charges would move.

Chapter 22 Gauss’s Law

Main Points of Chapter 22 Electric flux Gauss’ law
Using Gauss’ law to determine electric fields Conductors and electric fields Testing Gauss’ and Coulomb’s laws

Section 22.1 Given a diagram where the electric field is represented by flux lines, how do we determine the direction of the field at a given point? identify the locations where the field is strong and where it is weak? identify where the positive and negative charges must be present? How do we use the relationship between the electric field and electric flux to calculate the flux of a uniform electric field E through an arbitrary surface?

Section 22.1 How do we use the relationship between the electric field and electric flux to calculate the flux of a uniform electric field E through a curved surface when E is uniform in magnitude and perpendicular to the surface? How do we use the relationship between the electric field and electric flux to calculate the flux of a uniform electric field E through a rectangle when E is perpendicular to the rectangle and a function of one coordinate only? How do we state and apply the relationship of between electric flux and lines of force?

22.1 Electric Flux Electric flux:
A Electric flux through an area is proportional to the total number of field lines crossing the area. A

22.1 Electric Flux Electric flux:
A When  = 0°, the flux (field lines) passing through the area is maximized and when  = 90°, the flux (field lines) is zero. Therefore, for mathematical simplicity, it is important to note that the area A of a surface will represented by a vector A whose magnitude is A but whose direction is perpendicular to it surface.

22.1 Electric Flux Flux through a closed surface:

EM Field 6 Using EM Field, determine the flux through a closed surface (represented by a 2-D cross-section) by counting the number of field lines entering the area and subtracting it from the number of field lines exiting the area. This is because the area vector A for a closed surface is always directed outward. i.e. cos (0°) = 1 and cos (180°) = -1

EM Field 6 How can we draw a surface that has more field lines entering or exiting the closed surface?

EM Field 6 The following are examples of the Flux calculations using EM Field 6. Besides its relation to the number of field lines entering and exiting the closed surface, how does the flux calculation relate to charge?

Summary How does flux relate to the charge enclosed by a closed surface? HW (Place in your agenda): “Foundational Mathematics’ Skills of Physics” Packet (Page 10) The final copies for all Uniformly Charged Objects: 1 derivation with reasons for each mathematical step and 3 additional derivations: (*Refer to rubric) Web Assign Final Copies: Chapter 21 Web Assign Problems: Chapter 22.1 & 22.2 How do we apply integration and the Principle of Superposition to uniformly charged objects? How do we identify and apply the fields of highly symmetric charge distributions? How do we describe and apply the electric field created by uniformly charged objects?

Monday (Day 2)

Warm-Up Mon, Feb 23 For an electric field raining straight down into an imaginary box . . . What is the direction of E? What is the direction of A for the (1) bottom, (2) top, and (3) side surfaces? What is the angle, , between E and A for the (1) bottom, (2) top, and (3) side surfaces? What is value of cos  for the (1) bottom, (2) top, and (3) side surfaces? Place your homework on my desk: “Foundational Mathematics’ Skills of Physics” Packet (Page 10) - POSTPONED The final copies for all Uniformly Charged Objects: 1 derivation with reasons for each mathematical step and 3 additional derivations: (*Refer to rubric) Web Assign Final Copies: Chapter 21 For future assignments - check online at

Application: Electric Flux
Warm-up: What is the direction of E? What is the direction of A for the (1) bottom, (2) top, and (3) side surfaces? What is the angle, , between E and A for the (1) bottom, (2) top, and (3) side surfaces? What is value of cos  for the (1) bottom, (2) top, and (3) side surfaces? A E E

Atop Atop Ein Aside Abottom Eout Abottom

Agenda Review “Foundational Mathematics’ Skills of Physics” Packet (Page 10) with answer guide. - POSTPONED Review electric flux Discuss the following: Gauss’s Law Applications of Gauss’s Law Spherical Conductor, Point Charge, & Line Charges Work Time: Web Assign Problems

Section 22.2 How do we state Gauss’s Law in integral form and apply it qualitatively to relate electric flux and electric charge for a specified surface?

22-2 What Does Gauss’ Law Do?
Imagine field lines emanating from a positive charge. Now imagine a sphere of tissue paper around the charge. How many field lines penetrate the tissue? It doesn’t really matter how many we draw in the first place, as long as we are consistent; they all go through. Now imagine the charge being off-center; all the lines still go through:

Suppose the tissue is some shape other than spherical, but still surrounds the charge. All the field lines still go through: Now, imagine the paper is crinkled and overlaps itself; how shall we deal with the lines that pierce the tissue three times? Notice that they go out twice and in once – if we subtract the “ins” from the “outs” we are left with one line going out, which is consistent with the other situations.

Now, look at an open (flat) sheet. If it is perpendicular to the field, the maximum number of lines penetrates: If it is at an angle, fewer lines penetrate:

The number of field lines piercing the surface is proportional to the surface area, the orientation, and the field strength. If we stop counting lines and just use the field strength itself, we can define the electric flux through an infinitesimal area: Integrating gives the total flux:

For a closed surface, we can uniquely define the direction of the normal to the surface as pointing outwards and define: Then: Note the circle on the integral sign, which means that the integration is over a closed surface.

Note that the surface does not have to be made of real matter – it is a surface that we can imagine, but that does not have to exist in reality. This kind of imaginary surface is called a Gaussian surface. We can imagine it to be any shape we want; it is very useful to choose one that makes the problem you are trying to solve as easy as possible.

22-2 Gauss’ Law Without further ado, we can state Gauss’ law:
The electric flux through a closed surface that encloses a net charge is equal to the net charge divided by the permittivity of free space.

22-2 Gauss’ Law The electric flux through a closed surface that encloses no net charge is zero.

22.2 Gauss’s Law Restating Gauss’ Law, the net number of field lines through the surface is proportional to the charge enclosed, and also to the flux: Gauss’ Law can easily be used to find the electric field in situations with a high degree of symmetry.

E-Field for a Sphere or Point Charge

Spherical Conductor E + + dAsphere + A2 + r2 r0 + + r1 + + A1

22-2 Gauss’ Law for a Point Charge
The electric field through a Gaussian sphere with a single point charge at the center is easily calculated using:

22-2 Gauss’ Law But the result would be the same if the surface was not spherical, or if the charge was anywhere inside it! Therefore, we can quickly generalize this to any surface and any charge distribution; all can be considered as a collection of point or infinitesimal charges: Here, Q is the total net charge enclosed by the surface.

Applications of Gauss’ Law

22-3 Using Gauss’ Law to Determine Electric Fields
Problem-solving techniques: Make a sketch. Identify any symmetries. Choose a Gaussian surface that matches the symmetry – that is, the electric field is either parallel to the surface or constant and perpendicular to it. The correct choice in 3 should allow you to get the field outside the integral. Then solve.

Section 22.3 How do we describe the electric field of a long, uniformly charged wire? How do we describe the electric field of a thin spherical shell? How do we use superposition to determine the electric fields of coaxial cylinders? How do we use superposition to determine the electric fields of concentric spheres?

Section 22.3 How do we apply Gauss’s Law, along with symmetry arguments, to determine the electric field inside a uniformly charged long cylinder? outside a uniformly charged long cylinder? inside a uniformly charged cylindrical shell? outside a uniformly charged cylindrical shell? inside a uniformly charged sphere? outside a uniformly charged sphere? inside a uniformly charged spherical shell? outside a uniformly charged spherical shell?

Section 22.3 How do we apply Gauss’s Law to determine the charge density on a surface in terms of the electric field near the surface? How do we apply Gauss’s Law to determine the total charge on a surface in terms of the electric field near the surface? How do we prove and apply the relationship between the surface charge density on a conductor and the electric field strength near its surface? How do we qualitatively explain why there can be no electric field in a charge-free region completely surrounded by a single conductor? How do we qualitatively explain why the electric field outside of a closed conducting surface cannot depend on the precise location of charge in the space enclosed by the conductor? What is the significance of why the electric field outside of a closed conducting surface cannot depend on the precise location of charge in the space enclosed by the conductor?

E-Field for a Line of Charge

Long Uniform Line of Charge
dAtop E dAtube E dAbottom

Uniformly Charged Vertical Wire (–∞+∞)

Summary What does the closed integral sign mean in Gauss’ Law?
HW (Place in your agenda): “Foundational Mathematics’ Skills of Physics” Packet (Page 10) †“Foundational Mathematics’ Skills of Physics” Packet (Page 11) Web Assign How do we apply integration and the Principle of Superposition to uniformly charged objects? How do we identify and apply the fields of highly symmetric charge distributions? How do we describe and apply the electric field created by uniformly charged objects?

Tuesday (Day 3)

Warm-Up Tues, Feb 24 Calculate the electric flux for a charged object on the following slide. Place your homework on my desk: “Foundational Mathematics’ Skills of Physics” Packet (Page 10) †“Foundational Mathematics’ Skills of Physics” Packet (Page 11) - POSTPONED Web Assign Problems For future assignments - check online at

Two objects, O1 and O2, have charges +1 C and -2.0 C, respectively, and a third object, O3, is electrically neutral. (a) What is the electric flux through the surface A1 that encloses all three objects? (b) What is the electric flux through the surface A2 that encloses the third object only?

Agenda Review “Foundational Mathematics’ Skills of Physics” Packet (Page 11) with answer guide. Discuss the following: Applications of Gauss’ Law Charged Plate(s), Charged Surfaces, & Charged Spherical Insulators Conductors in Electric Fields Experimental Basis of Gauss’ and Coulomb’s Law Work Time: Web Assign Problems

Section 22.3 How do we apply Gauss’s Law, along with symmetry arguments, to determine the electric field? near a large, uniformly charged plane?

Section 22.3 How do we apply Gauss’s Law to determine the charge density on a surface in terms of the electric field near the surface? How do we apply Gauss’s Law to determine the total charge on a surface in terms of the electric field near the surface? How do we prove and apply the relationship between the surface charge density on a conductor and the electric field strength near its surface? How do we qualitatively explain why there can be no electric field in a charge-free region completely surrounded by a single conductor? How do we qualitatively explain why the electric field outside of a closed conducting surface cannot depend on the precise location of charge in the space enclosed by the conductor? What is the significance of why the electric field outside of a closed conducting surface cannot depend on the precise location of charge in the space enclosed by the conductor?

Infinite Plane of Charge (nonconducting uniform )
dA E A dAtube E dA

Uniformly Charged Disk (0∞)

Uniformly Charged Infinite Plate
No radius?!? What does that mean?!?

21-7 The Field of a Continuous Distribution
From the electric field due to a nonconducting uniform sheet of charge, we can calculate what would happen if we put two oppositely-charged sheets next to each other: The result: The individual fields: The superposition: How do we apply integration and the Principle of Superposition to uniformly charged objects? How do we identify and apply the fields of highly symmetric charge distributions? How do we describe and apply the electric field created by uniformly charged objects?

21.8 Field Lines The electric field between two closely spaced, oppositely charged parallel plates is constant.

E-Field at the Surface of a Conductor

Surface of a Conductor dA + A + + + + + E + + + + dAtube + + + + + +

So why are the E-Fields different for the plate and the surface?
Nonconducting: Charge remains localized (ie. +2Q remains fixed) Conducting: Like charges repel (ie. +2Q total -> +Q move to each side and conduct=1/2 insulator ) Therefore, both E-fields will have the same magnitude and direction.

EM Field 6 Find the field lines for:
2 Parallel Oppositely Charged Plates 2 Parallel Plates of the Same Charge

Spherical Insulator (Uniformly distributed charge)
dA1= dAsphere + + + + + + + + r2 + r0 + + + + + + A2 A1 + + dA2 r1

22-3 Conductors and Electric Fields
In conductors, the charges are free to move if there is an external electric field exerting a force on them. Therefore, in equilibrium, there is no static field inside a conductor. This also means that the external electric field is perpendicular to the conductor at its surface.

What if the conductor is charged – where does the excess charge go? By making a Gaussian surface very close to, but just under, the surface of the conductor, we see that any excess charge must lie on the outside of the conductor.

What if there is a cavity inside the conductor, and that cavity has charges in it? The field inside the conductor must still be zero, so charges will be induced on the inner surface of the cavity and the outer surface of the conductor:

Electrostatic Fields Near Conductors Looking at the electrostatic field very near a conductor, we find: and therefore: The electric field is perpendicular to the surface, and where the charge density is higher, the field is larger.

To summarize: 1. The electrostatic field inside a conductor is zero. 2. The electrostatic field immediately outside a conductor is perpendicular to the surface and has the value σ/ε0 where σ is the local surface charge density. 3. A conductor in electrostatic equilibrium—even one that contains nonconducting cavities—can have charge only on its outer surface, as long as the cavities contain no net charge. If there is a net charge within the cavity, then an equal and opposite charge will be distributed on the surface of the conductor that surrounds the cavity.

22-4 Are Gauss’ and Coulomb’s Laws Correct?
An experiment to validate Gauss’ law (that there is no charge within a conductor) can be done as follows: Need a hollow conducing sphere, a small conducting ball on an insulating rod, and an electroscope attached to the surface of the conductor.

Charge the small sphere and hold it inside the shell without touching. Induced charge will be on outside of shell and on electroscope. Now touch the inside of the shell with the small sphere. Charge will flow onto it until it is neutral, leaving the shell with a net positive charge.

Finally, remove the rod. The electroscope leaves do not move, indicating that the excess charge resides on the outside of the shell.

This table shows the results of such experiments looking for a deviation from an inverse-square law:

One problem with the above experiments is that they have all been done at short range, 1 meter or so. Other experiments, more sensitive to cosmic-scale distances, have been done, testing whether Coulomb’s law has the form: No evidence for a nonzero μ has been found.

Summary of Chapter 22 Electric flux due to field intersecting a surface S: Gauss’ law relates flux through a closed surface to charge enclosed: Can use Gauss’ law to find electric field in situations with a high degree of symmetry

Summary of Chapter 22 Electric flux: Gauss’s law:

Summary of Chapter 22 (con’t)
Properties of conductors: Electric field is zero inside Field just outside conductor is perpendicular to surface Excess charge resides on the outside of a conductor, unless there is a nonconducting cavity in it; in that case, there is an induced charge on both surfaces Gauss’ law has been verified to a very high degree of accuracy

Summary In a conductor, what happens to an electron in the presence of an electric field? What happens to the electric field in the presence of an conductor? HW (Place in your agenda): “Foundational Mathematics’ Skills of Physics” Packet (Page 10) †“Foundational Mathematics’ Skills of Physics” Packet (Page 11) LAST ONE!!!!!!!! Web Assign How do we apply integration and the Principle of Superposition to uniformly charged objects? How do we identify and apply the fields of highly symmetric charge distributions? How do we describe and apply the electric field created by uniformly charged objects?

Wednesday (Day 4)

Warm-Up Wed, Feb 25 Two thin concentric spherical shells of radii r1 and r2 (r1 < r2) contain a uniform surface charge densities 1 and 2, respectively. Determine the electric field for (a) r < r1, (b) r1 < r < r2, and (c) r > r2. (d) Under what conditions will E = 0 for r > r2? (e) Under what conditions will E = 0 for r1 < r < r2? Place your homework on my desk: †“Foundational Mathematics’ Skills of Physics” Packet (Page 11) Web Assign Problems For future assignments - check online at

HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS AND APPLY IT TO VARIOUS SITUATIONS? How do we describe and apply the concept of electric field? How do we describe and apply the electric field created by uniformly charged objects? How do we describe and apply the relationship between the electric field and electric flux? How do we describe and apply Gauss’s Law? How do we describe and apply the nature of electric fields in and around conductors? How do we describe and apply the concept of induced charge and electrostatic shielding?

Agenda Review “Foundational Mathematics’ Skills of Physics” Packet (Page 11) with answer guide. Complete the Gauss’ Law lab using EM Field 6 Complete Web Assign Problems

Summary What did you learn about drawing Gaussian surfaces when using EM Field? HW (Place in your agenda): Electrostatics Lab #4 Report (Due in 5 Classes) Web Assign How do we apply integration and the Principle of Superposition to uniformly charged objects? How do we identify and apply the fields of highly symmetric charge distributions? How do we describe and apply the electric field created by uniformly charged objects?

Thursday (Day 5)

Warm-Up Complete Chapter 22 Graphic Organizers Thurs, Feb 26
Place your homework on your desk: Web Assign Problems For future assignments - check online at

HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS AND APPLY IT TO VARIOUS SITUATIONS? How do we describe and apply the concept of electric field? How do we describe and apply the electric field created by uniformly charged objects? How do we describe and apply the relationship between the electric field and electric flux? How do we describe and apply Gauss’s Law? How do we describe and apply the nature of electric fields in and around conductors? How do we describe and apply the concept of induced charge and electrostatic shielding?

Agenda Complete the Gauss’ Law lab using EM Field 6
Complete Web Assign Problems

Summary Would Gauss’ Law be helpful in determining the electric field due to an electric dipole? HW (Place in your agenda): Electrostatics Lab #4 Report (Due in 4 Classes) Web Assign How do we apply integration and the Principle of Superposition to uniformly charged objects? How do we identify and apply the fields of highly symmetric charge distributions? How do we describe and apply the electric field created by uniformly charged objects?

Physics II: Electricity & Magnetism

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