# Gauss's Law

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## Video Clips

#### Gauss's Law

RealVideo®
5:39 minutes (7:11 - 12:50)

Calculating the flux through a sphere around a point charge, leading to the statement of Gauss's Law that flux through a closed surface is proportional to the net charge enclosed by the surface.

Prior Knowledge: Electric Field (beginning of video lecture 2), Electric Flux (beginning of video lecture 3)
Instructor: Prof. Walter Lewin

#### Gauss's Law With Spherical Symmetry

RealVideo®
10:53 minutes (12:50 - 23:43)

Need symmetric situations to find electric field using Gauss's Law. Example of finding the electric field inside and outside a hollow sphere with uniformly distributed charge. Comparison to gravitational field.

Prior Knowledge: Gauss's Law (7:11 of video lecture 3)
Instructor: Prof. Walter Lewin

#### Gauss's Law With Planar Symmetry

RealVideo®
6:22 minutes (23:43 - 30:05)

Finding the electric field around a plane with uniformly distributed charge. Result is stated, but some of the work is left to students as an exercise.

Prior Knowledge: Gauss's Law (7:11 of video lecture 3)
Instructor: Prof. Walter Lewin

#### Gauss's Law - Cylindrical Symmetry

RealVideo®
8:44 minutes (3:03 - 11:47)

Finding the magnitude and direction of the electric field created by a long cylinder with uniform charge density.

Prior Knowledge: Gauss's Law (7:11 of video lecture 3)
Instructor: Prof. Walter Lewin

## Lecture Notes

#### Gauss's Law

PDF
Page 7 to page 41

Gauss's Law defined; electric flux; open and closed surfaces; choosing gaussian surfaces and examples with spherical, cylindrical, and planar symmetry.

Prior Knowledge: Electric Field (OT2.4-2.6), Charge Density (OT2.9)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Determining Potential from Gauss's Law

PDF
Page 42 to page 47

Examples utilizing integrals and Gauss's Law to determine potential of a point charge and solid charged sphere.

Prior Knowledge: Electric Potential (OT3.1), Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Electric Flux

PDF
Page 7 to page 11

Definition; flux through cylinder and closed surfaces; flux with contained charge; Gauss's Law.

Prior Knowledge: Electric field
Instructor: Prof. Gabriella Sciolla

#### Gauss's Law

PDF
Page 12 to page 14

Restatement; application to sphere of charge, shell of charge, plane of charge.

Prior Knowledge: Electric flux
Instructor: Prof. Gabriella Sciolla

#### Proof of Gauss's Law

PDF
Page 2 to page 3

Theoretical and experimental verification of Gauss's Law.

Prior Knowledge: Gauss's Law
Instructor: Prof. Gabriella Sciolla

#### Divergence

PDF
Page 8 to page 11

Application of divergence to Gauss's Law; divergence theorem; definition with limits and derivatives; finding E from V.

Prior Knowledge: Gauss's Law
Instructor: Prof. Gabriella Sciolla

## Online Textbook Chapters

#### Electric Flux

PDF
Page 2 to page 3

Introduction to the concept of flux and its application to the electric field; flux through generalized surfaces in 3-D.

Prior Knowledge: Electric Field (OT2.4-2.6)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Gauss's Law

PDF
Page 3 to page 15

Gauss's Law defined; introduction to Gaussian sufaces for multiple shapes; worked examples for an infinite rod, infinite plane, spherical shell, and solid sphere of uniform charge.

Prior Knowledge: Electric Flux (OT4.1), Charge Density (OT2.9)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Problem Solving Strategies: Using Gauss' Law

PDF
Page 30 to page 31

Introduces step-by-step method to determine electric field using Gauss's Law for spheres, infinite planes, and infinite lines of charge.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

## Practice Problems

#### Finding Field and Charge of a Slab from Potential

PDF
Problem on page 27 to page 29

Given an expression for the electric potential, determine the electric field, location and charge density of the distribution. Solution is included after problem.

Prior Knowledge: Finding Electric Field from Electric Potential (OT3.5), Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Field of Two Parallel Plates of Charge

PDF
Problem on page 32 to page 33

Find the electric field everywhere due to two parallel oppositely charged infinite planes. Solution is included after problem.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Electric Flux Through a Square Surface

PDF
Problem on page 33 to page 35

Find the electric flux due to a point charge through a square surface and through a cube. Solution is included after problem.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Gauss's Law for Gravity

PDF
Problem on page 35

Find the gravitational field inside a spherical mass shell, using Gauss's Law. Solution is included after problem.

Prior Knowledge: Gauss's Law (OT4.2), Newton's Law of Gravity
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Electric Potential of a Solid Sphere

PDF
Problem on page 35 to page 36

Find the electric potential everywhere due to a uniformly charged sphere using Gauss's Law. Solution is included after problem.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Non-Conducting Solid Sphere with a Cavity

PDF
Problem on page 37

Find the electric field within a spherical cavity in a uniformly charged solid sphere.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Field and Potential of a Non-Uniform Sphere

PDF
Problem on page 38

Find electric field and electric potential everywhere due to a sphere with spherically symmetric non-uniform charge density; also, find total potential energy.

Prior Knowledge: Gauss's Law (OT4.2), Configuration Energy (OT3.3.1)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Electric Field of a Charged Slab

PDF
Problem on page 38

Find electric field everywhere due to a charged slab using Gauss's Law.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Electric Potential Energy of a Uniform Sphere

PDF
Problem on page 39

Find the total potential energy of a uniformly charged solid sphere.

Prior Knowledge: Gauss's Law (OT4.2), Configuration Energy (OT3.3.1)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Calculating Electric Field from Potential

PDF
Problem on page 39

Given an expression for the electric potential, determine the electric field and the appropriate charge density distribution.

Prior Knowledge: Finding Electric Field from Electric Potential (OT3.5), Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Electric Flux through Surfaces

PDF
Problem on page 9 to page 10

Determine the sign of electric flux through open and closed surfaces.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Applying Gauss's Law

PDF
Problem on page 11

Determine in what situations Gauss's Law should be utilized.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Quick Gauss's Law Questions

PDF
Problem on page 12 to page 14

Characterize electric field inside charged spherical shell and inside thick charged slab.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Conceptual Questions on Gauss's Law and Conductors

PDF
Problem on page 37

Questions on the interpretation of Gauss's Law and on charge and potential at conductor surfaces.

Prior Knowledge: Gauss's Law (OT4.2), Conductors (OT4.3)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Electric Field From Charged Disk

PDF
Problem 4

5-part E-field problem; calculating and plotting E-field along z-axis; limiting cases; connection to Coulomb's law.

Prior Knowledge: None
Instructor: Prof. Walter Lewin

#### Infinite Charged Sheet

PDF
Problem 6

4-part Gauss's law problem; finding E-field in 3 cases; plotting along z-axis.

Prior Knowledge: None
Instructor: Prof. Walter Lewin

#### Point Charge in a Hollow Sphere

PDF
Problem 1

3-part Gauss's law problem; determining qualitative charge distributions for different locations of the charge.

Prior Knowledge: None
Instructor: Prof. Walter Lewin

#### Electric Field From Cylinder

PDF
Problem 2

3-part Gauss's law problem; finding radial E-field and potential.

Prior Knowledge: None
Instructor: Prof. Walter Lewin

#### Uniform Plane Charge

PDF
Problem 1

Drawing and explaining electric field near sheet of charge.

Prior Knowledge: None
Instructors: Dr. Peter Dourmashkin, Prof. Gunther Roland

## Exam Questions

#### Charged Cylinder

PDF
Problem 3

4-part problem; finding charge distribution, electric field, and potential for charged cylinder, then again with a dielectric.

Prior Knowledge: None
Instructor: Prof. Walter Lewin