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FOR SCIENTISTS AND ENGINEERS

A STRATEGIC APPROACH

4/E

PHYSICS

RANDALL D. KNIGHT

Chapter 24 Lecture

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Chapter 24 Gauss’s Law

IN THIS CHAPTER, you will learn about and apply Gauss’s law.

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Chapter 24 Preview

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Chapter 24 Preview

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Chapter 24 Preview

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Chapter 24 Preview

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Chapter 24 Preview

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Chapter 24 Reading Questions

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The amount of electric field passing through a surface is called

  1. Electric flux.
  2. Gauss’s Law.
  3. Electricity.
  4. Charge surface density.
  5. None of the above.

Reading Question 24.1

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The amount of electric field passing through a surface is called

  1. Electric flux.
  2. Gauss’s Law.
  3. Electricity.
  4. Charge surface density.
  5. None of the above.

Reading Question 24.1

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Gauss’s law is useful for calculating electric fields that are

  1. Symmetric.
  2. Uniform.
  3. Due to point charges.
  4. Due to continuous charges.

Reading Question 24.2

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Gauss’s law is useful for calculating electric fields that are

  1. Symmetric.
  2. Uniform.
  3. Due to point charges.
  4. Due to continuous charges.

Reading Question 24.2

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Gauss’s law applies to

  1. Lines.
  2. Flat surfaces.
  3. Spheres only.
  4. Closed surfaces.

Reading Question 24.3

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Gauss’s law applies to

  1. Lines.
  2. Flat surfaces.
  3. Spheres only.
  4. Closed surfaces.

Reading Question 24.3

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The electric field inside a conductor in electrostatic equilibrium is

  1. Uniform.
  2. Zero.
  3. Radial.
  4. Symmetric.

Reading Question 24.4

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The electric field inside a conductor in electrostatic equilibrium is

  1. Uniform.
  2. Zero.
  3. Radial.
  4. Symmetric.

Reading Question 24.4

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Chapter 24 Content, Examples, and QuickCheck Questions

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Electric Field of a Charged Cylinder

  • Suppose we knew only two things about electric fields:
    1. The field points away from positive charges, toward negative charges.
    2. An electric field exerts a force on a charged particle.
  • From this information alone, what can we deduce about the electric field of an infinitely long charged cylinder?
  • All we know is that this charge is positive, and that it has cylindrical symmetry.

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Cylindrical Symmetry

  • An infinitely long charged cylinder is symmetric with respect to
    • Translation parallel to the cylinder axis.
    • Rotation by an angle about �the cylinder axis.
    • Reflections in any plane containing or perpendicular �to the cylinder axis.
  • The symmetry of the electric �field must match the symmetry �of the charge distribution.

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Electric Field of a Charged Cylinder

  • Could the field look like the figure below? (Imagine this picture rotated about the axis.)
  • The next slide shows what the field would look like reflected in a plane perpendicular to the axis (left to right).

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Electric Field of a Charged Cylinder

  • This reflection, which does not make any change in the charge distribution itself, does change the electric field.
  • Therefore, the electric field of a cylindrically symmetric charge distribution cannot have a component parallel to the cylinder axis.

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Electric Field of a Charged Cylinder

  • Could the field look like the figure below? (Here we’re looking down the axis of the cylinder.)
  • The next slide shows what the field would look like reflected in a plane containing the axis (left to right).

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Electric Field of a Charged Cylinder

  • This reflection, which does not make any change in the charge distribution itself, does change the electric field.
  • Therefore, the electric field of a cylindrically symmetric charge distribution cannot have a component tangent to the circular cross section.

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Electric Field of a Charged Cylinder

  • Based on symmetry arguments alone, an infinitely long charged cylinder must have a radial electric field, as shown below.
  • This is the one electric field shape that matches the symmetry of the charge distribution.

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Planar Symmetry

  • There are three fundamental symmetries; the first is planar symmetry.
  • Planar symmetry involves symmetry with respect to:
    • Translation parallel to the plane.
    • Rotation about any line perpendicular to the plane.
    • Reflection in the plane.

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Cylindrical Symmetry

  • There are three fundamental symmetries; the second is cylindrical symmetry.
  • Cylindrical symmetry involves symmetry with respect to
    • Translation parallel to �the axis.
    • Rotation about the axis.
    • Reflection in any plane containing or perpendicular to the axis.

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Spherical Symmetry

  • There are three fundamental symmetries; the third is spherical symmetry.
  • Spherical symmetry involves symmetry with respect to
    • Rotation about any axis that passes through the center point.
    • Reflection in any plane containing the center point.

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The Concept of Flux

  • Consider a box surrounding a region of space.
  • We can’t see into the box, but we know there is an outward-pointing electric field passing through every surface.
  • Since electric fields point away from positive charges, �we can conclude �that the box must contain net positive electric charge.

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The Concept of Flux

  • Consider a box surrounding a region of space.
  • We can’t see into the box, but we know there is an inward-pointing electric field passing through every surface.
  • Since electric fields �point toward negative charges, we can conclude that the �box must contain net negative electric charge.

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The Concept of Flux

  • Consider a box surrounding a region of space.
  • We can’t see into the box, but we know that the electric field points into the box on the left, and an equal electric field points out of the box on the right.
  • Since this external electric field is not altered by the contents of the box, the box must contain zero �net electric charge.

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Gaussian Surfaces

  • A closed surface through which an electric field passes is called a Gaussian surface.
  • This is an imaginary, mathematical surface, �not a physical surface.

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Gaussian Surfaces

  • A Gaussian surface is most useful when it matches the shape and symmetry of the field.
  • Figure (a) below shows a cylindrical Gaussian surface.
  • Figure (b) simplifies the drawing by showing two-dimensional end and side views.
  • The electric field is everywhere perpendicular to the side wall and no field passes through the top and bottom surfaces.

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Gaussian Surfaces

  • Not every surface is useful for learning about charge.
  • Consider the spherical surface in the figure.
  • This is a Gaussian surface, and the protruding electric field tells us there’s a positive charge inside.
  • It might be a point charge located on the left side, but we can’t really say.
  • A Gaussian surface that doesn’t match the symmetry of the charge distribution isn’t terribly useful.

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The Basic Definition of Flux

  • Imagine holding a rectangular wire loop of area A in front of a fan.
  • The volume of air flowing through the loop each second depends on the angle between the loop and the direction of flow.
  • The flow is maximum through a loop that is perpendicular to the airflow.

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The Basic Definition of Flux

  • Imagine holding �a rectangular wire loop of area A in front of a fan.
  • No air goes through the same loop if �it lies parallel to �the flow.

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The Basic Definition of Flux

  • Imagine holding �a rectangular wire loop of area A in �front of a fan.
  • The volume of air flowing through the loop each second depends on the angle θ between the loop normal and the velocity of the air:

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The Electric Flux

  • The electric flux Φe measures the amount of electric field passing through a surface of area A whose normal to the surface is tilted at angle θ from the field.

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The electric flux through the shaded surface is

QuickCheck 24.1

  1. 0
  2. 200 N m/C
  3. 400 N m2/C
  4. Flux isn’t defined for an open surface.

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The electric flux through the shaded surface is

QuickCheck 24.1

  1. 0
  2. 200 N m/C
  3. 400 N m2/C
  4. Flux isn’t defined for an open surface.

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The electric flux through the shaded surface is

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  1. 0
  2. 200 N m/C
  3. 400 N m2/C
  4. Some other value.

​

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The electric flux through the shaded surface is

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  1. 0
  2. 200 N m/C
  3. 400 N m2/C
  4. Some other value.

​

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The electric flux through the shaded surface is

QuickCheck 24.3

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  1. 0
  2. 400cos20º N m2/C
  3. 400cos70º N m2/C
  4. 400 N m2/C
  5. Some other value.

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The electric flux through the shaded surface is

QuickCheck 24.3

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  1. 0
  2. 400cos20º N m2/C
  3. 400cos70º N m2/C
  4. 400 N m2/C
  5. Some other value.

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The Area Vector

  • Let’s define an area vector to be a vector in the direction of , perpendicular to the surface, with a magnitude A equal to the area of the surface.
  • Vector has units of m2.

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The Electric Flux

  • An electric field passes through a surface of area A.
  • The electric flux can be defined as the dot-product:

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Example 24.1 The Electric Flux Inside a Parallel-Plate Capacitor

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Example 24.1 The Electric Flux Inside a Parallel-Plate Capacitor

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Example 24.1 The Electric Flux Inside a Parallel-Plate Capacitor

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The Electric Flux of a Nonuniform Electric Field

  • Consider a surface in a nonuniform electric field.
  • Divide the surface into many small pieces of area δA.
  • The electric flux through each small piece is
  • The electric flux through the whole surface is the surface integral:

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The Flux Through a Curved Surface

  • Consider a curved surface in an electric field.
  • Divide the surface into many small pieces of area δA.
  • The electric flux through each small piece is
  • The electric flux through the whole surface is the surface integral:

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Electric Fields Tangent to a Surface

  • Consider an electric field that is everywhere tangent, or parallel, to a curved surface.
  • is zero at every point on the surface, because is perpendicular to at every point.
  • Thus Φe = 0.

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Electric Fields Perpendicular to a Surface

  • Consider an electric field that is everywhere perpendicular to the surface and has the same magnitude E at every point.
  • In this case,

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Tactics: Evaluating Surface Integrals

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Surfaces A and B have the same shape and the same area. Which has the larger electric flux?

QuickCheck 24.4

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  1. Surface A has more flux.
  2. Surface B has more flux.
  3. The fluxes are equal.
  4. It’s impossible to say without knowing more about the electric field.

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Surfaces A and B have the same shape and the same area. Which has the larger electric flux?

QuickCheck 24.4

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  1. Surface A has more flux.
  2. Surface B has more flux.
  3. The fluxes are equal.
  4. It’s impossible to say without knowing more about the electric field.

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QuickCheck 24.5

Which surface, A or B, has the larger electric flux?

  1. Surface A has more flux.
  2. Surface B has more flux.
  3. The fluxes are equal.
  4. It’s impossible to say without knowing more about the electric field.

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QuickCheck 24.5

Which surface, A or B, has the larger electric flux?

  1. Surface A has more flux.
  2. Surface B has more flux.
  3. The fluxes are equal.
  4. It’s impossible to say without knowing more about the electric field.

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The Electric Flux Through a Closed Surface

  • The electric flux through a closed surface is
  • The electric flux is still the summation of the fluxes through a vast number of tiny pieces, pieces that now cover a closed surface.
  • NOTE: For a closed surface, we use the convention that the area vector dA is defined to always point toward the outside.

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Tactics: Finding the Flux Through a Closed Surface

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QuickCheck 24.6

These are cross sections of 3D closed surfaces. The top and bottom surfaces, which are flat, are in front of and behind the screen. The electric field is everywhere parallel to the screen. Which closed surface or surfaces have zero electric flux?

​

  1. Surface A
  2. Surface B
  3. Surface C
  4. Surfaces B and C
  5. All three surfaces

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QuickCheck 24.6

These are cross sections of 3D closed surfaces. The top and bottom surfaces, which are flat, are in front of and behind the screen. The electric field is everywhere parallel to the screen. Which closed surface or surfaces have zero electric flux?

​

  1. Surface A
  2. Surface B
  3. Surface C
  4. Surfaces B and C
  5. All three surfaces

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Electric Flux of a Point Charge

  • The flux integral through a spherical Gaussian surface centered on a single point charge is
  • The surface area of a sphere is Asphere = 4πr2.
  • Using Coulomb’s law for E, we find

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Electric Flux of a Point Charge

  • The electric flux through a spherical surface centered on a single positive point charge is Φe = q/ϵ0.
  • This depends on the amount of charge, but not on the radius of the sphere.
  • For a point charge, electric flux is independent of r.

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Electric Flux of a Point Charge

  • The electric flux through any arbitrary closed surface surrounding a point charge q may be broken up into spherical and radial pieces.
  • The total flux through the spherical pieces must be the same as through a single sphere:

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Electric Flux of a Point Charge

  • The electric flux through �any arbitrary closed surface entirely outside a point charge q may also be �broken up into spherical �and radial pieces.
  • The total flux through �the concave and convex spherical pieces must �cancel each other.
  • The net electric flux is �zero through a closed �surface that does not �contain any net charge.

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Electric Flux of Multiple Charges

  • Consider an arbitrary Gaussian surface and a group of charges q1, q2, q3, …
  • The contribution to the total flux for any charge qi inside the surface is qi /ϵ0.
  • The contribution for any charge outside the surface �is zero.
  • Defining Qin to be the sum �of all the charge inside the surface, we find Φe = Qin /ϵ0.

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Gauss’s Law

  • For any closed surface enclosing total charge Qin, the net electric flux through the surface is
  • This result for the electric flux is known as Gauss’s Law.

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QuickCheck 24.7

The electric field is constant �over each face of the box. �The box contains

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  1. Positive charge.
  2. Negative charge.
  3. No net charge.
  4. Not enough information to tell.

​

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QuickCheck 24.7

The electric field is constant �over each face of the box. �The box contains

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  1. Positive charge.
  2. Negative charge.
  3. No net charge.
  4. Not enough information to tell.

​

Net flux is outward.

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QuickCheck 24.8

Which spherical Gaussian surface has the larger electric flux?

​

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  1. Surface A
  2. Surface B
  3. They have the same flux.
  4. Not enough information to tell.

​

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QuickCheck 24.8

Which spherical Gaussian surface has the larger electric flux?

​

​

  1. Surface A
  2. Surface B
  3. They have the same flux.
  4. Not enough information to tell.

​

Flux depends only on the enclosed charge, not the radius.

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QuickCheck 24.9

Spherical Gaussian surfaces of �equal radius R surround two �spheres of equal charge Q. �Which Gaussian surface has �the larger electric field?

​

  1. Surface A
  2. Surface B
  3. They have the same electric field.
  4. Not enough information to tell.

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QuickCheck 24.9

Spherical Gaussian surfaces of �equal radius R surround two �spheres of equal charge Q. �Which Gaussian surface has �the larger electric field?

​

  1. Surface A
  2. Surface B
  3. They have the same electric field.
  4. Not enough information to tell.

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Using Gauss’s Law

  1. Gauss’s law applies only to a closed surface, called �a Gaussian surface.
  2. A Gaussian surface is not a physical surface. It need not coincide with the boundary of any physical object (although it could if we wished). It is an imaginary, mathematical surface in the space surrounding one or more charges.
  3. We can’t find the electric field from Gauss’s law alone. We need to apply Gauss’s law in situations where, from symmetry and superposition, we already can guess the shape of the field.

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Problem-Solving Strategy: Gauss’s Law

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QuickCheck 24.10

A spherical Gaussian surface surrounds an electric dipole. The net enclosed charge is zero. Which is true?

​

  1. The electric field is �zero everywhere on �the Gaussian surface.
  2. The electric field is not �zero everywhere on the Gaussian surface.
  3. Whether or not the field is zero on the surface depends on where the dipole is inside the sphere.

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QuickCheck 24.10

A spherical Gaussian surface surrounds an electric dipole. The net enclosed charge is zero. Which is true?

​

  1. The electric field is �zero everywhere on �the Gaussian surface.
  2. The electric field is not �zero everywhere on the Gaussian surface.
  3. Whether or not the field is zero on the surface depends on where the dipole is inside the sphere.

The flux is zero, but that doesn’t require the field to be zero.

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QuickCheck 24.11

The electric flux is shown through two Gaussian surfaces. In terms of q, what are charges q1 and q2?

​

  1. q1 = 2q; q2 = q
  2. q1 = q; q2 = 2q
  3. q1 = 2q; q2 = –q
  4. q1 = 2q; q2 = –2q
  5. q1 = q/2; q2 = q/2

​

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QuickCheck 24.11

The electric flux is shown through two Gaussian surfaces. In terms of q, what are charges q1 and q2?

​

  1. q1 = 2q; q2 = q
  2. q1 = q; q2 = 2q
  3. q1 = 2q; q2 = –q
  4. q1 = 2q; q2 = –2q
  5. q1 = q/2; q2 = q/2

​

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Example 24.3 Outside a Sphere of Charge

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Example 24.3 Outside a Sphere of Charge

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Example 24.3 Outside a Sphere of Charge

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Example 24.3 Outside a Sphere of Charge

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Example 24.3 Outside a Sphere of Charge

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Example 24.3 Outside a Sphere of Charge

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Example 24.6 The Electric Field of a Plane of Charge

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Example 24.6 The Electric Field of a Plane of Charge

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Example 24.6 The Electric Field of a Plane of Charge

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Example 24.6 The Electric Field of a Plane of Charge

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Example 24.6 The Electric Field of a Plane of Charge

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Example 24.6 The Electric Field of a Plane of Charge

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QuickCheck 24.12

A cylindrical Gaussian surface �surrounds an infinite line of charge. �The flux Φe through the two flat �ends of the cylinder is

​

  1. 0
  2. 2×2πrE
  3. 2×πr2E
  4. 2×rLE
  5. It will require an integration to find out.

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QuickCheck 24.12

A cylindrical Gaussian surface �surrounds an infinite line of charge. �The flux Φe through the two flat �ends of the cylinder is

​

  1. 0
  2. 2×2πrE
  3. 2×πr2E
  4. 2×rLE
  5. It will require an integration to find out.

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QuickCheck 24.13

A cylindrical Gaussian surface �surrounds an infinite line of charge. �The flux Φe through the wall �of the cylinder is

​

  1. 0
  2. 2πrLE
  3. πr2LE
  4. rLE
  5. It will require an integration to find out.

​

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QuickCheck 24.13

A cylindrical Gaussian surface �surrounds an infinite line of charge. �The flux Φe through the wall �of the cylinder is

​

  1. 0
  2. 2πrLE
  3. πr2LE
  4. rLE
  5. It will require an integration to find out.

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Conductors in Electrostatic Equilibrium

  • The figure shows a Gaussian surface just inside a conductor’s surface.
  • The electric field must be �zero at all points within the conductor, or else the electric field would cause the charge carriers to move and it wouldn’t be in equilibrium.
  • By Gauss’s Law, Qin = 0

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Conductors in Electrostatic Equilibrium

  • The external electric �field right at the surface of a conductor must be perpendicular to that surface.
  • If it were to have a tangential component, �it would exert a force on the surface charges and cause a surface current, and the conductor would not be in electrostatic equilibrium.

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Electric Field at the Surface of a Conductor

where η is the surface charge density of the conductor.

  • A Gaussian surface extending through the surface of a conductor has a flux only through the outer face.
  • The net flux is �Φe = AEsurface = Qin/ϵ0.
  • Here Qin = ηA, so the electric field at the surface of a conductor is

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QuickCheck 24.14

A point charge q is located distance r from the center of a neutral metal sphere. The electric field at the �center of the sphere is

​

  4. 0
  5. It depends on what the metal is.

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QuickCheck 24.14

A point charge q is located distance r from the center of a neutral metal sphere. The electric field at the �center of the sphere is

​

  4. 0
  5. It depends on what the metal is.

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Conductors in Electrostatic Equilibrium

  • The figure shows a charged conductor with a hole inside.
  • Since the electric field is zero inside the conductor, we must conclude that Qin = 0 for the interior surface.
  • Furthermore, since there’s �no electric field inside the conductor and no charge inside the hole, the electric field in the hole must be zero.

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Faraday Cages

  • The use of a conducting box, or Faraday cage, to exclude electric fields from a region of space is called screening.

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Conductors in Electrostatic Equilibrium

  • The figure shows a �charge q inside a hole within a neutral conductor.
  • Net charge –q moves to the inner surface and net charge +q is left behind �on the exterior surface.

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QuickCheck 24.15

Charge +3 nC is in a hollow cavity �inside a large chunk of metal that �is electrically neutral. The total �charge on the exterior surface �of the metal is

​

  1. 0 nC
  2. +3 nC
  3. –3 nC
  4. Can’t say without knowing the shape and location of the hollow cavity.

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QuickCheck 24.15

Charge +3 nC is in a hollow cavity �inside a large chunk of metal that �is electrically neutral. The total �charge on the exterior surface �of the metal is

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  1. 0 nC
  2. +3 nC
  3. –3 nC
  4. Can’t say without knowing the shape and location of the hollow cavity.

​

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Tactics: Finding the Electric Field of a Conductor in Electrostatic Equilibrium

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Example 24.7 The Electric Field at the Surface of a Charged Metal Sphere

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Example 24.7 The Electric Field at the Surface of a Charged Metal Sphere

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Example 24.7 The Electric Field at the Surface of a Charged Metal Sphere

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Chapter 24 Summary Slides

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General Principles

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General Principles

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Important Concepts

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Important Concepts

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Important Concepts

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Applications

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