Gauss's Law and Electric Flux

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Questions and Answers

Electric flux is best described as being proportional to which quantity concerning electric field lines passing through a surface?

  • The intensity of color of the lines.
  • The average speed of the lines.
  • The total length of the lines.
  • The number of lines. (correct)

If a closed surface encloses a charge, what determines the net number of electric field lines passing through it?

  • The shape of the surface.
  • The size of the enclosed charge. (correct)
  • The material properties of the surface.
  • The color of the surface.

When an electric field is uniform and oriented perpendicular to a rectangular surface of area A, how is the electric flux $Φ$ calculated?

  • $Φ = E + A$
  • $Φ = E/A$
  • $Φ = EA$ (correct)
  • $Φ = A/E$

If a surface with area A is oriented at an angle $\theta$ to a uniform electric field E, the electric flux through the surface is proportional to:

<p>The cosine of theta. (A)</p> Signup and view all the answers

When calculating electric flux through a non-uniform field incident on an irregular surface, what approach is used?

<p>Dividing the surface into small elements where the field is nearly constant. (B)</p> Signup and view all the answers

How is the total electric flux ($\Phi$) through a surface calculated if the electric field ($\vec{E}$) is constant over small area elements ($d\vec{A}$)?

<p>$\Phi = \int \vec{E} \cdot d\vec{A}$ over the surface (D)</p> Signup and view all the answers

For a closed surface, how do you determine the net electric flux?

<p>Subtract the flux entering the surface from the flux leaving it. (A)</p> Signup and view all the answers

Consider a sphere. If the electric field lines are entering the sphere are equal to the number of lines leaving it, what can be said about the net flux?

<p>It is zero. (D)</p> Signup and view all the answers

According to Gauss's Law, what is the relationship between the net electric flux through a closed surface and the charge enclosed by that surface?

<p>Proportional (C)</p> Signup and view all the answers

Gauss's Law relates which of the following?

<p>The net electric flux through a closed surface to the enclosed charge. (B)</p> Signup and view all the answers

If a charge q is located at the center of a sphere, the electric field at the surface of the sphere is proportional to which of the following?

<p>The permittivity of free space. (A)</p> Signup and view all the answers

If a closed spherical surface encloses a charge of q, how does the net electric flux change if the radius of the sphere is doubled?

<p>It remains the same. (C)</p> Signup and view all the answers

A closed surface encloses a charge q. If the surface is changed from a sphere to a cube, what happens to the net electric flux through the surface?

<p>It remains the same. (C)</p> Signup and view all the answers

A point charge is located outside a closed surface of arbitrary shape. What is the net electric flux through the surface?

<p>It is zero. (C)</p> Signup and view all the answers

What determines the electric field of a system with discrete charges or a continuous charge distribution?

<p>Their symmetry. (B)</p> Signup and view all the answers

If the net flux through a Gaussian surface is zero, which statement must be true?

<p>The net charge inside the surface is zero. (C)</p> Signup and view all the answers

A spherical Gaussian surface surrounds a point charge q. If the charge is tripled, what happens to the flux through the surface?

<p>It gets tripled. (C)</p> Signup and view all the answers

How does the electric flux through a Gaussian surface change if the volume of the enclosed sphere is doubled while the charge remains constant?

<p>It remains constant. (D)</p> Signup and view all the answers

If the surface enclosing a charge is changed from a sphere to a cube, how does the electric flux through the surface change?

<p>It remains the same. (D)</p> Signup and view all the answers

A charge is moved to another location inside a Gaussian surface. How does this affect the electric flux through the surface?

<p>It remains the same. (D)</p> Signup and view all the answers

What parameter is characterized using these terms: volume charge density, surface charge density and linear charge density?

<p>Charge distribution (B)</p> Signup and view all the answers

If a charge Q is uniformly distributed over a volume V, how is the volume charge density ($\rho$) defined?

<p>$\rho = Q/V$ (B)</p> Signup and view all the answers

The electric field due to a point charge can be calculated using Gauss's Law by selecting what?

<p>A spherical surface centered at the charge. (D)</p> Signup and view all the answers

When applying Gauss's Law to calculate the electric field due to a point charge, what geometrical relationship between the electric field and the Gaussian surface simplifies the calculation?

<p>The electric field is perpendicular to the surface. (B)</p> Signup and view all the answers

For a uniformly charged sphere, how does the electric field outside the sphere compare to that of a point charge?

<p>They are equivalent if the point charge is located at the center of the sphere. (A)</p> Signup and view all the answers

Inside of a uniformly charged sphere, what happens to the electric field as the distance from the center increases?

<p>Increases linearly (A)</p> Signup and view all the answers

If a thin spherical shell has a total charge Q distributed uniformly over its surface, what is the electric field inside the shell?

<p>It is zero. (D)</p> Signup and view all the answers

Outside of a thin spherical shell with total charge $Q$ and radius R, what is the electric field a distance $r$ from the center?

<p>$\frac{k_eQ}{r^2}$ (D)</p> Signup and view all the answers

What is the appropriate Gaussian surface to calculate the electric field at a distance r from a uniform positive line charge using Gauss's Law?

<p>A cylinder of radius r and length l, coaxial with the line charge. (A)</p> Signup and view all the answers

Given a uniform positive line charge, how is the electric field oriented with respect to a cylindrical Gaussian surface used to calculate it?

<p>Perpendicular to the curved surface and parallel to the end caps. (B)</p> Signup and view all the answers

How is a Gaussian surface constructed to determine the electric field due to a non-conducting, infinite plane of positive charge?

<p>A cylinder with its axis perpendicular to the plane. (C)</p> Signup and view all the answers

For a cylinder-shaped Gaussian surface when finding the electric field due to a plane of charge, how is the electric field oriented?

<p>Perpendicular to the plane and parallel to the end surfaces. (A)</p> Signup and view all the answers

What condition defines electrostatic equilibrium in a conductor?

<p>No net motion of charge within the conductor. (A)</p> Signup and view all the answers

What is the electric field at everywhere inside a conductor in electrostatic equilibrium?

<p>It is zero. (D)</p> Signup and view all the answers

Where does any net charge reside on an isolated conductor?

<p>Exclusively on its surface. (D)</p> Signup and view all the answers

What is the direction of the electric field just outside a charged conductor in electrostatic equilibrium?

<p>Perpendicular to the surface. (A)</p> Signup and view all the answers

On an irregularly shaped conductor, where does charge tend to accumulate?

<p>At sharp locations. (A)</p> Signup and view all the answers

According to Gauss's law relating to conductors, where is the field strongest?

<p>On the surface of a conductor with great curvature (A)</p> Signup and view all the answers

In a conducting slab placed in an external electric field, what happens to free electrons?

<p>They drift to one side, creating a surface charge. (C)</p> Signup and view all the answers

In a conductor, a state will be reached when the internal electric field balances that of the external field. What is the implication of this balance?

<p>A net field of zero exists inside. (C)</p> Signup and view all the answers

Flashcards

Electric Flux (Φ)

A measure of the number of electric field lines passing through a surface.

Flux & Enclosed Charge

The electric flux through a surface is proportional to the enclosed charge, regardless of the surface's shape.

Electric Flux (Uniform E)

In a uniform electric field (E) perpendicular to a rectangular surface of area (A), electric flux (Φ) equals EA.

Flux (Non-Perpendicular)

When the surface is not perpendicular to the uniform field, electric flux equals E * A * cos(θ).

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Electric Field (Irregular Surface)

Divide into many small elements , the electric field E can be considered constant inside this small area element.

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Closed Surface

Flux through a closed surface that divides space into inside and outside regions.

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Net Flux (Closed Surface)

For a closed surface, the net flux through it is given by Φ_c = Φ_leaving - Φ_entering.

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Gauss's Law

Relates the net electric flux through a closed surface to the charge enclosed by that surface.

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Surface Shape & Flux

A closed surface's shape does not affect net electrical flux.

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Charge Location and Flux

A surface doesn't enclose a point charge, the net electric flux is zero.

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Net Electric Flux Formula

The net electric flux through any closed surface is equal to the net charge inside it divided by ε₀.

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Zero Net Flux Implication

If net flux through a Gaussian surface is zero, the net charge is zero inside the surface.

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Volume Charge Density (ρ)

Charge per unit volume.

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Surface Charge Density (σ)

Charge per unit area.

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Linear Charge Density (λ)

Charge per unit length.

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Definition of conductors

Materials containing free electrons that moves about within it are conductors.

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Electrostatic Equilibrium

No net motion of charge within a conductor

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Zero Electric Field

The electric field is zero everywhere inside the conductor is property of conductors in electrostatic equilibrium

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Charge Accumulation

Charge tends to accumulate at sharp locations on an irregularly shaped conductor.

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Study Notes

Chapter 24: Gauss's Law

  • Chapter introduces the concept of electric flux and its applications using Gauss's Law.

Electric Flux

  • Described qualitatively using electric field lines.

Electric Flux (Φ)

  • Represents the number of electric field lines penetrating a surface.
  • Electric flux is proportional to the charge (q) as N ∝ q, where N is the number of electric field lines.
  • When a surface encloses a charge, the net number of lines that pass through the surface is proportional to the net charge, regardless of the surface's shape, so Φ₁ = Φ₂ = Φ₃.

Calculation of Electric Flux

  • For a uniform electric field (E) and a rectangular surface (A) where the electric field lines are perpendicular, the number of lines per unit area is proportional to E.
  • The number of lines is proportional to EA, and the electric flux is defined as Φ = EA.
  • The units for electric flux are [Φ] = [E][A] = N/C * m² = N*m²/C
  • If the surface is not perpendicular to the field, the normal to the surface of area "A" makes an angle θ with the uniform field, the electric flux is Φ = E * A' = E * A cos θ.
  • The flux can also be expressed as Φ = E • A.
  • The number of lines crossing A equals the number of lines crossing the projected area A'.
  • If the electric field is not uniform and the surface is irregular, the surface is divided into small elements, each with an area of ΔA.
  • ΔA is a vector representing the area of the i-th element.
  • It's assumed that the electric field Eᵢ is constant inside this small area element.
  • The flux through the element is ΔΦᵢ = Eᵢ * ΔAᵢ cos θᵢ, or simply ΔΦᵢ = Eᵢ • ΔAᵢ.
  • The total electric flux is given by Φ = Σ ΔΦᵢ = Σ Eᵢ • ΔAᵢ.
  • For infinitesimally small elements, Φ = ∫Surface E • dA.

Electric Flux Through a Closed Surface

  • A closed surface divides space into inside and outside regions.
  • The net flux through the surface is given by Φc = ΦLeaving - ΦEntering, where Φc = § E • dA = § Eₙ dA.
  • Eₙ = E cos θ, representing the normal component of E.

Example 24.1: Electric Flux Through a Sphere

  • Find the electric flux through a sphere of radius 1.00 m with a charge of +1.00 μC at its center.
  • The magnitude of the electric field is calculated, followed by use of the formula Φ = E • A.
  • This gives Φ = EA = E(4πr²) = 9x10³(4π(1.0)²) = 1.13×10⁵ N•m²/C.

Gauss's Law

  • This is used to solve this problem more easily.

Gauss’s Law

  • Relates the net electric flux through a closed surface to the charge enclosed by the surface.
  • For a positive charge q at the center of a sphere of radius r, the electric field at the surface is E = ke q/r².
  • Φc = § E • dA = § E cos θdA
  • Фс = § E dA, where En = E.
  • The flux is Φc = § E dA, where A = 4πr² and E = keq/r², so Φc = 4πkeq.
  • This formula means that the net electric flux through a closed spherical surface is proportional to the charge inside it.
  • The net electric flux through a closed spherical surface is independent of its radius "r".
  • This is because (E) is proportional to (1/r²) and (A) is proportional to (r²).
  • Because of the shape of the surface, N1 = N2 = N3.
  • The net flux is the same due to each surface, Φ1 = Φ2 = Φ3.
  • This means Φc = q/ε₀ regardless of the surface shape.

Effect of Charge Location

  • Electric charge located outside the surface.

Conceptual Example 24.3a

Answers with Gauss’s law to conceptual problems.

Properties of Conductors in Electrostatic Equilibrium

  • Electric field is zero everywhere inside the conductor.
  • Any charge on an isolated conductor resides on its surface.
  • The electric field just outside a charged conductor is perpendicular to the surface and equals σ/ε₀, where σ is the charge per unit area.
  • On an irregularly shaped conductor, charge accumulates at sharp locations.

Applications of Gauss’s Law

  • Point charge
  • Spherically symmetric charge distribution
  • Outside the charged sphere
  • Inside the charged sphere
  • Thin spherical shell
  • Outside the shell
  • Inside the shell
  • Cylindrically symmetric charge distribution.
  • Non conducting infinite plane.

Charge Density

  • When applying Gauss's Law, it's helpful to understand charge density concepts.

Volume Charge Density (ρ)

  • If a charge Q is uniformly distributed over a volume V, the charge per unit volume is ρ = Q/V (C/m³).

Surface Charge Density (σ)

  • Charge Q is uniformly distributed over a surface of area A, the charge per unit area is σ = Q/A (C/m²).

Linear Charge Density (λ)

  • Charge Q is uniformly distributed over a line of length ℓ, the charge per unit length is λ = Q/ℓ (C/m).

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