Gauss' Law and Divergence Theorem Quiz
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Questions and Answers

What is the unit of measurement for electric flux?

  • Newtons per square meter
  • Coulombs per square meter
  • Volts per meter
  • Coulombs (correct)

    • What describes the direction of electric flux density D at a point?

  • The direction of the surface area vector
  • The direction of the flux lines at that point (correct)
  • The direction of the charge Q
  • The direction of the electric field E

    • How is the magnitude of electric flux density D calculated?

  • By counting the total number of charge carriers crossing the surface
  • By dividing the electric flux by the surface area (correct)
  • By dividing the total charge by the surface area
  • By multiplying the electric field by the surface area

    • Which equation correctly shows the relationship between electric flux density D and electric field intensity E in free space?

    <p>D = ε0 E</p> Signup and view all the answers

    What does Gauss' Law state about electric flux through a closed surface?

    <p>It is proportional to the total charge enclosed by the surface.</p> Signup and view all the answers

    What special conditions must a closed surface satisfy for Gauss' Law integration?

    <p>D must be either normal or tangential to the surface.</p> Signup and view all the answers

    Which of the following describes the differential flux crossing a surface area dS?

    <p>dψ = D ∙ dS</p> Signup and view all the answers

    At a point located at distance r from a point charge Q, what is the formula for electric flux density D?

    <p>D = Q / (4πε0r^2) a</p> Signup and view all the answers

    What is the expression for the electric displacement field D created by an infinite plane of sheet with surface charge density 𝜌𝑠?

    <p>𝐃 = rac{𝜌𝑠}{2𝜖𝑜} 𝐚𝑛</p> Signup and view all the answers

    In the region between two infinite parallel-plate capacitors where the plates are charged, what is the electric field E?

    <p> rac{𝜌𝑠}{2𝜖𝑜} 𝐚𝑦</p> Signup and view all the answers

    What happens to the electric field E outside the region of two charged parallel plates?

    <p>It decreases to zero.</p> Signup and view all the answers

    When applying Gauss' law to a small volume with a non-uniform distribution of D, what shape is typically chosen for the Gaussian surface?

    <p>A small cuboid</p> Signup and view all the answers

    In the context of the infinite plane of sheet problem, how does the charge density affect the displacement field?

    <p>It increases the magnitude of D.</p> Signup and view all the answers

    What is the expression for the total charge of a surface charge distribution?

    <p>$Q = \int \rho_S , dS$</p> Signup and view all the answers

    In the context of Gauss' Law for a point charge, what represents the relation between the electric displacement field $D$ and the total charge $Q$?

    <p>$D = \frac{Q}{4\pi r^2} \hat{r}$</p> Signup and view all the answers

    Which component of the electric displacement field D crosses normally through the faces of a Gaussian surface in a differential volume?

    <p>Only one component crosses normally.</p> Signup and view all the answers

    When using Gauss' law, what is the relationship between the total electric flux through a closed surface and the enclosed charge?

    <p>Flux equals the enclosed charge divided by ε₀.</p> Signup and view all the answers

    For an infinite line charge along the z-axis, how is the electric displacement field $D$ expressed?

    <p>$D = D_\rho \hat{\rho}$</p> Signup and view all the answers

    Which of the following integrals represents the total charge contained in a cylindrical volume with a line charge?

    <p>$Q = \rho_L L$</p> Signup and view all the answers

    For a Gaussian surface enclosing a charge, what is the mathematical expression used to calculate the electric flux?

    <p>∮ 𝐷 ∙ 𝑑𝐒</p> Signup and view all the answers

    In spherical coordinates, how do you express the electric field $E$ in terms of the electric displacement field $D$ for a point charge?

    <p>$E = \frac{D}{4\pi\epsilon_o r^2}$</p> Signup and view all the answers

    What is the main characteristic of the Gaussian surface used for an infinite line charge?

    <p>It is cylindrical in shape.</p> Signup and view all the answers

    How is the integration of $D$ performed over the surfaces of the cylindrical Gaussian surface for an infinite line charge?

    <p>Only on the curved surface.</p> Signup and view all the answers

    What does the expression $D = \frac{\rho_L}{2\pi\rho} \hat{\rho}$ represent in the context of an infinite line charge?

    <p>Electric displacement field due to line charge</p> Signup and view all the answers

    What is indicated by a positive divergence of the vector flux density 𝐃?

    <p>A source exists at that point.</p> Signup and view all the answers

    Which mathematical expression represents the divergence of 𝐃 based on the given content?

    <p>div 𝐃 = lim (∮ 𝐃 ∙ 𝑑𝐒) / Δ𝑣</p> Signup and view all the answers

    What does a negative divergence of the vector flux density 𝐃 indicate?

    <p>The existence of a sink.</p> Signup and view all the answers

    In evaluating the divergence of 𝐃, what occurs as the volume element Δ𝑣 approaches zero?

    <p>The outflow of flux is measured per unit volume.</p> Signup and view all the answers

    Which relation corresponds to the contribution from the front face of the surface integral for 𝐃?

    <p>∫front = [D_x + Δ𝑥] Δ𝑦 Δ𝑧</p> Signup and view all the answers

    Which of the following expressions correctly represents the total contribution from all faces of the closed surface?

    <p>∮ 𝐃 ∙ 𝑑𝐒 = (∂D_x/∂x + ∂D_y/∂y + ∂D_z/∂z) Δ𝑥 Δ𝑦 Δ𝑧</p> Signup and view all the answers

    What happens to the divergence of 𝐃 when there is no source or sink present?

    <p>It equals zero.</p> Signup and view all the answers

    What is the expression for divergence in rectangular coordinates?

    <p>$\frac{\partial D_x}{\partial x} + \frac{\partial D_y}{\partial y} + \frac{\partial D_z}{\partial z}$</p> Signup and view all the answers

    In cylindrical coordinates, what forms the differential volume element?

    <p>$\rho d\rho d\phi dz$</p> Signup and view all the answers

    Which coordinate system requires the term $r^2 \sin(\theta)$ in its volume differential for the divergence calculation?

    <p>Spherical</p> Signup and view all the answers

    What does the divergence theorem state about total flux crossing a closed surface?

    <p>It equals the total charge within the surface.</p> Signup and view all the answers

    What is the correct point form of Maxwell's first equation?

    <p>$\nabla \cdot D = \rho_v$</p> Signup and view all the answers

    Which expression represents the divergence in spherical coordinates?

    <p>$\frac{1}{r^2} \frac{\partial(r^2 D_r)}{\partial r} + \frac{1}{r \sin(\theta)} \frac{\partial(sin(\theta) D_{\theta})}{\partial \theta} + \frac{1}{r \sin(\theta)} \frac{\partial D_{\phi}}{\partial \phi}$</p> Signup and view all the answers

    How is the Del operator defined in Cartesian coordinates?

    <p>$\nabla = \frac{\partial}{\partial x} \hat{a_x} + \frac{\partial}{\partial y} \hat{a_y} + \frac{\partial}{\partial z} \hat{a_z}$</p> Signup and view all the answers

    What relationship does Gauss’ law illustrate regarding electric displacement and charge?

    <p>The outward flux is proportional to the total charge enclosed.</p> Signup and view all the answers

    Study Notes

    Gauss' Law and Divergence Theorem

    • Electric Flux Lines (ψ): A scalar field where Q coulombs of charges produce ψ (=Q) lines of electric flux. Direct proportionality exists between electric flux and charge (ψ = Q). Unit is coulombs.
    • Electric Flux Density (D): A vector field. Direction of D at a point is the direction of flux lines at that point. Magnitude is the number of flux lines crossing a surface normal to the lines, divided by the surface area. Measured in coulombs per square meter (C/m²) or lines per square meter.
    • Differential Flux: Differential flux (dψ) crossing a differential area (dS) normal to its direction is given by dψ = D * dS * cos θ, where θ is the angle between D and the normal.
    • Gauss' Law: The total electric flux passing through any closed surface equals the total charge enclosed by that surface. Mathematically represented as ∫ D ⋅ dS = Qenc.
    • Special Gaussian Surfaces: To simplify integration, choose a closed surface where D is either normal or tangential to the surface, and D is constant over the relevant portion.
    • Enclosed Charge: This is based on charge distribution. Point charge: Q; Multiple point charges: ∑Qn; Line charge: ∫ρL dL; Surface charge: ∫ρS dS; and Volume charge: ∫ρv dv, where ρL, ρS, ρv are linear, surface, and volume charge densities respectively.
    • Relation Between D and E: A point charge (Q) produces flux lines directed outward. Electric flux density (D) at a point (r) for the flux passing symmetrically through an imaginary spherical surface (area 4πr²) is given by D = Q/(4πr²). In free space, D = ε0E.

    Application of Gauss' Law

    • Point Charge: Gaussian surface is a sphere centered at the charge. D is normal and constant across the surface, allowing D to be factored out of the integral, which yields D = Q/(4πr²).
    • Infinite Line Charge: Gaussian surface is a cylinder, where D is only in the radial direction. This permits the integration, concluding with D = ρL/(2πr), where ⍴L is line charge density and r is the variable radius.
    • Infinite Plane of Sheet: Gaussian surface is a closed cylinder, normal to the plane, with flux only through the top and bottom surfaces. This leads to D = ρs / (2ε0), where ρs is surface charge density and ε0 is permittivity of free space.

    Divergence

    • Differential Volume: The divergence of D at a point is calculated by finding the total flux leaving a small, closed surface around that point per unit volume (as the surface shrinks to zero). The resulting formula is div D = (∂Dx/∂x) + (∂Dy/∂y) + (∂Dz/∂z)(or , div D = ∇·D).
    • Divergence Theorem: The total flux leaving a closed surface is equal to the volume integral of the divergence of D throughout the enclosed volume (∫S D⋅dS = ∫v(∇ ⋅ D)dV).

    Maxwell's First Equation

    • Integral Form: ∫S D ⋅ dS = Qenc, where D is electric flux density, dS is an infinitesimal surface area element, Qenc is the enclosed charge.
    • Point Form: ∇ ⋅ D = ρv, where ρv is the volume charge density.

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    Gauss' Law and Divergence Theorem PDF

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    Test your understanding of Gauss' Law and the Divergence Theorem with this comprehensive quiz. Explore concepts like electric flux, flux density, and the mathematical principles governing these laws. Perfect for students in advanced physics courses.

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