Gauss' Law and Divergence Theorem Quiz

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

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

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

<p>dψ = D ∙ dS (A), dψ = D dS cos θ (C)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The total amount of electric flux passing through a closed surface.

A closed surface used for applying Gauss's Law.

A scalar quantity representing the number of electric flux lines associated with a charge.

A vector quantity representing the density of electric flux lines at a point.

The direction of the electric flux density at a point is the same as the direction of the electric field line at that point.

The relationship between Electric Flux Density (D) and Electric Field Intensity (E) in free space.

The total electric flux passing through a closed surface is equal to the total charge enclosed by that surface.

A Gaussian surface is chosen to simplify the integration in Gauss's Law by aligning its surface to be either normal or tangential to the electric flux density.

The electric flux density (D) is constant over the portion of the surface where D dot dS is non-zero, meaning the electric flux lines are perpendicular to the surface.

The total charge enclosed within a closed surface is equal to the integral of the electric flux density (D) over the entire surface.

The electric field intensity (E) due to a point charge Q at a distance r from the charge is given by E = Q / (4πε₀r²), where ε₀ is the permittivity of free space.

The electric field intensity (E) due to an infinite line charge with uniform charge density ρL at a radial distance ρ from the line charge is given by E = ρL / (2πε₀ρ), where ε₀ is the permittivity of free space.

The electric flux density (D) is constant over a closed surface, allowing for easier calculation of the total charge enclosed.

The integral of the electric flux density (D) over a closed surface is equal to the total charge enclosed within the surface. This is known as Gauss's Law.

The electric field lines are perpendicular to the surface, and the electric field strength is constant over the surface. This simplifies the calculation of the electric flux.

The electric field lines are radial and decrease in strength as the distance from the line charge increases.

The electric field intensity (E) due to an infinite plane of charge with a constant surface charge density (ρs) is directly proportional to the charge density and inversely proportional to the permittivity of free space (εo).

Divergence of a vector field, in this context, represents the outward flux density per unit volume at a point. It measures how much the field 'diverges' or 'spreads out' at that point.

The divergence theorem is a mathematical theorem that relates the divergence of a vector field over a volume to the flux of the field across the surface enclosing the volume. It's particularly useful in electromagnetism to relate the electric field to the enclosed charge.

To calculate the divergence of the electric field at a point, we apply Gauss's Law to a small cuboid enclosing that point. The electric flux through each face of the cuboid is related to the component of the electric field perpendicular to that face, and the divergence is calculated from the sum of these fluxes.

The integral of the dot product of the electric flux density (D) and the differential surface area (dS) over a closed surface. It represents the total electric flux passing through the surface.

The divergence of the electric flux density (D) is defined as the outflow of flux from a small closed surface per unit volume, as the volume approaches zero.

The volume charge density is the amount of charge per unit volume. It represents how densely charge is distributed in space.

A positive divergence indicates a source of electric flux at that point. This implies that more electric flux is flowing out of the point than into it.

A negative divergence indicates a sink of electric flux at that point. This means that more electric flux is flowing into the point than out of it.

A zero divergence indicates a point where there is no net source or sink of electric flux. This implies that the electric flux entering and leaving the point is equal.

The integral of D ⋅ dS over the back face of a small volume element. It can be approximated by the product of the electric flux density (D) on the back face, the area of the back face (ΔyΔz), and the negative change in the x-direction (−Δx).

The divergence of a vector field, 𝐃, represents the outward flux per unit volume at a point in space. It quantifies how much the vector field is expanding or contracting at that point.

The divergence of a vector field 𝐃 in rectangular coordinates is calculated as the sum of partial derivatives of each component of 𝐃 with respect to its corresponding coordinate.

The divergence of a vector field 𝐃 in cylindrical coordinates is calculated using partial derivatives of 𝐃 with respect to radial distance (𝜌), azimuthal angle (𝜙), and the z-axis.

The divergence of a vector field 𝐃 in spherical coordinates is calculated using partial derivatives of 𝐃 with respect to radial distance (r), polar angle (𝜃), and azimuthal angle (𝜙).

The Del operator (𝛁) is a vector operator used in vector calculus, particularly in electromagnetic theory, for operations like gradient, divergence, and curl.

The divergence of a vector field 𝐃 can be expressed using the dot product of the Del operator (𝛁) and 𝐃.