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Find the value of divergence theorem for the field D = 2xy i + x2 j for the rectangular parallelepiped given by x = 0 and 1, y = 0 and 2, z = 0 and 3.
To find the value of the divergence theorem for the given vector field ( mathbf{D} = 2xy mathbf{i} + x^2 mathbf{j} ), over the rectangular parallelepiped bounded by (x = 0) and (1), (y = 0) and (2), (z = 0) and (3), we first need to understand and apply the divergence theorem itself. The divergenceRead more
To find the value of the divergence theorem for the given vector field ( mathbf{D} = 2xy mathbf{i} + x^2 mathbf{j} ), over the rectangular parallelepiped bounded by (x = 0) and (1), (y = 0) and (2), (z = 0) and (3), we first need to understand and apply the divergence theorem itself. The divergence theorem relates the flow (flux) of a vector field through a closed surface to the divergence of the field in the volume enclosed by the surface. Mathematically, it is represented as:
[ intintint_V (nabla cdot mathbf{D}) dV = intint_S mathbf{D} cdot mathbf{n} dS ]
Where:
– (V) is the volume inside the surface (S),
– (mathbf{D}) is the vector field,
– (nabla cdot mathbf{D}) represents the divergence of (mathbf{D}),
– (dV) is a volume element, and
– (dS) is an element of the surface area with (mathbf{n}) being the outward facing normal.
Given the vector field (mathbf{D} = 2xy mathbf{i} + x^2 mathbf
See lessFind the value of divergence theorem for A = xy2 i + y3 j + y2z k for a cuboid given by 0<x<1, 0<y<1 and 0<z<1
To find the value of the divergence of vector field ( mathbf{A} = xy^2 hat{i} + y^3 hat{j} + y^2z hat{k} ) over the cuboid defined by the intervals (0 < x < 1), (0 < y < 1), and (0 < z < 1) using the divergence theorem, we first need to compute the divergence of ( mathbf{A} ).The divergence of a vecRead more
To find the value of the divergence of vector field ( mathbf{A} = xy^2 hat{i} + y^3 hat{j} + y^2z hat{k} ) over the cuboid defined by the intervals (0 < x < 1), (0 < y < 1), and (0 < z < 1) using the divergence theorem, we first need to compute the divergence of ( mathbf{A} ).
The divergence of a vector field ( mathbf{A} = Phat{i} + Qhat{j} + Rhat{k} ) is given by:
[ nabla cdot mathbf{A} = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z} ]
For ( mathbf{A} = xy^2 hat{i} + y^3 hat{j} + y^2z hat{k} ),
[ P = xy^2, Q = y^3, R = y^2z ]
Computing the partial derivatives,
[ frac{partial P}{partial x} = y^2 ]
[ frac{partial Q}{partial y} = 3y^2 ]
[ frac{partial R}{partial z} = y^2
See lessCompute divergence theorem for D= 5r2 /4 i in spherical coordinates between r=1 and r=2
To compute the divergence theorem for the given vector field ( textbf{D} = frac{5r^2}{4} hat{i} ) in spherical coordinates between ( r=1 ) and ( r=2 ), we first need to express the vector field in spherical coordinates and then apply the divergence theorem accordingly.### Step 1: Convert to SphericaRead more
To compute the divergence theorem for the given vector field ( textbf{D} = frac{5r^2}{4} hat{i} ) in spherical coordinates between ( r=1 ) and ( r=2 ), we first need to express the vector field in spherical coordinates and then apply the divergence theorem accordingly.
### Step 1: Convert to Spherical Coordinates
The problem presents a vector field in presumably Cartesian coordinates (given the use of ( hat{i} ), typically representing the unit vector in the x-direction in Cartesian coordinates). In spherical coordinates, positions are given by ( (r, theta, phi) ), where:
– (r) is the radial distance from the origin,
– (theta) is the polar angle measured from the z-axis,
– (phi) is the azimuthal angle in the xy-plane from the x-axis.
To convert ( frac{5r^2}{4} hat{i} ) into spherical coordinates, we acknowledge that in spherical coordinates, the Cartesian (x) component relates to (r) as (x = r sin(theta) cos(phi)). However, the vector field provided does not directly correlate with standard spherical components since it’s prescribed in the ( hat{i} ) direction. Therefore, we’re a bit at an impasse regarding conventions; the given description suggests a simplification or a misunderstanding in the application of the vector field
See lessCompute the Gauss law for D= 10ρ3 /4 i, in cylindrical coordinates with ρ= 4m, z=0 and z=5
To compute Gauss's law using the given electric flux density, (D = frac{10rho^3}{4} hat{i}) in cylindrical coordinates, we follow the integral form of Gauss's law for electric fields, which states:[oint_S mathbf{D} cdot dmathbf{A} = Q_{text{enc}}]Where:- (oint_S mathbf{D} cdot dmathbf{A}) is the eleRead more
To compute Gauss’s law using the given electric flux density, (D = frac{10rho^3}{4} hat{i}) in cylindrical coordinates, we follow the integral form of Gauss’s law for electric fields, which states:
[
oint_S mathbf{D} cdot dmathbf{A} = Q_{text{enc}}
]
Where:
– (oint_S mathbf{D} cdot dmathbf{A}) is the electric flux through a closed surface (S),
– (Q_{text{enc}}) is the total electric charge enclosed by the surface.
Given that (D) is in the direction of (hat{i}), which implies it’s purely radial in cylindrical coordinates, this simplifies the problem since we only need to consider the component of (mathbf{D}) that is normal to the surface (S). For a cylindrical surface with radius (rho) and height from (z=0) to (z=5), the relevant surface area for the flux calculation is the curved surface area, because the electric flux density is radially outward and will not penetrate the top and bottom surfaces normally.
The curved surface area of a cylinder is (A = 2pirho h), where (rho) is the radius of the cylinder and (h) is the height. For (rho = 4
See lessEvaluate Gauss law for D = 5r2 /4 i in spherical coordinates with r = 4m and θ = π/2
To evaluate Gauss's Law using the provided electric flux density (D = frac{5r^2}{4} hat{i}) in spherical coordinates for (r = 4 ,m) and (theta = frac{pi}{2}), let's first clarify the given parameters and the application of Gauss's Law within this context.First, it's important to note that the electrRead more
To evaluate Gauss’s Law using the provided electric flux density (D = frac{5r^2}{4} hat{i}) in spherical coordinates for (r = 4 ,m) and (theta = frac{pi}{2}), let’s first clarify the given parameters and the application of Gauss’s Law within this context.
First, it’s important to note that the electric flux density (mathbf{D}) given as (D = frac{5r^2}{4} hat{i}) seems to be expressed in a form that mixes variables from Cartesian coordinates ((hat{i}) is a unit vector in the direction of the x-axis in Cartesian coordinates) with those of spherical coordinates ((r) is the radial distance in spherical coordinates). For a more consistent application related to Gauss’s Law in spherical coordinates, it’s essential to correctly interpret or transform this expression into spherical coordinates fully. However, assuming the intent was to express (mathbf{D}) in a purely spherical context and aiming to perform an evaluation as requested, one needs to correct the misunderstanding about the vector notations and how Gauss’s Law applies.
—
Given the context, let’s reframe the statement into a scenario Gauss’s Law could be applied to directly:
If by (D = frac{5r^2}{4}) you meant the magnitude of the electric flux density in spherical coordinates without specifying a direction, and considering (
See lessA field in which a test charge around any closed surface in static path is zero is called
A field in which the net electric flux through any closed surface is zero is characteristic of an electrostatic field in equilibrium. Specifically, this principle is encapsulated by Gauss's Law in electrostatics, which states that the total electric flux through a closed surface is proportional to tRead more
A field in which the net electric flux through any closed surface is zero is characteristic of an electrostatic field in equilibrium. Specifically, this principle is encapsulated by Gauss’s Law in electrostatics, which states that the total electric flux through a closed surface is proportional to the charge enclosed by the surface. Therefore, if a test charge moves around any closed path in such a field and the net work done is zero, it implies that the electric field is conservative. However, the precise terminology you’re looking for, which describes the scenario where the net work done on a test charge over a closed path is zero, refers to a conservative electric field. Yet, your question hints at the concept encapsulated by Gauss’s Law for electrostatics, where the net electric flux out of any closed surface is proportional to the charge enclosed by the surface. If there’s no charge within the closed surface or if the positive and negative charges inside cancel out, the net electric flux would be zero, indicating an electrostatic equilibrium condition within that closed surface.
See lessdentify the nature of the field, if the divergence is zero and curl is also zero
If a vector field has both zero divergence and zero curl, this means it is a conservative field or a potential field. In physics, this typically represents a situation where there are no sources or sinks in the field (zero divergence) and no circular or rotational motion within the field (zero curl)Read more
If a vector field has both zero divergence and zero curl, this means it is a conservative field or a potential field. In physics, this typically represents a situation where there are no sources or sinks in the field (zero divergence) and no circular or rotational motion within the field (zero curl). An example of this would be the electric field surrounding a set of electric charges in static equilibrium. In such a field, the work done to move a test charge from one point to another is independent of the path taken between the two points, and there is a scalar potential function from which the field can be derived.
See less