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Compute the charge enclosed by a cube of 2m each edge centered at the origin and with the edges parallel to the axes. Given D = 10y3 /3 j
To compute the charge enclosed by the cube using the given electric flux density ( mathbf{D} = frac{10y^3}{3} hat{mathbf{j}} ) C/m(^2), we need to apply Gauss's Law in integral form, which relates the electric flux through a closed surface to the charge enclosed by that surface. However, because we'Read more
To compute the charge enclosed by the cube using the given electric flux density ( mathbf{D} = frac{10y^3}{3} hat{mathbf{j}} ) C/m(^2), we need to apply Gauss’s Law in integral form, which relates the electric flux through a closed surface to the charge enclosed by that surface. However, because we’re dealing with the electric flux density (mathbf{D}) directly, we can integrate (mathbf{D}) over the surface of the cube to find the total charge enclosed without explicitly invoking Gauss’s Law.
Given that the cube has its edges parallel to the axes and is centered at the origin with edge length 2 m, it extends from (-1) m to (1) m along the x, y, and z axes.
Since (mathbf{D}) only has a y-component ((frac{10y^3}{3} hat{mathbf{j}})), the flux through the cube will only occur through the faces perpendicular to the y-axis, i.e., the faces at (-1) m and (1) m in the y-direction. The areas of the faces through which (mathbf{D}) passes are parallel to the xz-plane.
The total charge enclosed ((Q_{text{enc}})) by the cube can be obtained by integrating the normal component of (math
See lessCompute the Gauss law for D = 10ρ3 /4 i, in cylindrical coordinates with ρ = 4m, z = 0 and z = 5, hence find charge using volume integral
To compute Gauss's law for a given electric displacement field ( mathbf{D} ) and to find the charge enclosed using a volume integral, we work step by step through the problem. Gauss's law in differential form relates the divergence of the electric displacement field ( mathbf{D} ) to the free chargeRead more
To compute Gauss’s law for a given electric displacement field ( mathbf{D} ) and to find the charge enclosed using a volume integral, we work step by step through the problem. Gauss’s law in differential form relates the divergence of the electric displacement field ( mathbf{D} ) to the free charge density ( rho_{free} ) present in the medium:
[
nabla cdot mathbf{D} = rho_{free}
]
Given:
[
mathbf{D} = frac{10rho^3}{4} hat{mathbf{i}}
]
This is given in cylindrical coordinates ((rho, phi, z)) but with a slight confusion in the notation since (hat{mathbf{i}}) is typically used for Cartesian coordinates. Assuming it’s meant to represent the radial component in cylindrical coordinates, it should correctly be (hat{rho}) instead of (hat{mathbf{i}}), so:
[
mathbf{D} = frac{10rho^3}{4} hat{rho}
]
To compute the charge enclosed within a cylindrical volume defined by ( rho = 4m ), between ( z = 0 ) and ( z = 5 ), we first compute the volume integral of the charge density ( rho_{free} ).
Since ( nabla
See lessCompute divergence theorem for D = 5r2 /4 i in spherical coordinates between r = 1 and r = 2 in volume integra
To compute the divergence of a vector field using the divergence theorem in spherical coordinates, we follow a systematic approach. Given a vector field (mathbf{D} = frac{5r^2}{4} hat{i}) in a coordinate system, we first need to express this field in spherical coordinates and then find its divergencRead more
To compute the divergence of a vector field using the divergence theorem in spherical coordinates, we follow a systematic approach. Given a vector field (mathbf{D} = frac{5r^2}{4} hat{i}) in a coordinate system, we first need to express this field in spherical coordinates and then find its divergence. However, a direct conversion of the given vector field into spherical coordinates poses a challenge since the field is given in a form that suggests it’s already partially in a non-Cartesian form ((frac{5r^2}{4} hat{i}) suggests a dependence on radial distance but uses (hat{i}), which is a Cartesian unit vector). Assuming the intention is to deal with a radially dependent vector field in a spherical context, we can reinterpret the vector field in spherical coordinates, focusing on its radial component only.
Spherical Coordinates Background
In spherical coordinates, a position in space is given by (r) (radial distance), (theta) (polar angle, measured from the positive z-axis), and (phi) (azimuthal angle, measured in the x-y plane from the positive x-axis). Vector fields in spherical coordinates are expressed in terms of these variables and their unit vectors (hat{r}), (hat{theta}), and (hat{phi}).
Given Vector Field
Given the ambiguities in the initial presentation of the vector field, but
See lessIf D = 2xy i + 3yz j + 4xz k, how much flux passes through x = 3 plane for which -1<y<2 and 0<z<4?
To solve this, we'll use the concept of flux through a surface. The flux (Phi) of a vector field (textbf{D} = Ptextbf{i} + Qtextbf{j} + Rtextbf{k}) through a surface (S) is given by the surface integral of (textbf{D} cdot textbf{n} dS), where (textbf{n}) is the unit normal to the surface and (dS) isRead more
To solve this, we’ll use the concept of flux through a surface. The flux (Phi) of a vector field (textbf{D} = Ptextbf{i} + Qtextbf{j} + Rtextbf{k}) through a surface (S) is given by the surface integral of (textbf{D} cdot textbf{n} dS), where (textbf{n}) is the unit normal to the surface and (dS) is a differential element of the surface area.
Given (textbf{D} = 2xy textbf{i} + 3yz textbf{j} + 4xz textbf{k}) and considering the plane (x = 3) with (-1 < y < 2) and (0 < z < 4), we'll calculate the flux through this plane area.
For the plane (x = 3), the normal vector is parallel to the (textbf{i}) direction since the plane is perpendicular to the x-axis. Therefore, only the component of (textbf{D}) in the direction of (textbf{i}) contributes to the flux through this plane.
The relevant component of (textbf{D}) here is (P = 2xy), and since (x = 3), we have (P = 6y). The
See lessFind 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 less