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A cube is defined by \(1

Short Answer

Expert verified
In summary: a) The total flux leaving the closed surface of the cube is 2.4. b) The divergence of D at the center of the cube is (4.84, 8.476). c) The total charge enclosed within the cube is approximately 0.0216 C.

Step by step solution

01

(a) Apply Gauss's law to find the total flux leaving the closed surface of the cube

To find the total flux leaving the closed surface of the cube, we need to integrate the field D over the six faces of the cube. Let's first consider the four faces perpendicular to the x and y axes: i. Face with x = 1: 11.211.2(2(1)2y)dydz=411.211.2ydydz ii. Face with x = 1.2: 11.211.2(2(1.2)2y)dydz=6.91211.211.2ydydz iii. Face with y = 1: 11.211.2(3x2(1)2)dxdz=311.211.2x2dxdz iv. Face with y = 1.2: 11.211.2(3x2(1.2)2)dxdz=4.3211.211.2x2dxdz The electric displacement field has no component in the z direction, so there will be no flux through the faces with z = 1 and z = 1.2. Now, the total flux through the surface is the sum of the fluxes through each face: Φ=11.211.2(4+6.912+3x2+4.32x2)dydz Evaluating this integral, we get: Φ=2.4
02

(b) Evaluate the divergence of D at the center of the cube

To evaluate the divergence of D at the center of the cube, we first need to calculate the divergence of D: D=(x(2x2y),y(3x2y2))=(4xy,6x2y) The coordinates of the center of the cube are (1.1,1.1,1.1). Substituting these values into the divergence of D: D|(1.1,1.1,1.1)=(4(1.1)(1.1),6(1.1)2(1.1))=(4.84,8.476)
03

(c) Estimate the total charge enclosed within the cube using Equation (8)

To estimate the total charge enclosed within the cube, we need to integrate the divergence of D over the volume of the cube: Q=11.211.211.2DdV Q=11.211.211.2(4xy+6x2y)dydxdz Evaluating this integral, we get: Q=0.0216C So the total charge enclosed within the cube is approximately 0.0216 C.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electrical Flux
Electrical flux, a cornerstone of electromagnetics, refers to the amount of electric field that passes through a given area. It is symbolically represented as \text\textGreek\textPhi_{E}\text and can be vividly imagined as the number of electric field lines penetrating a surface. In mathematical terms, the electric flux through a surface S is defined by the integral of the electric field E over this surface, represented as \textGreek\textPhiE=\textintolimitsSE\textcdotdA, where dA specifies an infinitesimal area on the surface with a direction normal to the surface.

In the context of Gauss's Law, this concept is essential as it equates the net flux through a closed surface to the charge enclosed by the surface, divided by the permittivity of the space. For our cube problem, calculating the flux required integrating the given electric displacement field D over the six faces of the cube, considering the direction and magnitude of the field on each face.
Divergence of Electric Field
Divergence in electromagnetics is a measure of the net 'outflow' of an electric field from an infinitesimal volume at a given point in space. Mathematically, it is denoted as abla\textcdotE for the electric field E. In essence, it tells us whether a particular region behaves like a 'source' or a 'sink' of the electric field.

The divergence of the field is a key factor in Gauss's Law for electricity, which is fundamentally linked to charge distribution. In our cube exercise, the divergence was calculated at a specific point - the center of the cube. After finding the partial derivatives of the electric displacement field D concerning each coordinate, these values were substituted to get the divergence at that central point. This step is vital for understanding how the electric field behaves at different points in space and is used in determining the charge density within a particular region.
Enclosed Charge Estimation
The estimation of the enclosed charge in a volume is an important application of electromagnetics that follows from Gauss's Law. It states that the total charge enclosed by a closed surface can be found by integrating the divergence of the electric field throughout the volume enclosed by the surface, or in the case of a given electric displacement field D, by Q=\textintabla\textcdotDdV.

For our cube problem, the process involved setting up a triple integral of the divergence of D over the volume of the cube to estimate the total charge inside it. The result of such an integral provides us with a numerical value of charge, influencing the electric and magnetic behavior of the system. Significantly, this concept helps in understanding how charges distribute in space and affect the surrounding electric field.
Electromagnetic Theory
Electromagnetic theory encompasses the study of electric and magnetic fields and their interactions with matter. At the heart of this theory are Maxwell's equations, which describe how electric charges and currents produce electric and magnetic fields, and conversely, how these fields interact with charges and currents.

One of Maxwell's equations is Gauss's Law, the focal point of our cube exercise. This law fundamentally relates electric fields, charge distributions, and the concept of an electric flux. Whether you are calculating the flux through a surface, evaluating the divergence of an electric field, or estimating the enclosed charge within a volume, you are essentially applying principles of electromagnetic theory to solve real-world problems. This theory is not only foundational for understanding the cube exercise but is also the backbone of modern electrical engineering, informing everything from telecommunications to power generation systems.

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Most popular questions from this chapter

A certain light-emitting diode (LED) is centered at the origin with its surface in the xy plane. At far distances, the LED appears as a point, but the glowing surface geometry produces a far-field radiation pattern that follows a raised cosine law: that is, the optical power (flux) density in watts /m2 is given in spherical coordinates by Pd=P0cos2θ2πr2ar watts /m2 where θ is the angle measured with respect to the direction that is normal to the LED surface (in this case, the z axis), and r is the radial distance from the origin at which the power is detected. (a) In terms of P0, find the total power in watts emitted in the upper half-space by the LED; (b) Find the cone angle, θ1, within which half the total power is radiated, that is, within the range 0<θ<θ1; ( c ) An optical detector, having a 1mm2 cross-sectional area, is positioned at r=1 m and at θ=45, such that it faces the LED. If one milliwatt is measured by the detector, what (to a very good estimate) is the value of P0 ?

Volume charge density is located as follows: ρv=0 for ρ<1 mm and for ρ>2 mm,ρv=4ρμC/m3 for 1<ρ<2 mm. (a) Calculate the total charge in the region \(0<\rho<\rho_{1}, 0

Given the flux density D=16rcos(2θ)aθC/m2, use two different methods to find the total charge within the region \(1

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The sun radiates a total power of about 3.86×1026 watts (W). If we imagine the sun's surface to be marked off in latitude and longitude and assume uniform radiation, (a) what power is radiated by the region lying between latitude 50N and 60N and longitude 12W and 27W?(b) What is the power density on a spherical surface 93,000,000 miles from the sun in W/m2?

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