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Calculate D at the point specified if (a)D=(1/z2)[10xyzax+ 5x2zay+(2z35x2y)az] at P(2,3,5);(b)D=5z2aρ+10ρzaz at P(3,45,5);(c)D=2rsinθsinϕar+rcosθsinϕaθ+rcosϕaϕ at P(3,45,45).

Short Answer

Expert verified
Question: Calculate the divergence of the vector field D at the specified points in Cartesian, cylindrical, and spherical coordinates. Solution: a) In Cartesian coordinates, the divergence at point (2,3,5) is D(2,3,5)=320. b) In cylindrical coordinates, the divergence at point (3,45,5) is D(3,45,5)=3403. c) In spherical coordinates, the divergence at point (3,45,45) is D(3,45,45)=6+1+23.

Step by step solution

01

Calculate the divergence of D in Cartesian coordinates#a)

To calculate the divergence of D in Cartesian coordinates, we need to compute the following partial derivatives: x(10xyz), y(5x2z), and z(2z35x2y). We find the following: x(10xyz)=10yz y(5x2z)=5x2 z(2z35x2y)=6z2 Now we compute the cartesian divergence of D as: D=10yz+5x2+6z2
02

Evaluate the cartesian divergence of D at the given point#a)

We find the cartesian divergence of D at the given point P(2,3,5) by substituting the coordinates of the point into the resulting equation from Step 1: D(2,3,5)=10(3)(5)+5(2)2+6(5)2=150+20+150=320
03

Calculate the divergence of D in cylindrical coordinates#b)

In cylindrical coordinates, we have D=5z2aρ+10ρzaz. To compute the divergence, we use: D=1ρρ(ρDρ)+z(Dz) Computing the partial derivatives and applying the formula, we get: D=1ρρ(5ρz2)+z(10ρz) D=1ρ(10z2)+10ρ D=10(z2ρ+ρ)
04

Evaluate the cylindrical divergence of D at the given point#b)

The given point in cylindrical coordinates is P(3,45,5). We convert the angle to radians: 45=π4. Now we substitute the coordinates into the divergence equation from Step 3: D(3,π4,5)=10(253+3)=10(343)=3403
05

Calculate the divergence of D in spherical coordinates#c)

In spherical coordinates, we have D=2rsinθsinϕar+rcosθsinϕaθ+rcosϕaϕ. To compute the divergence, we use: D=1r2r(r2Dr)+1rsinθθ(sinθDθ)+1rsinθϕ(Dϕ) Computing the partial derivatives and applying the formula, we get: D=1r2r(r2(2rsinθsinϕ))+1rsinθθ(sinθ(rcosθsinϕ))+1rsinθϕ(rcosϕ) D=1r2(6r2sinθsinϕ)+1rsinθ(sin2θsinϕ)+1rsinθ(rsinϕ) D=6sinθsinϕ+sinθsinϕsinϕr
06

Evaluate the spherical divergence of D at the given point#c)

The given point in spherical coordinates is P(3,45,45). We convert the angles to radians: 45=π4 and 45=π4. Now we substitute the coordinates into the divergence equation from Step 5: D(3,π4,π4)=6sin(π4)sin(π4)+sin(π4)sin(π4)sin(π4)3 D(3,π4,π4)=6(12)(12)+(12)(12)(12)3 D(3,π4,π4)=6+1+23 So the divergences at the specified points are: a) D(2,3,5)=320 b) D(3,45,5)=3403 c) D(3,45,45)=6+1+23

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Key Concepts

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

Cartesian Coordinate System
In the study of vector calculus, the Cartesian coordinate system is fundamental. It involves three orthogonal axes: the x-axis, y-axis, and z-axis, which allow for representation of points in a three-dimensional space using a set of three coordinates (x,y,z).

In this exercise, we calculate the divergence of a vector field in Cartesian coordinates, which requires the computation of partial derivatives with respect to each of these three axes. For example, finding the divergence at a point involves substituting the specific x, y, and z values into the divergence formula. This system is especially useful because it simplifies to straightforward partial derivatives and simple multiplication when applying the divergence operator.
Cylindrical Coordinate System
When dealing with symmetries like those found in cylinders, the cylindrical coordinate system is more beneficial. It expresses a point in space with three coordinates (ρ,ϕ,z), where ρ is the radial distance from the origin, ϕ is the azimuthal angle about the z-axis, and z is the same as in the Cartesian system.

In this system, to calculate divergence, we need to know how to compute the derivatives with respect to ρ and z and understand how to adjust for the coordinate system's radial nature. This is evident in the exercise where specific formulas for divergence in cylindrical coordinates are employed, which take into account these nuances to evaluate the vector field.
Spherical Coordinate System
For problems involving spherical symmetry, the spherical coordinate system is extremely effective. Here, a point in space is described by three coordinates (r,θ,ϕ), where r is the radial distance from the origin, θ is the polar angle from the z-axis, and ϕ is the azimuthal angle in the x-y plane.

The divergence in spherical coordinates requires an understanding of this system's complexity, as seen in the provided exercise. The divergence operator in spherical coordinates includes derivatives with respect to each of these coordinates, and it requires specific multiplicative factors, such as r2 and rsinθ to account for the changing volume element sizes in spherical coordinates.
Scalar Field Divergence
The divergence of a scalar field is not conventionally defined because divergence, by definition, applies to vector fields. However, the concept of divergence can be thought of as a measure of the rate at which the density of a physical quantity, represented by the vector field, changes in space.

In our exercise, divergence is calculated for a vector field across different coordinate systems. This divergence helps us understand how much the vector field is 'spreading out' from a point, which would be analogous to the spreading of a scalar quantity like temperature or pressure if they were modeled as vector fields.
Vector Calculus
Vector calculus is a mathematical discipline that deals with the differentiation and integration of vector fields, often in two or three-dimensional Euclidean space. Key operations in vector calculus include gradient, divergence, and curl.

In the context of this exercise, the focus is on the divergence operator, denoted as abla, which measures the magnitude of a source or sink at a given point in a vector field. This is a crucial concept for understanding physical phenomena such as the flow of fluids or the electromagnetic field generated by charges.
Partial Derivatives
Partial derivatives represent the rate at which a function changes as one variable changes, whilst keeping the other variables constant. In the context of the Cartesian coordinate system, we compute partial derivatives with respect to x and y for a vector field D as shown in the exercise steps.

Determining the partial derivatives is a necessary step in calculating divergence in any coordinate system because it allows us to measure how the vector field varies in each direction around a point. It’s also essential in both cylindrical and spherical coordinates, although the variables with respect to which we differentiate change according to the system used.

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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 ?

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?

Use Gauss's law in integral form to show that an inverse distance field in spherical coordinates, D=Aar/r, where A is a constant, requires every spherical shell of 1 m thickness to contain 4πA coulombs of charge. Does this indicate a continuous charge distribution? If so, find the charge density variation with r.

An infinitely long cylindrical dielectric of radius b contains charge within its volume of density ρv=aρ2, where a is a constant. Find the electric field strength, E, both inside and outside the cylinder.

(a) Use Maxwell's first equation, D=ρv, to describe the variation of the electric field intensity with x in a region in which no charge density exists and in which a nonhomogeneous dielectric has a permittivity that increases exponentially with x. The field has an x component only; (b) repeat part (a), but with a radially directed electric field (spherical coordinates), in which again ρv=0, but in which the permittivity decreases exponentially with r.

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