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Chapter 2: Problem 30

For fields that do not vary with \(z\) in cylindrical coordinates, the equations of the streamlines are obtained by solving the differential equation \(E_{p} / E_{\phi}=\) \(d \rho /(\rho d \phi)\). Find the equation of the line passing through the point \(\left(2,30^{\circ}, 0\right)\) for the field \(\mathbf{E}=\rho \cos 2 \phi \mathbf{a}_{\rho}-\rho \sin 2 \phi \mathbf{a}_{\phi}\).

Short Answer

Expert verified
In the given cylindrical coordinate system, the equation of the streamline passing through the point (2, 30°, 0) for the given field is ρ^2 = cos(2φ) + 1.

Step by step solution

01

Calculate the ratio Eρ / Eφ

First, we need to find the ratio of Eρ and Eφ components of the given field. The field components are: Eρ = ρcos(2φ) Eφ = -ρsin(2φ) So the ratio is: Eρ / Eφ = (ρcos(2φ)) / (-ρsin(2φ)) Since the field is not depending on 'z', we can cancel the 'ρ' term both in numerator and denominator: Eρ / Eφ = cos(2φ) / (-sin(2φ)) Next, we need to use this ratio to find the streamlines equation.
02

Solve the differential equation

We have the following differential equation: Eρ / Eφ = dρ / (ρdφ) Replacing the ratio Eρ / Eφ found in step 1: cos(2φ) / (-sin(2φ)) = dρ / (ρdφ) To solve this differential equation, separate the variables. Move the ρ-related terms to the left, and φ-related terms to the right. ρdρ = -cos(2φ)dφ / sin(2φ) Integrate both sides: ∫ρdρ = -∫cos(2φ)dφ / sin(2φ) Let's assume c is the constant of integration. (ρ^2) / 2 + c = 1/2 ∫ -d (cos(2φ)) Thus we have: ρ^2 = cos(2φ) + C
03

Find the constant of integration (C) using the given point

We are given the point (ρ, Φ) = (2, 30°). Use this point to find out the value of the constant, C. ρ = 2 φ = 30° = π/6 Plug these values into the equation: (2^2) = cos(2 * π/6) + C Solving for C, we get: C = 1 Now we have found the constant of integration, and our streamline equation is: ρ^2 = cos(2φ) + 1 Therefore, the equation of the line passing through the point (2,30°,0) for the given field is ρ^2 = cos(2φ) + 1.

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Most popular questions from this chapter

Point charges of \(50 \mathrm{nC}\) each are located at \(A(1,0,0), B(-1,0,0), C(0,1,0)\), and \(D(0,-1,0)\) in free space. Find the total force on the charge at \(A\).

A \(100-n C\) point charge is located at \(A(-1,1,3)\) in free space. \((a)\) Find the locus of all points \(P(x, y, z)\) at which \(E_{x}=500 \mathrm{~V} / \mathrm{m} \cdot(b)\) Find \(y_{1}\) if \(P\left(-2, y_{1}, 3\right)\) lies on that locus.

The electron beam in a certain cathode ray tube possesses cylindrical symmetry, and the charge density is represented by \(\rho_{v}=-0.1 /\left(\rho^{2}+10^{-8}\right)\) \(\mathrm{pC} / \mathrm{m}^{3}\) for \(0<\rho<3 \times 10^{-4} \mathrm{~m}\), and \(\rho_{v}=0\) for \(\rho>3 \times 10^{-4} \mathrm{~m} .(a)\) Find the total charge per meter along the length of the beam; \((b)\) if the electron velocity is \(5 \times 10^{7} \mathrm{~m} / \mathrm{s}\), and with one ampere defined as \(1 \mathrm{C} / \mathrm{s}\), find the beam current.

An electric dipole (discussed in detail in Section 4.7) consists of two point charges of equal and opposite magnitude \(\pm Q\) spaced by distance \(d\). With the charges along the \(z\) axis at positions \(z=\pm d / 2\) (with the positive charge at the positive \(z\) location), the electric field in spherical coordinates is given by \(\mathbf{E}(r, \theta)=\left[Q d /\left(4 \pi \epsilon_{0} r^{3}\right)\right]\left[2 \cos \theta \mathbf{a}_{r}+\sin \theta \mathbf{a}_{\theta}\right]\), where \(r>>d\). Using rectangular coordinates, determine expressions for the vector force on a point charge of magnitude \(q(a)\) at \((0,0, z) ;(b)\) at \((0, y, 0)\).

Two identical uniform sheet charges with \(\rho_{s}=100 \mathrm{nC} / \mathrm{m}^{2}\) are located in free space at \(z=\pm 2.0 \mathrm{~cm}\). What force per unit area does each sheet exert on the other?
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