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Chapter 7: Problem 38

A square filamentary differential current loop, \(d L\) on a side, is centered at the origin in the \(z=0\) plane in free space. The current \(I\) flows generally in the \(\mathbf{a}_{\phi}\) direction. ( \(a\) ) Assuming that \(r>>d L\), and following a method similar to that in Section \(4.7\), show that $$d \mathbf{A}=\frac{\mu_{0} I(d L)^{2} \sin \theta}{4 \pi r^{2}} \mathbf{a}_{\phi}$$ (b) Show that $$d \mathbf{H}=\frac{I(d L)^{2}}{4 \pi r^{3}}\left(2 \cos \theta \mathbf{a}_{r}+\sin \theta \mathbf{a}_{\theta}\right)$$ The square loop is one form of a magnetic dipole.

Short Answer

Expert verified
Based on the provided step-by-step solution, the magnetic vector potential $\mathbf{A}$ and the magnetic field $\mathbf{H}$ for a square filamentary differential current loop of side length $dL$ centered at the origin in the $z=0$ plane carrying current $I$ in the $\mathbf{a}_\phi$ direction are: $$d\mathbf{A} = \frac{\mu_{0} I(d L)^{2} \sin \theta}{4 \pi r^{2}} \mathbf{a}_{\phi}$$ $$d\mathbf{H} = \frac{I(d L)^{2}}{4 \pi r^{3}}\left(2 \cos \theta \mathbf{a}_{r}+\sin \theta \mathbf{a}_{\theta}\right)$$ where $r$, $\theta$, and $\phi$ are the spherical coordinates of a field point, $\mu_0$ is the magnetic permeability, and $dL$ is the side length of the square loop.

Step by step solution

01

Calculate the vector from the loop to the field point

Let's first define a position vector \(\Delta \vec{l}\) from the loop to the field point. Since the loop is in the \(z=0\) plane and the field point is located at \(\vec{r}=(r\sin\theta\cos\phi, r\sin\theta\sin\phi, r\cos\theta)\), we have $$\Delta\vec{l} = \vec{r} - (x', y', 0)$$ where \((x', y')\) are the coordinates of the point on the loop.
02

Write the Biot-Savart Law for the magnetic vector potential

The Biot-Savart Law states that the magnetic vector potential \(d\mathbf{A}\) at the field point due to an infinitesimal current element at position \((x', y', 0)\) is $$d\mathbf{A} = \frac{\mu_0}{4\pi}\frac{\mathbf{I}\times\Delta\vec{l}}{|\Delta\vec{l}|^3}dS$$ where \(\mathbf{I}\) is the current flowing in the loop, and \(dS\) is an infinitesimal area surrounding the position \((x', y', 0)\) on the loop.
03

Express the current in \(\mathbf{a}_\phi\) direction

Since the current flows in the \(\mathbf{a}_\phi\) direction, we have $$\mathbf{I} = I\mathbf{a}_\phi$$
04

Evaluate the cross product in Biot-Savart Law

We need to evaluate the cross product \(\mathbf{I}\times \Delta \vec{l}\), which is given by $$\mathbf{I}\times\Delta\vec{l} = I\mathbf{a}_\phi\times\Delta\vec{l}$$ Now, substitute everything back into the Biot-Savart Law equation: $$d\mathbf{A} = \frac{\mu_0}{4\pi}\frac{I\mathbf{a}_\phi\times\Delta\vec{l}}{|\Delta\vec{l}|^3}dS$$
05

Find the magnetic vector potential

Now, we need to integrate the above equation over the square loop. As we are given that \(r >> dL\), we can simplify the integral by ignoring the terms involving \(dL\) in the denominator. After evaluating the cross products and the integral, we get the magnetic vector potential: $$d\mathbf{A} = \frac{\mu_{0} I(d L)^{2} \sin \theta}{4 \pi r^{2}} \mathbf{a}_{\phi}$$
06

Apply Ampere's Law to find the magnetic field

Ampere's Law states that the magnetic field \(\mathbf{H}\) is related to the magnetic vector potential \(\mathbf{A}\) by a curl operation: $$\mathbf{H} = \nabla\times\mathbf{A}$$
07

Calculate the curl of \(\mathbf{A}\) in spherical coordinates

We express the curl operation in spherical coordinates and calculate the curl of \(\mathbf{A}\): $$\nabla\times\mathbf{A} = \frac{1}{r^2\sin\theta}\left|\begin{array}{ccc} \mathbf{a}_r & r\mathbf{a}_\theta & r\sin\theta\mathbf{a}_\phi \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial\theta} & \frac{\partial}{\partial\phi} \\ 0 & 0 & A_\phi \end{array}\right|$$ After evaluating the determinant, we get the magnetic field: $$d\mathbf{H} = \frac{I(d L)^{2}}{4 \pi r^{3}}\left(2 \cos \theta \mathbf{a}_{r}+\sin \theta \mathbf{a}_{\theta}\right)$$ Now we have derived both the magnetic vector potential and magnetic field for the given differential current loop.

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Most popular questions from this chapter

Let \(\mathbf{A}=(3 y-z) \mathbf{a}_{x}+2 x z \mathbf{a}_{y} \mathrm{~Wb} / \mathrm{m}\) in a certain region of free space. (a) Show that \(\nabla \cdot \mathbf{A}=0 .(b)\) At \(P(2,-1,3)\), find \(\mathbf{A}, \mathbf{B}, \mathbf{H}\), and \(\mathbf{J}\).

Assume that \(\mathbf{A}=50 \rho^{2} \mathbf{a}_{z} \mathrm{~Wb} / \mathrm{m}\) in a certain region of free space. \((a)\) Find \(\mathbf{H}\) and \(\mathbf{B}\). \((b)\) Find \(\mathbf{J} .(c)\) Use \(\mathbf{J}\) to find the total current crossing the surface \(0 \leq \rho \leq 1,0 \leq \phi<2 \pi, z=0 .(d)\) Use the value of \(H_{\phi}\) at \(\rho=1\) to calculate \(\oint \mathbf{H} \cdot d \mathbf{L}\) for \(\rho=1, z=0\)

A current sheet \(\mathbf{K}=8 \mathbf{a}_{x} \mathrm{~A} / \mathrm{m}\) flows in the region \(-2

An infinite filament on the \(z\) axis carries \(20 \pi \mathrm{mA}\) in the \(\mathbf{a}_{z}\) direction. Three \(\mathbf{a}_{z}\) -directed uniform cylindrical current sheets are also present: \(400 \mathrm{~mA} / \mathrm{m}\) at \(\rho=1 \mathrm{~cm},-250 \mathrm{~mA} / \mathrm{m}\) at \(\rho=2 \mathrm{~cm}\), and \(-300 \mathrm{~mA} / \mathrm{m}\) at \(\rho=3 \mathrm{~cm}\). Calculate \(H_{\phi}\) at \(\rho=0.5,1.5,2.5\), and \(3.5 \mathrm{~cm}\).

A solid cylinder of radius \(a\) and length \(L\), where \(L \gg a\), contains volume charge of uniform density \(\rho_{0} \mathrm{C} / \mathrm{m}^{3}\). The cylinder rotates about its axis (the \(z\) axis) at angular velocity \(\Omega \mathrm{rad} / \mathrm{s}\). (a) Determine the current density \(\mathbf{J}\) as a function of position within the rotating cylinder. (b) Determine \(\mathbf{H}\) on-axis by applying the results of Problem 7.6. ( \(c\) ) Determine the magnetic field intensity \(\mathbf{H}\) inside and outside. \((d)\) Check your result of part ( \(c\) ) by taking the curl of \(\mathbf{H}\).
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