Question

Question: In Figure, current $\mathit{I}\mathbf{=}\mathbf{56}\mathbf{.}\mathbf{2}\mathbf{}\mathbf{mA}$is set up in a loop having two radial lengths and two semicircles of radii$\mathit{a}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{72}\mathbf{}\mathbf{cm}$ and$\mathit{b}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{36}\mathbf{}\mathbf{cm}$ with a common center$\mathit{P}$(a) What are the magnitude and (b) What are the direction (into or out of the page) of the magnetic field at P and the (c) What is the magnitude of the loop’s magnetic dipole moment? and (d) What is the direction of the loop’s magnetic dipole moment?

Short answer

1. Magnitude of magnetic field at point P is $4.97\times {10}^{-7}\mathrm{T}$.
2. Direction of magnetic field at point P is into the page.
3. Magnitude of the loop’s magnetic dipole moment is $1.06\times {10}^{-3}{\mathrm{Am}}^2$.
4. Direction of dipole moment is into the page.

Step-by-step solution

Identification of given data

1. Current is$56.2\mathrm{mA}$
2. Radius$a=5.72\mathrm{cm}$
3. Radius is$b=9.36\mathrm{cm}$

Understanding the concept of magnetic field for semicircle

Semicircle wire generates a magnetic field that is half that of circle wire. We use the formula for magnetic field for semicircle, and the direction is given by right hand rule.

Formula:

$\mathbf{\%}\mathbf{}\mathit{M}\mathit{a}\mathit{t}\mathit{h}\mathit{T}\mathit{y}\mathit{p}\mathit{e}\mathbf{!}\mathit{T}\mathit{r}\mathit{a}\mathit{n}\mathit{s}\mathit{l}\mathit{a}\mathit{t}\mathit{o}\mathit{r}\mathbf{!}\mathbf{2}\mathbf{!}\mathbf{1}\mathbf{!}\mathit{A}\mathit{M}\mathit{S}\mathbf{}\mathit{L}\mathit{a}\mathit{T}\mathit{e}\mathit{X}\mathbf{.}\mathit{t}\mathit{d}\mathit{l}\mathbf{!}\mathit{A}\mathit{M}\mathit{S}\mathit{L}\mathit{a}\mathit{T}\mathit{e}\mathit{X}\mathbf{!}\mathbf{}\mathbf{\%}\mathbf{}\mathit{M}\mathit{a}\mathit{t}\mathit{h}\mathit{T}\mathit{y}\mathit{p}\mathit{e}\mathbf{!}\mathit{M}\mathit{T}\mathit{E}\mathit{F}\mathbf{!}\mathbf{2}\mathbf{!}\mathbf{1}\mathbf{!}\mathbf{+}\mathbf{-}\mathbf{}\mathbf{\%}\mathbf{}\mathit{f}\mathit{e}\mathit{a}\mathit{a}\mathit{g}\mathit{K}\mathit{a}\mathit{r}\mathit{t}\mathbf{1}\mathit{e}\mathit{v}\mathbf{2}\mathit{a}\mathit{q}\mathit{a}\mathit{t}\mathit{C}\mathit{v}\mathit{A}\mathit{U}\mathit{f}\mathit{e}\mathit{B}\mathit{S}\mathit{j}\mathit{u}\mathit{y}\mathit{Z}\mathit{L}\mathbf{2}\mathit{y}\mathit{d}\mathbf{9}\mathit{g}\mathit{z}\mathit{L}\mathit{b}\mathit{v}\mathit{y}\mathit{N}\mathit{v}\mathbf{2}\mathit{C}\mathit{a}\mathit{e}\mathit{r}\mathit{b}\mathit{b}\mathit{j}\mathit{x}\mathit{A}\mathit{H}\mathit{X}\mathbf{}\mathbf{\%}\mathbf{}\mathit{g}\mathit{a}\mathit{r}\mathit{m}\mathit{W}\mathit{u}\mathbf{51}\mathit{M}\mathit{y}\mathit{V}\mathit{X}\mathit{g}\mathit{a}\mathit{r}\mathit{u}\mathit{a}\mathit{v}\mathit{P}\mathbf{1}\mathit{w}\mathit{z}\mathit{Z}\mathit{b}\mathit{I}\mathit{t}\mathit{L}\mathit{D}\mathit{h}\mathit{i}\mathit{s}\mathbf{9}\mathit{w}\mathit{B}\mathit{H}\mathbf{5}\mathit{g}\mathit{a}\mathit{r}\mathit{u}\mathit{W}\mathit{q}\mathit{V}\mathit{v}\mathit{N}\mathit{C}\mathit{P}\mathit{v}\mathit{M}\mathit{C}\mathit{G}\mathbf{4}\mathit{u}\mathit{z}\mathbf{}\mathbf{\%}\mathbf{}\mathbf{3}\mathit{b}\mathit{q}\mathit{e}\mathit{e}\mathbf{0}\mathit{e}\mathit{v}\mathit{G}\mathit{u}\mathit{e}\mathit{E}\mathbf{0}\mathit{j}\mathit{x}\mathit{y}\mathit{a}\mathit{i}\mathit{b}\mathit{a}\mathit{i}\mathit{e}\mathit{Y}\mathit{l}\mathit{f}\mathbf{9}\mathit{i}\mathit{r}\mathit{V}\mathit{e}\mathit{e}\mathit{u}\mathbf{0}\mathit{d}\mathit{X}\mathit{d}\mathit{h}\mathbf{9}\mathit{v}\mathit{q}\mathit{q}\mathit{j}\mathbf{-}\mathit{h}\mathit{E}\mathit{e}\mathit{e}\mathit{u}\mathbf{0}\mathit{x}\mathit{X}\mathit{d}\mathit{b}\mathit{b}\mathbf{}\mathbf{\%}\mathbf{}\mathit{a}\mathbf{9}\mathit{f}\mathit{r}\mathit{F}\mathit{j}\mathbf{0}\mathbf{-}\mathit{O}\mathit{q}\mathit{F}\mathit{f}\mathit{e}\mathit{a}\mathbf{0}\mathit{d}\mathit{X}\mathit{d}\mathit{d}\mathbf{9}\mathit{v}\mathit{q}\mathit{a}\mathit{q}\mathbf{-}\mathit{J}\mathit{f}\mathit{r}\mathit{V}\mathit{k}\mathit{F}\mathit{H}\mathit{e}\mathbf{9}\mathit{p}\mathit{g}\mathit{e}\mathit{a}\mathbf{0}\mathit{d}\mathit{X}\mathit{d}\mathit{a}\mathit{r}\mathbf{-}\mathit{J}\mathit{b}\mathbf{9}\mathit{h}\mathit{s}\mathbf{0}\mathit{d}\mathit{X}\mathit{d}\mathbf{}\mathbf{\%}\mathbf{}\mathit{b}\mathit{P}\mathit{Y}\mathit{x}\mathit{e}\mathbf{9}\mathit{v}\mathit{r}\mathbf{0}\mathbf{-}\mathit{v}\mathit{r}\mathbf{0}\mathbf{-}\mathit{v}\mathit{q}\mathit{p}\mathit{W}\mathit{q}\mathit{a}\mathit{a}\mathit{e}\mathit{a}\mathit{a}\mathit{b}\mathit{i}\mathit{G}\mathit{a}\mathit{a}\mathit{i}\mathit{a}\mathit{a}\mathit{c}\mathit{a}\mathit{W}\mathit{a}\mathit{b}\mathit{e}\mathit{a}\mathit{a}\mathit{e}\mathit{a}\mathit{W}\mathit{a}\mathit{a}\mathit{u}\mathit{a}\mathit{a}\mathit{a}\mathit{O}\mathit{q}\mathit{a}\mathit{a}\mathit{a}\mathbf{}\mathbf{\%}\mathbf{}\mathit{b}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathit{p}\mathit{e}\mathit{G}\mathit{a}\mathit{a}\mathit{m}\mathit{O}\mathit{q}\mathit{a}\mathit{i}\mathit{a}\mathit{b}\mathit{g}\mathbf{2}\mathit{d}\mathit{a}\mathbf{9}\mathit{m}\mathit{a}\mathit{a}\mathit{l}\mathit{a}\mathit{a}\mathit{a}\mathit{p}\mathit{a}\mathit{q}\mathit{a}\mathit{a}\mathbf{8}\mathit{q}\mathit{a}\mathit{c}\mathit{q}\mathit{a}\mathit{H}\mathbf{8}\mathit{o}\mathit{q}\mathit{B}\mathit{c}\mathit{a}\mathit{W}\mathit{G}\mathbf{}\mathbf{\%}\mathbf{}\mathit{j}\mathit{b}\mathit{G}\mathit{a}\mathit{e}\mathit{q}\mathit{i}\mathit{U}\mathit{d}\mathit{e}\mathit{h}\mathit{a}\mathit{p}\mathit{a}\mathit{q}\mathit{a}\mathit{a}\mathbf{8}\mathit{q}\mathit{a}\mathit{c}\mathit{a}\mathit{a}\mathit{I}\mathbf{0}\mathit{a}\mathit{G}\mathit{a}\mathit{e}\mathit{q}\mathit{i}\mathit{W}\mathit{d}\mathit{a}\mathit{N}\mathit{a}\mathit{a}\mathit{m}\mathit{O}\mathit{u}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathit{a}\mathbf{!}\mathbf{41}\mathit{C}\mathbf{1}\mathbf{!}\mathbf{}\mathbf{\$}\mathit{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\backslash }\mathit{f}\mathit{r}\mathit{a}\mathit{c}\mathbf{\left\{}\mathbf{\right\{}\mathbf{\backslash }\mathit{m}\mathit{u}\mathbf{}\mathit{I}\mathbf{\backslash }\mathit{t}\mathit{h}\mathit{e}\mathit{t}\mathit{a}\mathbf{}\mathbf{\left\}}\mathbf{\right\}}\mathbf{\left\{}\mathbf{\right\{}\mathbf{4}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathit{R}\mathbf{\left\}}\mathbf{\right\}}\mathbf{\$}\mathbf{\%}\mathbf{}\mathit{M}\mathit{a}\mathit{t}\mathit{h}\mathit{T}\mathit{y}\mathit{p}\mathit{e}\mathbf{!}\mathit{E}\mathit{n}\mathit{d}\mathbf{!}\mathbf{2}\mathbf{!}\mathbf{1}\mathbf{!}$uncaught exception: Invalid chunk

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('4f6d14a8e2365d3...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('

(a) Determining the Magnitude of the magnetic field at point P

Now magnetic field for semicircle is as follows

$$\begin{gathered} B_1=\frac{{\mu }_{o}I\theta }{4\mathrm{\pi a}} \\ =\frac{{\mu }_{o}I\mathrm{\pi }}{4\mathrm{\pi a}} \\ =\frac{{\mu }_{o}I}{4a} \end{gathered}$$

And for radius b it is as follows

$B_2=\frac{{\mu }_{o}I}{4b}$

So total magnetic field is as follows

role="math" localid="1662799565933" $$\begin{gathered} B=B_1+B_2 \\ =\frac{{\mathrm{\mu }}_{\mathrm{o}}\mathrm{I}}{4\mathrm{a}}+\frac{{\mathrm{\mu }}_{\mathrm{o}}\mathrm{I}}{4\mathrm{b}} \\ =\frac{{\mathrm{\mu }}_{\mathrm{o}}\mathrm{I}}{4}(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}) \\ =(4\mathrm{\pi }\times {10}^{-7}{\mathrm{NA}}^{-2})\times (56.2\times {10}^{-3}\mathrm{A})(\frac{1}{0.0572\mathrm{m}}+\frac{1}{0.0936\mathrm{m}}) \\ =4.97\times {10}^{-7}\mathrm{T} \end{gathered}$$

(b) Determining the direction of the magnetic field at point P

The direction of the magnetic field is given by the right-hand rule, and it is into the page.

(c) Determining the magnitude of the loop’s magnetic dipole moment

Now dipole moment is given as follows

$\mu =IA$

Here

role="math" localid="1662799996915" $$\begin{gathered} A=\frac{\mathrm{\pi }\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)}{2} \\ =\frac{\mathrm{\pi }\left({\left(0.0572\mathrm{m}\right)}^2+{\left(0.0936\mathrm{m}\right)}^2\right)}{2} \\ =0.0189{\mathrm{m}}^2 \end{gathered}$$

Now

$$\begin{gathered} \mu =\left(56.2\times {10}^{-3}\mathrm{A}\right)\left(0.0189{\mathrm{m}}^2\right) \\ =1.06\times {10}^{-3}{\mathrm{Am}}^2 \end{gathered}$$

(d) Determining the direction of dipole moment

Since the direction is the same as the magnetic field, so it is into the page.