Question

The diagrams show three circuits consisting of concentric circular arcs (either half or quarter circles of radii r, 2r, and 3r) and radial lengths. The circuits carry the same current. Rank them according to the magnitudes of the magnetic fields they produce at C, least to greatest

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Answer 1

Answer:

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Answer 2

Solution:

Know that, Radial segments don't produce magnetic field at C, so consider arcs.

Assume that the current is counterclockwise and the magnetic field to be positive pointing out of the page.

Understand that, magnetic field at the center from an arc $\[\varphi \]$of radius R is $\[\frac{{{\mu _0}i\phi }}{{4\pi R}}\]$.

Therefore,

(1)

$\[\begin{array}{l}B = \frac{{{\mu _0}i\pi }}{{4\pi \left( {3r} \right)}} + \frac{{{\mu _0}i\pi }}{{4\pi r}}\\ \Rightarrow B = \frac{1}{3}\frac{{{\mu _0}i}}{r}\end{array}\]$

(2)

$\[\begin{array}{l}B = \frac{{{\mu _0}i\pi }}{{4\pi \left( {3r} \right)}} - \frac{{{\mu _0}i\pi }}{{4\pi r}}\\ \Rightarrow B = - \frac{1}{6}\frac{{{\mu _0}i}}{r}\end{array}\]$

(3)

$\[\begin{array}{l}B = \frac{{{\mu _0}i\pi }}{{4\pi \left( {3r} \right)}} - \frac{{{\mu _0}i\left( {\frac{\pi }{2}} \right)}}{{4\pi r}} - \frac{{{\mu _0}i\left( {\frac{\pi }{2}} \right)}}{{4\pi \left( {2r} \right)}}\\ \Rightarrow B = - \frac{5}{{48}}\frac{{{\mu _0}i}}{r}\end{array}\]$

Therefore, the magnitude of the magnetic fields at C from least to greatest are (3), (2), (1).