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A closed loop carrying I=5 A is made of two concentric semicircular arcs of radii 10 cm and 20 cm, both centred at O and lying on the same side of…

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A closed loop carrying

I=5\ \mathrm{A}

is made of two concentric semicircular arcs of radii

10\ \mathrm{cm}

and

20\ \mathrm{cm}

, both centred at

O

and lying on the same side of the line through

O

, joined at their ends by straight radial segments. The magnitude of the magnetic field at

O

Show answer and solution

Answer: (D) $2.5\pi\ \mu\mathrm{T}$

The radial segments pass through

O

and give no field. A semicircular arc of radius

r

gives

B=\dfrac{\mu_0I}{4r}

. Going round the closed loop, the two arcs circulate in opposite senses about

O

, so the fields subtract:

B=\dfrac{\mu_0I}{4}\left(\dfrac1{0.1}-\dfrac1{0.2}\right)=\dfrac{4\pi\times10^{-7}\times5}{4}\times5=2.5\pi\times10^{-6}\ \mathrm{T}

. Adding them gives the

7.5\pi\ \mu\mathrm{T}

trap.

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