Magnetic Effects Of Current Question 423
Question: The current density $ \vec{j} $ inside a long, solid, cylindrical wire of radius a=12 mm is in the direction of the central axis, and its magnitude varies linearly with radial distance r from the axis according to $ J=\frac{J_{0}r}{a} $ , where $ J_{0}=\frac{10^{5}}{4\pi }A/m^{2}. $ Find the magnitude of the magnetic field at in $ \mu T $
Options:
A) $ 10\mu T $
B) $ 4\mu T $
C) $ 5\mu T $
D) $ 3\mu T $
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Answer:
Correct Answer: A
Solution:
[a] Current in the element $ =J(2\pi r.dr) $ Current enclosed by ampere loop of radius $ a/2 $ $ I=\int\limits_{0}^{a/2}{\frac{J_{0}r}{a}.2\pi }r.dr $ $ =\frac{2\pi J_{0}}{3a}{{( \frac{a}{2} )}^{3}}=\frac{\pi J_{0}a^{3}}{12} $ Applying ampere’s law $ B.2\pi .\frac{a}{2}={\mu_{0}}.\frac{\pi J_{0}a^{2}}{12}\Rightarrow B=\frac{{\mu_{0}}J_{0}a}{12} $ On putting values, $ B=10\mu T $
BETA
