Electro Magnetic Induction And Alternating Currents Question 473

Question: A copper rod of length 0.19 m is moving parallel to a long wire with a uniform velocity of 10 m/s. The long wire carries 5 ampere current and is perpendicular to the rod. The ends of the rod are at distances 0.01 m and 0.2 m from the wire. The emf induced in the rod will be-

Options:

A) $ 10,\mu V $

B) $ 20,\mu V $

C) $ 30,\mu V $

D) $ 40,\mu V $

Show Answer

Answer:

Correct Answer: C

Solution:

  • EMF induced in an element of length dx at a distance x from wire = $ Bvdx $
    $ \therefore $ Total EMF induced in the rod $ E=\int\limits_{0.01}^{0.2}{Bv}dx=\int\limits_{0.01}^{0.2}{\frac{{\mu_{0}}iv}{2\pi x}} $

    $ dx=\frac{{\mu_{0}}iv}{2\pi } $

    $ \int\limits_{0.01}^{0.2}{\frac{1}{x}},dx $

    $ E=\frac{\mu_0iv}{2\pi }[ \log_ex ]_{0.01}^{0.2}$=

    $\frac{\mu_{0}iv}{2\pi }[\log_10(0.2)-{\log_{10}} $ $ (0.01)]\times 2.303 $

    $ E=\frac{4\pi \times {{10}^{-7}}\times 5\times 10}{2\pi }[1.301]\times 2.303 $

    $ =2.99\times {{10}^{-5}},V\approx 30\mu V $



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