JEE Advance - Physics (2012 - Paper 1 Offline - No. 20)
A cylinder cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the magnitude of the magnetic field at the point P is given by $${N \over {12}}{\mu _0}aJ$$, then the value of N is ______________.
Explanation
BR = BT $$-$$ BC
R = Remaining portion
T = Total portion and
C = cavity
$${B_R} = {{{\mu _0}{I_T}} \over {2a\pi }} - {{{\mu _0}{I_C}} \over {2(3a/2)\pi }}$$ ..... (i)
$${I_T} = J(\pi {a^2})$$
$${I_C} = J\left( {{{\pi {a^2}} \over 4}} \right)$$
Substituting the values in Eq. (i), we have
$${B_R} = {{{\mu _0}} \over {a\pi }}\left[ {{{{I_T}} \over 2} - {{{I_C}} \over 3}} \right]$$
$$ = {{{\mu _0}} \over {a\pi }}\left[ {{{\pi {a^2}J} \over 2} - {{\pi {a^2}J} \over {12}}} \right]$$
$$ = {{5{\mu _0}aJ} \over {12}}$$
$$\therefore$$ $$N = 5$$
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