Question

A long cylindrical shell, with inner radius a and outer radius b. has a uniform current density and total current I directed out of the page, as shown. Which graph best shows the magnitude of the magnetic field as a function of distance from the center of the shell. A wire loop of area A is placed in a time-varying but spatially uniform magnetic field that is perpendicular to the plane of the loop, as shown at right The induced emf in the loop is given by e - bAt^1\2 where b is a constant The time varying magnetic field could be given by 1/2bAt^-1/2 1/2 bt^-1/2 1/2br^1/2 2/3bAr^3/2 2/3bt^3/2

Answer 1

6). Current Density for the given cylinder, $J= \frac{I}{\pi(b^2-a^2)}$ ................................(1)

Now for the Magetic Field inside the cylinder(0r<a), Using Ampere's Circuital Law,

$\oint \vec{B}.\vec{dl}= \mu_o I_{encl}=0$  (Since no current in enclosed in this region)

thus   $B_1= 0$ for region 0<r<a.........................................(A)

Now for the region of the cylinder, a<r<b. Using Ampere's Circuital Law,

$\oint \vec{B_2}.\vec{dl}= \mu_o I_{encl}$

$B_2( 2 \pi r)= \mu_o (J \pi (r^2-a^2))$

$B_2= \mu_o\frac{J (r^2-a^2)}{2r}$

Using equation 1 in above,

  $B_2=\frac{I \mu_o (r^2-a^2)}{2r\pi (b^2-a^2)}$

thus we can say that $B_2\propto r$for region a<r<b..................................(B)

Now for the region of the cylinder, r>b. Using Ampere's Circuital Law,

$\oint \vec{B_3}.\vec{dl}= \mu_o I_{encl}$

$B_3( 2 \pi r)= \mu_o I$ (SInce then whole of the current lies in the region)

$B_3=\frac{\mu _0 I}{2 \pi r}$

thus we can say that $B_3\propto \frac{1}{r}$ for region r>b..................................(C)

Then following equation A, B and C we can say that graph (D). depicts correctly the variation of Magnetic Field with the distance.