[2013EM/HMW-10]

PHY 102: Introduction to Physics-2
Tutorial-10
(Review)

1. Poynting theorem: An inductor made up of a very long air-core solenoid of length $I$, radius $r$ and $n$ turns per unit length. For $\left({d{\pmb I}\over dt}\right) >0$, verify that $$ {dU_B\over dt} = -\int_{\rm Surface} \vec{S}\cdot d\vec{A} $$
2. Magnetic field in a current loop:
   • [(i)] Find the magnetic field at the center of the loop carrying a current $I$ and made of two concentric semicircles of radii $a$ and $b$ and two straight segments along the diameter as shown in figure.
   • [(ii)] Find $B$ if $I=2A$, $a=1$m, $b=2$m.
   Electromotive force (EMF):  A rod of mass $m$ and length $w$ and resistance $R$ starts from rest and slides on two parallel rails of zero resistance as shown in figure 2. A battery of voltage $V$ is connected as shown. (i) Argue that the net EMF in the loop is $V-Bvw$ when the rod has speed $v$. (ii) Write down $F=m\cdot(dv/dt)$ and integrate it so show that $$ \nu(t) = {V\over Bw}\left(1-e^{B^2w^2t/mR}\right) $$ Hint: Find the limiting speed and separate that out from the total $v$.