The magnetic field in the cylindrical region shown in figure increase at a constant rate of $$20.0mT/s$$. Each side of the square loop $$abcd$$ and $$defa$$ has a length of $$1.00cm$$ and a resistance of $$4.00\Omega$$. Find the current (magnitude and sense) in the wire $$ad$$ if (a) the switch $${S}_{1}$$ is closed but $${S}_{2}$$ is opne, (b) $${S}_{1}$$ is open but $${S}_{2}$$ is closed, (c) both $${S}_{1}$$ and $${S}_{2}$$ are open and (d) both $${S}_{1}$$ and $${S}_{2}$$ are closed.
$${S}_{1}$$ closed, $${S}_{2}$$ open
net $$R=4\times 4=16\Omega$$
$$e=\cfrac { d\phi }{ { dt }^{ } } =A\cfrac { dB }{ dt } ={ 10 }^{ -4 }\times 2\times { 10 }^{ -4 }=2\times { 10 }^{ -6 }V\quad $$
i through $$ad=\cfrac { e }{ R } =\cfrac { 2\times { 10 }^{ -6 } }{ 16 } =1.25\times { 10 }^{ -7 }A$$ along ad
(b)$$R=16\Omega$$
$$e=A\cfrac { dB }{ dt } =2\times { 10 }^{ -5 }V\quad $$
$$i=\cfrac { 2\times { 10 }^{ -6 } }{ 16 } =1.25\times { 10 }^{ -7 }A$$
(c) Since both $${S}_{1}$$ and $${S}_{2}$$ are open, no current is passed as circuit is open ie $$i=0$$
(d) Since both $${S}_{1}$$ and $${S}_{2}$$ are closed, the circuit forms a balanced wheat stone bridge and no current will flow along $$ad$$ $$i=0$$
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