Physics

The flux of magnetic field through a closed conducting loop changes with time according to the equation,$$ \Phi ={ at }^{ 2 }+bt+c$$.$$(a)$$ Write the $$SI$$ units of $$a,b$$ and $$c$$ $$(b)$$ If the magnitudes of $$a,b$$ and $$c$$ are $$0\cdot 20, 0.40$$ and $$0.60$$ respectively, find the including emf at $$t=2\ s$$.


SOLUTION
Given:
$$ \Phi ={ at }^{ 2 }+bt+c$$
Same dimensional quantity can be added, subtracted and equated.
(a) $$a=\left[ \cfrac { \phi  }{ { t }^{ 2 } }  \right] =\left[ \cfrac { \phi /t }{ t }  \right] =\cfrac { volt }{ sec } ; \ \ \ b=\left[ \cfrac { \phi  }{ { t }^{  } }  \right] =volt; \ \ \ c=\left[ \phi  \right] =Weber$$

(b) $$a=0.2,b=0.4,c=0.6,t=2s$$
$$\quad E=\cfrac { d\phi  }{ { dt }^{  } } =2at+b=2\times 0.2\times 2+0.4=1.2volt$$
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