Question 288: Current Electricity

Question

In a meter bridge, the wire of length 1 m has a non-uniform cross-section such that, the variation $${{dR} \over {d\ell }}$$ of its resistance R with length $$\ell $$  is $${{dR} \over {d\ell }}$$ $$ \propto $$ $${1 \over {\sqrt \ell  }}$$. Two equal resistances are connected as shown in the figure. The
galvanometer has zero deflection when the jockey is at point P. What is the length AP ?

Accepted Answer

For the given wire :
dR = C $${{d\ell } \over {\sqrt \ell  }}$$,
where C = constant.
Let resistance of part
AP is R
1
and PB is R
2
$$ 	herefore $$  $${{R'} \over {R'}} = {{{R_1}} \over {{R_2}}}$$   or    R
1
= R
2
By balanced WSB concept.
Now    $$\int {dR = c\int {{{d\ell } \over {\sqrt \ell  }}} } $$
$$ 	herefore $$  R
1
= C$$\int\limits_0^\ell  {{\ell ^{ - 1/2}}d\ell } $$ = C.2.$${\sqrt \ell  }$$
R
2
= C$$\int\limits_\ell ^1 {{\ell ^{ - 1/2}}d\ell } $$ = C.(2 $$-$$ 2$${\sqrt \ell  }$$)
Putting R
1
= R
2
C2$${\sqrt \ell  }$$ = C(2 $$-$$ 2$${\sqrt \ell  }$$)
$$ 	herefore $$  2$${\sqrt \ell  }$$ = 1
$${\sqrt \ell  }$$ = $${1 \over 2}$$
i.e. $$\ell $$ = $${1 \over 4}$$ m     $$ \Rightarrow $$  0.25 m

Answer

0.3 m

Answer

0.25 m

Answer

0.35 m

Answer

0.2 m

Current Electricity

Choose the correct answer: