JEE MAIN - Physics (2017 - 8th April Morning Slot - No. 23)
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A 9 V battery with internal resistance of 0.5 $$\Omega $$ is connected across an infinite network as shown in the figure. All ammeters A1, A2, 3 and voltmeter V are ideal.
Choose correct statement.
Reading of A1 is 2 A
Reading of A1 is 18 A
Reading of V is 9 V
Reading of V is 7 V
Explanation
Assume equivalent resistance of the infinite network is x.
The circuit can be redrawn like this,
_8th_April_Morning_Slot_en_23_2.png)
Here 4 $$\Omega $$ and x $$\Omega $$ are parallel,
$$\therefore\,\,\,$$ $${1 \over {{x_{eq}}}}$$ = $${1 \over 4}$$ + $${1 \over x}$$
$$ \Rightarrow $$$$\,\,\,$$ xeq = $${{4x} \over {4 + x}}$$
$$\therefore\,\,\,$$ Equivalent resistance between point A and B is ,
x = $${{4x} \over {4 + x}}$$ + 2
$$ \Rightarrow $$$$\,\,\,$$ x = $$ {{8 + 6x} \over {4 + x}}$$
$$ \Rightarrow $$$$\,\,\,$$ 4x + x2 = 8 + 6x
$$ \Rightarrow $$$$\,\,\,$$ x2 $$-$$ 2x $$-$$ 8 = 0
$$ \Rightarrow $$ $$\,\,\,$$ x = $${{2 \pm \sqrt {4 - 4\left( 1 \right)\left( { - 8} \right)} } \over 2}$$
$$ \Rightarrow $$$$\,\,\,$$x = $${{2 \pm 6} \over 2}$$
$$ \Rightarrow $$$$\,\,\,$$ = 4 $$\Omega $$
(as negative value of x is not acceptable)
$$\therefore\,\,\,$$ Reading of Aonmeter, A1 = $${V \over {R + r}}$$
A1 = $${9 \over {4\,\, + 0.5}}$$ = 2 A
The circuit can be redrawn like this,
Here 4 $$\Omega $$ and x $$\Omega $$ are parallel,
$$\therefore\,\,\,$$ $${1 \over {{x_{eq}}}}$$ = $${1 \over 4}$$ + $${1 \over x}$$
$$ \Rightarrow $$$$\,\,\,$$ xeq = $${{4x} \over {4 + x}}$$
$$\therefore\,\,\,$$ Equivalent resistance between point A and B is ,
x = $${{4x} \over {4 + x}}$$ + 2
$$ \Rightarrow $$$$\,\,\,$$ x = $$ {{8 + 6x} \over {4 + x}}$$
$$ \Rightarrow $$$$\,\,\,$$ 4x + x2 = 8 + 6x
$$ \Rightarrow $$$$\,\,\,$$ x2 $$-$$ 2x $$-$$ 8 = 0
$$ \Rightarrow $$ $$\,\,\,$$ x = $${{2 \pm \sqrt {4 - 4\left( 1 \right)\left( { - 8} \right)} } \over 2}$$
$$ \Rightarrow $$$$\,\,\,$$x = $${{2 \pm 6} \over 2}$$
$$ \Rightarrow $$$$\,\,\,$$ = 4 $$\Omega $$
(as negative value of x is not acceptable)
$$\therefore\,\,\,$$ Reading of Aonmeter, A1 = $${V \over {R + r}}$$
A1 = $${9 \over {4\,\, + 0.5}}$$ = 2 A
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