Current Electricity

From diagram

${i_p} = {E \over {2 + {r \over 2}}}$ and ${i_s} = {{2E} \over {2 + 2r}}$

given ${i_p} = {i_s}$

${1 \over {2 + {r \over 2}}} = {1 \over {1 + r}}$

$1 + r = 2 + {r \over 2}$

$r = 2\,\Omega $

R₁₀ = 2 = R₀(1 + $\alpha$ $\times$ 10)

R₃₀ = 3 = R₀(1 + $\alpha$ $\times$ 30)

On solving

$\alpha$ = 0.033/$^\circ$C

From the given setup

$y \times {R_G} = (x - y)({R_S})$

$ \Rightarrow y \times 72 = (x - y) \times 8$

$ \Rightarrow 9y = x - y$

$ \Rightarrow y = {x \over {10}}$ or 10% of x

Initially 5 $\Omega$ and 5 $\Omega$ are in series and then in parallel with 10 $\Omega$ this pattern continues thus

R_net = 5 $\Omega$

$R = {{\rho l} \over A}$

Also volume will remain constant

i.e., Al = constant $ \Rightarrow A \propto {1 \over l}$

$\therefore$ $R \propto {l^2}$

${{\Delta R} \over R} = 2{{\Delta l} \over l} = 0.8$

$I = {{2\varepsilon } \over {R + {r_1} + {r_2}}}$

As per the question,

${{2\varepsilon } \over {R + {r_1} + {r_2}}} \times {r_2} - \varepsilon = 0$

$ \Rightarrow R = {r_2} - {r_1}$

According to the information, current through galvanometer = nK

$ \Rightarrow {S \over {S + G}}i = nK$

$ \Rightarrow i = {{nK(S + G)} \over S}$

The circuit for the given situation is:

Since G and S are in parallel,

$\Rightarrow$ ${i \over 3}$ $\times$ G = ${2i \over 3}$ $\times$ S

$\Rightarrow$ G = 2S

$\Rightarrow$ G equals twice the value of shunt resistance.

${R_{eq}} = {{2 \times 4} \over {2 + 6}} + 6 = {{22} \over 3}$

$\Rightarrow$ A and B are in parallel and C is in series.

${{1.5 \times 10} \over {10 + {r \over 2}}} = 1.2$

$\Rightarrow$ r = 5 $\Omega$

Given all resistances have same resistance R.

Now, we can redraw the circuit as below

Let resistances be R₁, R₂, R₃ and R₄.

$\because$ ${{{R_1}} \over {{R_3}}} = {{{R_2}} \over {{R_4}}}$

So, circuit will behave as a Wheatstone bridge and no current will flow through middle resistor.

$\therefore$ ${R_{eq}} = {{({R_1} + {R_2})({R_3} + {R_4})} \over {({R_1} + {R_2}) + ({R_3} + {R_4})}}$

$ = {{(R + R)(R + R)} \over {(R + R) + (R + R)}}$

$ = R$