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Physics · 2025 · 2 marks
CBSE 2025 · Region 2 · Set 2 · Q21
(a) Two wires of the same material and the same radius have their lengths in the ratio 2: 3. They are connected in parallel to a battery which…
Two wires of the same material and the same radius have their lengths in the ratio $\displaystyle 2: 3$. They are connected in parallel to a battery which supplies a current of $\displaystyle 15$ A . Find the current through the wires.
In the circuit three ideal cells of e.m.f. $\displaystyle \mathrm{V}, \mathrm{V}$ and $\displaystyle 2$ V are connected to a resistor of resistance R , a capacitor of capacitance C and another resistor of resistance $\displaystyle 2 R$ as shown in figure. In the steady state find
(i)
the potential difference between P and Q and (ii) potential difference across capacitor $\displaystyle C$.
Marking-scheme solution
$\displaystyle R_{1}=\frac{\rho l_{1}}{A} ; R_{2}=\frac{\rho l_{2}}{A}$
$\displaystyle \frac{l_{1}}{l_{2}}=\frac{2}{3} \Rightarrow \frac{R_{1}}{R_{2}}=\frac{2}{3}$
$\displaystyle I \propto \frac{1}{R}$
$\displaystyle \Rightarrow \frac{I_{1}}{I_{2}}=\frac{3}{2}$
$\displaystyle \Rightarrow I_{1}=\frac{3}{5} \times 15=9 \mathrm{~A}$
$\displaystyle \Rightarrow I_{2}=\frac{2}{5} \times 15=6 \mathrm{~A}$
In steady state:
$\displaystyle 2 V-V=i(2 R+R)$
$\displaystyle i=\frac{V}{3 R}$
(i)
$\displaystyle V_{P}-V_{Q}=-V-i R$
$\displaystyle =-V-\frac{V}{3}$
$\displaystyle V_{P}-V_{Q}=-\frac{4 V}{3}$
(ii)
$\displaystyle V_{P}-V_{Q}=-V+V_{C}$
$\displaystyle -\frac{4 V}{3}=-V+V_{C}$
$\displaystyle V_{C}=-\frac{V}{3}$
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