Question

Find the potential difference between X and Y in volts is :-

A. 1
B. -1
C. 2
D. -2

Answer 1

Find the effective resistance of the upper and lower combination of the resistance by using the formula for the effective resistance of two resistances in series connection. Then use Ohm’s law to find the potential difference between X and Y.

Formula used:
$V=iR$
where $V$ is the potential difference across the resistance and $i$ is the current in the resistance $R$.

Complete step by step answer:
Let us calculate the effective resistance of the circuit.When two resistances ${{R}_{1}}$ and ${{R}_{2}}$ are connected in series, the effective resistance of the two is equal to ${{R}_{eff}}={{R}_{1}}+{{R}_{2}}$.
We can see that the two resistances $2\Omega$ and $3\Omega$, meeting at point X, are in series connected. Therefore, the effective resistance of the two is equal to,
${{R}_{eff,1}}=2+3=5\Omega$
Similarly, the effective resistance of the other pair of $2\Omega$ and $3\Omega$ resistances is equal to ${{R}_{eff,2}}=2+3=5\Omega$.

The two $5\Omega$ resistances are between the same potential differences (say V).
From Ohm’s law, we know that $V=iR$, where V is the potential difference across the resistance and i is the current in the resistance R.
Since the potential differences and the resistances are the same, the current in both resistance will be the same.
And by junction law at point Y, we get that $2=i+i$.
$\Rightarrow i=1A$.
Now, the potential difference between A and X is equal to ${{V}_{X}}-{{V}_{A}}=(1)(2)=2V$ …. (i).
The potential difference between A and Y is equal to ${{V}_{A}}-{{V}_{Y}}=(1)(3)=3V$….. (ii).
Subtract (ii) by (i).
$\Rightarrow {{V}_{A}}-{{V}_{Y}}-\left( {{V}_{A}}-{{V}_{X}} \right)=3-2$
$\therefore {{V}_{X}}-{{V}_{Y}}=1\,V$

Hence, the correct option is A.

Note: When two resistances are connected parallel to each other, the reciprocal of the effective resistance of the two is equal to the sum of the reciprocal of the individual resistances i.e. $\dfrac{1}{{{R}_{eff}}}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}}$.The major difference between series and the parallel circuit is the amount of current that flows through each of the components in the circuit. In a series circuit, the same amount of current flows through all the components placed in it. On the other hand, in parallel circuits, the components are placed in parallel with each other due to this the circuit splits the current flow.