Question

The given figure shows a part of an electrical circuit. The potentials at the points $a,b$ and $c$ are $30V,12V$ and $2V$ respectively. Find the currents through the three resistors.

Answer 1

Hint: We have to consider the node as $d$ and then suppose a potential of $V$ exists there. As $b$ and $c$ are in parallel so the current from $a$ splits up into two equal values. We have to assume a current is flowing in the circuit $a$. Then by comparing the equations with the help of Kirchhoff’s Law we will find the value of the unknown potential. After that we will be using Ohm’s Law to find the values of current.

Complete step by step solution:

Let the potential at point $d$ as shown in figure be ${V_d}$. Let a current $i$ flows from $a$ to $d$.
Since, $d$ to $b$ circuit and the $d$ to $c$ circuit are in parallel so both have the same current flowing through it.
It means that current $i$ is being split up into two equal magnitudes of current $\dfrac{i}{2}$. So, $\dfrac{i}{2}$ current is flowing through the parallel circuit.
The potentials at the points $a,b$ and $c$ are $30V,12V$ and $2V$ respectively.
So, ${V_a} = 30V$, ${V_b} = 12V$ and ${V_c} = 2V$.
And the given resistances are, ${R_a} = 10\Omega $, ${R_b} = 20\Omega $ and ${R_c} = 30\Omega $
Now applying Kirchhoff’s Law we get,
${V_a} - {V_d} - \left( {i \times {R_a}} \right) = 0$
$ \Rightarrow 30 - {V_d} = 10i - - - - - \left( 1 \right)$
Similarly,
${V_d} - 12 = 20 \times \dfrac{i}{2} = 10i - - - - - \left( 2 \right)$
${V_d} - 2 = 30 \times \dfrac{i}{2} = 15i - - - - - \left( 3 \right)$
Comparing equation $\left( 1 \right)$ and $\left( 2 \right)$ we get,
$30 - {V_d} = {V_d} - 12$
$\therefore {V_d} = 25$
So, the potential at point $d$ is $25V$.
Now, we know that the currents flowing through resistances ${R_a}$, ${R_b}$ and ${R_c}$ is $i$, $\dfrac{i}{2}$ and $\dfrac{i}{2}$ respectively.
Now by using Ohm’s Law we get,
${V_a} - {V_d} = i \times {R_{}}$
$30 - 25 = i \times 10$
So the value of $i = 0.5{\text{ }}A$
Therefore, $\dfrac{i}{2} = 0.25{\text{ }}A$.
So, the currents flowing through resistances ${R_a}$, ${R_b}$ and ${R_c}$ is $0.5{\text{ }}A$, $0.25{\text{ }}A$ and $0.25{\text{ }}A$.

Note: It must be noted that we must consider a common potential point in this type of question. As the potential at point $d$ is greater than $b$ and $c$ so, the current flows outwards from $d$ with respect to $b$ and $c$ but as the potential at point $a$ is greater than $d$, the currents flows inwards or towards $d$.