Question

In the given circuit the maximum power that can be delivered safely to any of the resistors is 24W. Maximum potential difference between X and Y so that all the resistance are safe is:
 

(A) 36V
(B) 32V
(C) 20V
(D) 12V

Answer 1

Hint: In this question between the two points X and Y we have five resistors, two are in parallel to one other and the rest three are parallel to each other. We will calculate the resultant resistance of these two and add them in series.
Since the power rating is given, we can find the voltage by
$P = \dfrac{{{V^2}}}{R}$
Where V is the voltage, P is the power and R is the resistance.

Complete answer:
Let’s take the first parallel connection with a resistor of 6 ohm and 12 ohm.

Now the equivalent resistance for the above circuit is given as
\[\dfrac{1}{{{R_1}}} = \dfrac{1}{6} + \dfrac{1}{{12}} \]
\[\dfrac{1}{{{R_1}}} = \dfrac{{2 + 1}}{{12}} = \dfrac{3}{{12}} = \dfrac{1}{4} \]
${R_1} = 4\Omega $
Let’s take the second parallel connection with a resistor of 10 ohm, 12 ohm and 15 ohm.

Now the equivalent resistance for the above circuit is given as
\[\dfrac{1}{{{R_2}}} = \dfrac{1}{{10}} + \dfrac{1}{{12}} + \dfrac{1}{{15}} \]
\[ \dfrac{1}{{{R_2}}} = \dfrac{{6 + 5 + 4}}{{60}} = \dfrac{{15}}{{60}} = \dfrac{1}{4} \]
$R{}_2 = 4\Omega $
Now the resistances ${R_1}$ and ${R_2}$ are in series so the net equivalent resistance is
${R_{eq}} = {R_1} + {R_2} = 4 + 4 = 8\Omega $
The net resistance of the circuit is 8 ohms.
So now the safety power for the circuit to remain safe is 24 watts
Now we know that
$P = \dfrac{{{V^2}}}{R}$
Where V is the voltage, P is the power and R is the resistance.
$24 = \dfrac{{{V^2}}}{8}$
Now by rearranging the equation we get that.
\[{V^2} = 24 \times 8 = 192 \]
\[V = \sqrt {192} = 13.8V \]
Now the maximum load of voltage is 13.8 volts so we need to choose that option which is less than or equal to 13.8 volts.
And the correction option is 12 volts.

The correct option is (D).

Note:
Now the circuit is rated as a 24 watts circuit, and we found out the maximum voltage we can apply to the point X and Y. If we somehow manage to apply excess voltage rather than volts then there is a chance that the circuit may get on fire due to a short circuit in. So, in daily life the electrical appliances are marked with this kind of voltage restriction along with them to prevent accidents.