Question

Two lamps, each with a resistance of \[50\Omega \], are connected in series. The lamps will fuse if a power of more than $200\;{\text{W}}$ is dissipated in it. Then the maximum voltage that can be applied to the circuit is:
A. $100V$
B. $200V$
C. \[150V\]
D. \[250V\]

Answer 1

Hint: Since, the lamps are connected in a series combination, the resistances of both of these lamps will add up to form net equivalent resistance. This question can easily be solved by using the basic equation of power and voltage.

Formula used: We will use the following formula to find out our required solution:
\[{P_{\max }} = \dfrac{{{V^2}_{\max }}}{R}\]
Where,
\[{P_{\max }}\] is the maximum power
\[{V_{\max }}\] is the maximum voltage
\[R\] is the net resistance

Complete step by step answer:
According to the question, it is provided to us that
\[{P_{\max }}\] before fusing \[ = 200W\]
Also, the lamps are connected in series. So, the net effective resistance will add up as they are in a series combination.
That is,
\[{\operatorname{R} _{eq}} = {R_1} + {R_2}\]
Where
\[{\operatorname{R} _{eq}}\] is the equivalent resistance of both the lamps
\[{R_1}\] is the resistance of the first lamp
\[{R_2}\] is the resistance of the second lamp
Therefore, we get
\[{\operatorname{R} _{eq}} = 50 + 50 = 100\Omega \]
Now, we will use our main formula that is to be used to find out the maximum voltage that can be applied so that the lamps do not get fused.
That is,
\[{P_{\max }} = \dfrac{{{V^2}_{\max }}}{R}\]
On rearranging the above formula, we get
\[{V^2}_{\max } = {P_{\max }} \times R\]
Now, we will substitute the values of maximum power and the net equivalent resistance in the above formula to get
\[{V^2}_{\max } = 200 \times 100 = 20000\]
On further solving this equation, we get
\[\therefore {V_{\max }} = 141.42V\]
Hence, the lamps will fuse if we apply a voltage greater than \[141.42V\]. So, we can apply any amount of voltage below this value so that the lams do not fuse.
Looking at the options provided to us, the only option less than \[141.42V\]is option (A.) which is \[100V\].
So, the correct option is (A).

Note: The rate at which work is done or energy is transformed into an electrical circuit is electric power. Simply put, it is a measure of how much energy over a period of time is used.