Question

The resistance of hot tungsten filament is about 10 times the cold resistance. What will be the resistance of 100 W and 200 V lamps when not in use?
A. \[40\,\Omega \]
B. \[20\,\Omega \]
C. \[400\,\Omega \]
D. \[200\,\Omega \]

Answer 1

Hint: Use the formula for power in the electrical circuit. Use Ohm’s law to express the power in terms of voltage and resistance. Calculate the resistance of the hot filament and then use it to calculate the resistance of cold filament using the given quantities.

Formula used:
\[P = {I^2}R\]
Here, I is the current and R is the resistance.

Complete step by step answer:
We have given the power of the resistance is 100 W and voltage applied across it is 200 V.
We assume the resistance of cold filament is \[{R_{cold}}\] and resistance of hot filament is \[{R_{hot}}\]. We have given that,
\[{R_{hot}} = 10{R_{cold}}\]
We know the relation between power, resistance and current is,
\[P = {I^2}R\] …… (1)
Here, I is the current and R is the resistance.
From Ohm’s law, we have,
\[V = IR\]
\[ \Rightarrow I = \dfrac{V}{R}\]
Substituting the above equation in equation (1), we get,
\[P = {\left( {\dfrac{V}{R}} \right)^2}R\]
\[ \Rightarrow P = \dfrac{{{V^2}}}{R}\]
Rearranging the above equation for R, we get,
\[R = \dfrac{{{V^2}}}{P}\]
Now, we can calculate the resistance of hot filament as follows,
\[{R_{hot}} = \dfrac{{{V^2}}}{P}\]
Substituting 200 V for V and 100 W for P in the above equation, we get,
\[{R_{hot}} = \dfrac{{{{\left( {200} \right)}^2}}}{{100}}\]
\[ \Rightarrow {R_{hot}} = 400\,\Omega \]
Since we have given that the resistance of hot filament is 10 times the resistance of the cold filament, we have,
\[{R_{hot}} = 10{R_{cold}}\]
\[ \Rightarrow {R_{cold}} = \dfrac{{{R_{hot}}}}{{10}}\]
Substituting \[400\,\Omega \] for \[{R_{hot}}\] in the above equation, we get,
\[{R_{cold}} = \dfrac{{400}}{{10}}\]
\[ \Rightarrow {R_{cold}} = 40\,\Omega \]
Therefore, the resistance of the cold filament is \[40\,\Omega \].

So, the correct answer is “Option A”.

Note:
Students can use Ohm’s law to express the power in terms of voltage and resistance and back to the current and resistance. The key is to remember Ohm’s law. Make sure that units of voltage and resistance are volt and ohm respectively. The power can be expressed in watts if and only if the voltage is in volts and resistance is in ohm.