Question

Two bulbs of \[500W\] and \[200W\] are manufactured to operate on the \[220V\] line. The ratio of heat produced in \[500W\] and \[200W\] in two cases, when firstly they are connected in parallel and secondary in series will be:
A) \[\dfrac{5}{2}, \dfrac{2}{5}\]
B) \[\dfrac{5}{2}, \dfrac{5}{2}\]
C) \[\dfrac{2}{5}, \dfrac{5}{2}\]
D) \[\dfrac{2}{5}, \dfrac{2}{5}\]

Answer 1

Hint: First we calculate ratio of heat produced by two bulbs connected in parallel. Then we find the ratio of heat produced by two bulbs connected in series. Finally we calculate the ratio of heat produced in both connections.

Formula used:
Two bulbs are connected first in series then parallel. So in series connection, heat produced by two bulbs is calculated by \[H = Pt\]. For parallel connection, different formulas are used. We use the \[H = {I^2}Rt\] formula for heat production. After that we convert resistance in the form of power. The relation between resistance and power is given by \[P = \dfrac{{{V^2}}}{R}\]. We write resistance in terms of power because in question only powers are given.

Complete step by step solution:
Given: two bulbs having power \[500W\] and \[200W\].
Hence, \[{P_1} = 500W\& {P_2} = 200W\]
Firstly, two bulbs connected in parallel,
Then the ratio of heat produced in two bulbs is given by,
\[
  \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{{P_1}t}}{{{P_2}t}} \\
   \Rightarrow \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{{P_1}}}{{{P_2}}} \\
  \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{500}}{{200}} \\
   \Rightarrow \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{5}{2} \\
 \]
Secondly, two bulbs connected in series,
Then the ratio of heat produced in two bulbs is given by,
\[
  \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{{I^2}{R_1}t}}{{{I^2}{R_2}t}} \\
   \Rightarrow \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{{R_1}}}{{{R_2}}} \\
 \]
\[
  \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{\dfrac{{{V^2}}}{{{P_1}}}}}{{\dfrac{{{V^2}}}{{{P_2}}}}} \\
   \Rightarrow \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{{P_2}}}{{{P_1}}} \\
 \]
\[
  \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{200}}{{500}} \\
   \Rightarrow \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{2}{5} \\
 \]
Therefore the ratio of heat generated in parallel to series is \[\dfrac{5}{2}, \dfrac{2}{5}\].

Hence, the correct option is A.

Additional information: When a current flow through a component is a circuit, then an electrical energy is converted into heat energy. The heat generated in a circuit is dissipated into the air around the components present in a circuit. Heat lamps are used for the application of generating heat. The relation between current, resistance, time is given by \[H = {I^2}Rt\]. This relation is called the joule’s law of heating. According to this law, the heat produced is proportional to square of current I. Heat is proportional to resistance of the circuit R and time t. Time is defined during the current flows through the circuit. Power is related to heat. The power is defined by the rate at which the heat is dissipated.

Note: Students must be clear about different formulas used in case of series as well as parallel connections. They must convert parameters in those which are given in question. Students must be clear about the formula used in both connections.