Question

Two heater wires of equal length are first connected in series and then in parallel. The ratio of heat produced in the two cases is________. 
1. 1 : 2
2. 2 : 1
3. 1 : 4
4. 4 : 1

Answer 1

Correct Answer - Option 3 : 1 : 4

CONCEPT:

• Joule's law of heating states that the amount of heat produced in material of resistance 'R'  when a current 'I' passes through it for time 't' is given by

⇒ H = I²Rt

Substituting \(I = \frac{V}{R}\)in the above equation  for a unit time can be rewritten as

\(⇒ H = \frac{V^{2}}{ R}\)

Where V = Applied voltage and R =Resistance 

CALCULATION :

Given - R = Resistance of both wires, V = Voltage applied to the wires in series and parallel connection

• The effective resistance in the series combination of two wires is given by

\(⇒ R_{Series} = R+R = 2 R\)

• The amount of heat generated on series combination is given by

\(⇒ H_{Series} = \frac{V^{2}}{2R}\)       ------ (1)

• The effective resistance in parallel combination is given by

\(\frac{1}{R_{parllel}} = \frac{1}{R}+\frac{1}{R}\)

\(⇒ R_{parallel} = \frac{R}{2}\)

• The amount of heat generated on the parallel combination is given by

\(⇒ H_{Parallel} = \frac{V^{2}}{\frac{R}{2}} = \frac{2V^{2}}{R}\)      ------ (2)

• The ratio of the amount of heat generated between parallel and series combination is given by

\(⇒ \frac{H_{Series}}{H_{Parllel}} = \frac{\frac{V^{2}}{2R}}{\frac{2V^{2}}{R}} = \frac{1}{4}\)

• The ratio between the heat produced is 1 : 4
• Hence, option 3 is the answer