Two heater wires of equal length are first connected in series an
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| 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________.
A. 1 : 2
B. 2 : 1
C. 1 : 4
D. 4 : 1
Please scroll down to see the correct answer and solution guide.
Right Answer is: C
SOLUTION
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