Question

Two identical resistance wires are first connected in series and then in parallel in an electric circuit. The ratio of heat produced in the two cases will be :

• A. 1:2
• B. 2:1
• C. 1:4
• D. 4:1

Hint:

When dealing with heat produced and comparing series vs. parallel connections, it's often easiest to use the $H = V^2/R \cdot t$ formula if the voltage source is assumed to be constant across the combinations. This avoids calculating currents for each case. Remember that series connection increases total resistance, while parallel connection decreases it.

Answer 1

The Correct Option is C

Step 1: Define the resistances and heat produced.
Let the resistance of each identical wire be $R$.
The heat produced ($H$) in an electric circuit is given by Joule's Law of Heating: $H = I^2Rt$, where $I$ is current, $R$ is resistance, and $t$ is time. Alternatively, $H = \frac{V^2}{R}t$ if voltage is constant, or $H = VIt$. We will assume the same voltage source $V$ is applied in both cases, which is typical for such problems. Step 2: Calculate equivalent resistance and heat produced in series connection.
When two identical resistors ($R$) are connected in series, the equivalent resistance ($R_S$) is:
$R_S = R + R = 2R$
The heat produced in the series circuit ($H_S$) when connected to a voltage source $V$ for time $t$ will be:
$H_S = \frac{V^2}{R_S}t = \frac{V^2}{2R}t$ Step 3: Calculate equivalent resistance and heat produced in parallel connection.
When two identical resistors ($R$) are connected in parallel, the equivalent resistance ($R_P$) is:
$\frac{1}{R_P} = \frac{1}{R} + \frac{1}{R} = \frac{2}{R}$ $R_P = \frac{R}{2}$
The heat produced in the parallel circuit ($H_P$) when connected to the same voltage source $V$ for time $t$ will be:
$H_P = \frac{V^2}{R_P}t = \frac{V^2}{(R/2)}t = \frac{2V^2}{R}t$ Step 4: Determine the ratio of heat produced (series to parallel).
Now, find the ratio $H_S : H_P$:
$\frac{H_S}{H_P} = \frac{\frac{V^2}{2R}t}{\frac{2V^2}{R}t}$
Cancel out common terms ($V^2$, $R$, $t$):
$\frac{H_S}{H_P} = \frac{1/2}{2} = \frac{1}{2 \times 2} = \frac{1}{4}$
So, the ratio of heat produced in the two cases (series to parallel) is 1:4. $$(3) 1:4$$