Question

Three conducting wires of same material and of equal lengths and equal diameters are first connected in series and then in parallel in a circuit across the same potential difference. Find the ratio of heat produced in series and parallel combination

Answer 1

Answer:

Two conducting wires of the same material and of equal lengths and equal diameter are first connected in series and then parallel in a circuit across the same potential difference. The ratio of heat produced in series and parallel combinations would be (a) 1:2. (b) 2:1.

Answer 2

The ratio of heat produced in series and the parallel combination is 0.11 i.e.,  $\frac{1}{9}$ .

Given:

Three conducting wires of the same material and equal lengths and equal diameters are connected in series and parallel in a circuit across the same potential difference of 'V' volts.

To Find:

The ratio of heat produced in series and parallel combination =?

Solution:

Let us assume, For the conducting wires;

The length of the wire is 'l'.

The area of the cross-section of the wire is 'A'.

The Resistivity of the material is 'ρ'.

Hence, we can write the Resistance of a single wire as follows;

Resistance (R) of the wire  =  ρ$\frac{l}{A}$.

We know, that the formula to calculate the heat produced in the circuit is;

H = $\frac{V^{2} }{R_{eq} }$ × t.

Case 1:

When a circuit is formed by connecting the three conducting wires in series across the potential difference of 'V' volts.

Here,

The equivalent resistance of the circuit is  $R_{eq}$;

∴  $R_{eq}$ = R + R + R = 3R

∴ The heat produced in the circuit when the resistances are arranged in a series is;

$H_{s}$ = $\frac{V^{2} }{3R}$ × t.                                       ----------(1)

Case 2:

When a circuit is formed by connecting the three conducting wires in parallel across the potential difference of 'V' volts.

Here,

The equivalent resistance of the circuit is  $R_{eq}$;

∴  $\frac{1}{R_{eq} }$  =  $\frac{1}{R}+$  $\frac{1}{R}+$  $\frac{1}{R}$

∴  $\frac{1}{R_{eq} }$  = $\frac{3}{R}$

∴   $R_{eq}$  =  $\frac{R}{3}$

∴ The heat produced in the circuit when the resistances are arranged in a parallel is;

$H_{p}$ = $\frac{V^{2} }{\frac{R}{3} }$ × t.                                      ----------(2)

Now, Dividing equation (1) by equation (2), we get;

$\frac{H_{s} }{H_{p} }$  =  $\frac{1}{9}$

Thus, The ratio of heat produced in series and parallel combination is 0.11.

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