Understanding Heat Production in Heating Coils

The amount of heat generated by an electrical component like a heating coil is directly proportional to the electrical power it dissipates. For a circuit where the voltage (V) is kept constant, the power (P) consumed by a resistance (R) is given by the formula:

$$P = \frac{V^2}{R}$$

This formula tells us that if the voltage V remains the same, the power P (and thus the heat produced) will be lowest when the total resistance R of the circuit is highest.

We are given four identical heating coils, each possessing the same resistance, let's call it R. The question asks for the connection method that results in minimum heat. This means we need to find the configuration that yields the maximum total equivalent resistance.

Calculating Equivalent Resistance for Coil Connections

1. All Coils Connected in Parallel

When four identical resistors (each with resistance R) are connected in parallel, the total equivalent resistance ($R_p$) is calculated using the formula:

$$\frac{1}{R_p} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} + \frac{1}{R}$$

Simplifying this equation:

$$\frac{1}{R_p} = \frac{4}{R}$$

Taking the reciprocal of both sides gives the total resistance:

$$R_p = \frac{R}{4}$$

2. All Coils Connected in Series

Connecting resistors in series simply involves adding their individual resistances. For four coils in series, the total resistance ($R_s$) is:

$$R_s = R + R + R + R = 4R$$

3. Two Parallel Pairs in Series

In this configuration, we first create two pairs of coils, with each pair having two coils connected in parallel. The resistance of one such parallel pair ($R_{p1}$) is:

$$R_{p1} = \frac{1}{\frac{1}{R} + \frac{1}{R}} = \frac{1}{\frac{2}{R}} = \frac{R}{2}$$

Similarly, the second parallel pair ($R_{p2}$) also has a resistance of $$ \frac{R}{2} $$.

These two parallel pairs are then connected in series. The total resistance ($R_{series\_pairs}$) is the sum of their resistances:

$$R_{series\_pairs} = R_{p1} + R_{p2} = \frac{R}{2} + \frac{R}{2} = R$$

4. One Pair in Parallel with the Other Two in Series

Here, one pair of coils is connected in parallel, giving a resistance ($R_{p1}$) of:

$$R_{p1} = \frac{R}{2}$$

The remaining two coils are connected in series, resulting in a resistance ($R_{s1}$) of:

$$R_{s1} = R + R = 2R$$

These two combinations ($R_{p1}$ and $R_{s1}$) are then connected in parallel. The overall total resistance ($R_{total}$) is found using the parallel resistance formula:

$$\frac{1}{R_{total}} = \frac{1}{R_{p1}} + \frac{1}{R_{s1}} = \frac{1}{R/2} + \frac{1}{2R}$$

Simplifying the terms:

$$\frac{1}{R_{total}} = \frac{2}{R} + \frac{1}{2R}$$

To add these fractions, we use a common denominator of 2R:

$$\frac{1}{R_{total}} = \frac{4}{2R} + \frac{1}{2R} = \frac{5}{2R}$$

Therefore, the total resistance for this configuration is:

$$R_{total} = \frac{2R}{5}$$

Comparison of Resistances for Different Coil Connections

Determining the Connection for Minimum Heat Production

To produce the minimum heat with a constant voltage, we need the connection that results in the maximum total resistance. Based on our calculations:

• Parallel connection: $$ 0.25R $$
• Series connection: $$ 4R $$
• Two parallel pairs in series: $$ R $$
• One pair parallel, two series: $$ 0.4R $$

The highest resistance value is $$ 4R $$, which is achieved when all four heating coils are connected in series.

Therefore, the four heating coils will produce minimum heat when connected all in series.