Question

Two resistors, one having a resistance of$\mathbf{900}\mathbf{}\mathbf{k\Omega }$, are connected in series to produce a total resistance of $\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{M\Omega }$. (a) What is the value of the second resistance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Short answer

(a)The value of the second resistance is${\mathrm{R}}_2=-400\mathrm{k\Omega }$.

(b)Value of the second resistor is negative.

(c)Resistance of the series is less than the resistance of the single resistor.

Step-by-step solution

Combination of series resistance

The resultant resistance of series combination resistors is given as,

${\mathbf{R}}_{\mathbf{s}}\mathbf{=}{\mathbf{R}}_{\mathbf{1}}\mathbf{+}{\mathbf{R}}_{\mathbf{2}}$

Here, ${\mathbf{R}}_{\mathbf{s}}$is the resultant resistance for series combination, ${\mathbf{R}}_{\mathbf{1}}$and ${\mathbf{R}}_{\mathbf{2}}$are the resistances of the resistors connected in series.

Calculation of the second resistance

(a)

Resistors value in series adds up. That means that:

$$\begin{gathered} {\mathrm{R}}_2={\mathrm{R}}_{\mathrm{s}}-{\mathrm{R}}_1 \\ =0.5\mathrm{M\Omega }-900\mathrm{k\Omega } \\ =0.5\mathrm{M\Omega }\left(\frac{1000\mathrm{k\Omega }}{1\mathrm{M\Omega }}\right)-900\mathrm{k\Omega } \\ =-400\mathrm{k\Omega } \end{gathered}$$

Therefore, the value of the second resistance is${\mathrm{R}}_2=-400\mathrm{k\Omega }$.

Unreasonable about the result

(b)

The unreasonable result is: negative value of the second resistor.

Inconsistence assumption

(c)

The assumptions are unreasonable or inconsistent is, resistance of the series is less than the resistance of the single resistor.