Question: A 42 Ω resistor and a 20 Ω resistor are connected in parallel, and the combination is connected across a 240-V dc line.

(a) What is the resistance of the parallel combination?

(b) What is the total current through the parallel combination?

(c) What is the current through each resistor?

Given data:

• R₁= 42 Ω
• R₂= 20 Ω
• V = 240 V

To find the equivalent resistance of R₁ and R₂:

The two resistors are in parallel and to calculate their combination we will use the equation :

${\mathit{R}}_{\mathbf{e}\mathbf{q}}\mathbf{=}\frac{{\mathbf{R}}_{\mathbf{1}}{\mathbf{R}}_{\mathbf{2}}}{{\mathbf{R}}_{\mathbf{1}}\mathbf{+}{\mathbf{R}}_{\mathbf{2}}}$

Substitute values to find R_eq as:

$$\begin{gathered} R_{\mathrm{eq}}=\frac{{\mathrm{R}}_1{\mathrm{R}}_2}{{\mathrm{R}}_1+{\mathrm{R}}_2} \\ =\frac{\left(42\Omega \right)\left(20\Omega \right)}{42\Omega +20\Omega } \\ =13.55\Omega \end{gathered}$$

The resistance of the parallel combination is 13.55 $\Omega$.

The potential difference V' across resistors connected in the parallel is the same for every resistor and equals the potential difference across the combination and we can use the value of the voltage to find the current across the combination by using Ohm's law as:

${\mathrm{I}}_{\mathrm{t}}=\frac{\mathrm{V}}{{\mathrm{R}}_{\mathrm{eq}}}$

Substitute values to find I_t

$$\begin{gathered} {\mathrm{I}}_{\mathrm{t}}=\frac{\mathrm{V}}{{\mathrm{R}}_{\mathrm{eq}}} \\ =\frac{240\hspace{0.17em}V}{13.55\Omega } \\ =17.70A \end{gathered}$$

The total current through the parallel combination is 17.70 A.

The result in Step 2 represents the total current in the resistors where the total current through resistors connected in parallel is the sum of the currents through the individual resistors R₁ and R₂.

$$\begin{gathered} {\mathrm{I}}_1=\frac{\mathrm{V}}{{\mathrm{R}}_1} \\ =\frac{240\hspace{0.17em}V}{42\Omega } \\ =5.71A \end{gathered}$$

But as we discussed in part (b) the voltage is the same and V₁ = V₂ = 240 V. Now use Ohm's law to get the current in each resistor as:

$$\begin{gathered} {\mathrm{I}}_2=\frac{\mathrm{V}}{{\mathrm{R}}_2} \\ =\frac{240\hspace{0.17em}V}{20\Omega } \\ =12.00A \end{gathered}$$

The current through resistor R₁ is 5.71 A and the current through resistor R₂ is 12.00 A.