Question

An electric appliance connected to $220V$line has two resistance coils $P$ and $Q$, each of $44\Omega$ resistance which may be used separately, in series and in parallel. What are the currents in three cases?

Answer 1

Step 1 - Given data

1. Resistance of coil P $R_P=44\Omega$
2. Resistance of coil Q $R_Q=44\Omega$
3. Supplied voltage $V=220V$

Step 2 - Calculate the current when used separately

Assume when the resistances are used separately then the current flowing through them is $i$.

According to Ohm's law, the formula to calculate the current for $P$ coil is given by

$i=\frac{V}{R_P}$

Substitute the value of resistance of coil P and supplied voltage into the equation to find $i.$

$$\begin{gathered} i=\frac{220V}{44\Omega } \\ i=5A \end{gathered}$$

Both coils have the same value of resistance and the same amount of voltage is supplied to them, therefore current flowing through both coils is the same.

Step 3 - Finding the equivalent resistance when connected in series.

Consider $R_{eq}$be the equivalent resistance of the series combination.

So the formula to calculate the equivalent resistance is given by

$R_{eq}=R_P+R_Q$

Substitute the values of resistances of the coil into the formula to calculate $R_{eq}$

$$\begin{gathered} R_{eq}=44\Omega +44\Omega \\ {R}_{eq}=88\Omega \end{gathered}$$

Step 4 - Calculate the current through the series combination.

According to Ohm's law, $i_s=\frac{V}{R_{eq}}$

Where $i_s$ is the current through the coils in series combination.

Substitute the values into the formula

$$\begin{gathered} i_s=\frac{220V}{88\Omega } \\ {i}_s=2.5A \end{gathered}$$

Step 5 - Finding the equivalent resistance when connected in a parallel combination.

Consider $R_{eq\text{'}}$ be the equivalent resistance of the parallel combination

The equivalent resistance in the parallel combination is given by

$$\begin{gathered} \frac{1}{R_{eq\text{'}}}=\frac{1}{R_P}+\frac{1}{R_Q} \\ \frac{1}{R_{eq\text{'}}}=\frac{1}{44\Omega }+\frac{1}{44\Omega } \\ \frac{1}{R_{eq\text{'}}}=\frac{1+1}{44\Omega } \\ {R}_{eq\text{'}}=\frac{44\Omega }{2} \\ {R}_{eq\text{'}}=22\Omega \end{gathered}$$

Step 6 - Calculate the current through the parallel circuit.

Consider $i_P$ be the current through the parallel combination

According to Ohm's law,

$i_P=\frac{V}{R_{eq\text{'}}}$.

Substitute the required values into the formula of $i_P$.

$$\begin{gathered} i_P=\frac{220V}{22\Omega } \\ =10A \end{gathered}$$

Final answer -

1. When coils are used separately, the current through each of them is $5A$.
2. In a series combination, the current flowing through the circuit is $2.5A$.
3. In parallel combination, the current flowing through the circuit current is $10A$.