Question

A 12-ohm resistor is connected in parallel with a series combination of resistors of 8 and 16 ohms. If the drop across the 8-ohm resistor is 48 volts, determine the total impressed emf and the total current.

Answer 1

To solve this problem, we first analyze the circuit's configuration based on the given information. We have a 12-ohm resistor in parallel with a series combination of an 8-ohm and a 16-ohm resistor. The drop across the 8-ohm resistor is 48 volts.

### Step 1: Determine Current through the 8-ohm Resistor
Since we know the voltage drop across the 8-ohm resistor is 48 volts, we can determine the current flowing through it using Ohm's law:

\[
I_8 = \frac{V}{R} = \frac{48 \text{ volts}}{8 \text{ ohms}} = 6 \text{ A}
\]

### Step 2: Determine Total Voltage Across the Series Combination
The total voltage across the series combination of the 8-ohm and 16-ohm resistors can be found by calculating the voltage drop across the 16-ohm resistor. Since the same current flows through both resistors in series, we can find the current through the 16-ohm resistor:

\[
I_{16} = I_8 = 6 \text{ A}
\]

Now we can find the voltage drop across the 16-ohm resistor:

\[
V_{16} = I_{16} \times R_{16} = 6 \text{ A} \times 16 \text{ ohms} = 96 \text{ volts}
\]

Thus, the total voltage across the series combination (8 ohms + 16 ohms) is:

\[
V_{\text{total series}} = V_{8} + V_{16} = 48 \text{ volts} + 96 \text{ volts} = 144 \text{ volts}
\]

### Step 3: Calculate the Impressed EMF
Since the 12-ohm resistor is in parallel with the series combination, the total impressed voltage (or EMF) is equal to the voltage across the parallel section. Therefore, the total impressed EMF is:

\[
E = V_{\text{total series}} = 144 \text{ volts}
\]

### Step 4: Calculate the Total Current
To find the total current flowing from the source, we need to determine the total current through both branches of the circuit.

#### Current through the 12-ohm Resistor
Using the voltage across the 12-ohm resistor (which is equal to the impressed EMF), we can calculate the current through the 12-ohm resistor:

\[
I_{12} = \frac{E}{R_{12}} = \frac{144 \text{ volts}}{12 \text{ ohms}} = 12 \text{ A}
\]

#### Total Current from the Source
The total current supplied by the source will be the sum of the currents in the parallel branches (the current through the 12-ohm resistor and the combined current through the series resistors):

\[
I_{\text{total}} = I_{12} + I_8 = 12 \text{ A} + 6 \text{ A} = 18 \text{ A}
\]

### Conclusion
The total impressed EMF is **144 volts**, and the total current is **18 A**.