Question

5. 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 the problem, we need to analyze the circuit involving a 12-ohm resistor connected in parallel with a series combination of an 8-ohm and a 16-ohm resistor. We also know that the voltage drop across the 8-ohm resistor is 48 volts.

### Step 1: Calculate the current through the 8-ohm resistor.
Using Ohm's Law (\( V = I \times R \)), we can calculate the current \( I_1 \) through the 8-ohm resistor:

\[
I_1 = \frac{V}{R} = \frac{48 \, \text{V}}{8 \, \Omega} = 6 \, \text{A}
\]

### Step 2: Calculate the voltage across the series combination.
The total voltage across the series combination of the 8-ohm and 16-ohm resistors can be found by using the fact that the same current flows through both resistors. The current \( I_1 \) is already calculated as 6 A.

\[
V_{total} = V_{8\Omega} + V_{16\Omega}
\]
\[
V_{8\Omega} = 48 \, \text{V}
\]
\[
V_{16\Omega} = I_1 \times R_{16\Omega} = 6 \, \text{A} \times 16 \, \Omega = 96 \, \text{V}
\]

Adding these voltages together gives the total voltage across the series combination:

\[
V_{total} = 48 \, \text{V} + 96 \, \text{V} = 144 \, \text{V}
\]

### Step 3: Find the impedance of the parallel combination.
Next, we will find the equivalent resistance of the parallel combination of the 12-ohm resistor and the equivalent resistance of the series combination of the 8-ohm and 16-ohm resistors.

The equivalent resistance of the series combination \( R_{series} \) is:

\[
R_{series} = 8 \, \Omega + 16 \, \Omega = 24 \, \Omega
\]

Now, we can determine the total equivalent resistance \( R_{eq} \) of the parallel circuit:

\[
\frac{1}{R_{eq}} = \frac{1}{R_{12}} + \frac{1}{R_{series}} = \frac{1}{12 \, \Omega} + \frac{1}{24 \, \Omega}
\]
Finding a common denominator (which is 24):
\[
\frac{1}{R_{eq}} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8}
\]
Thus,
\[
R_{eq} = 8 \, \Omega.
\]

### Step 4: Calculate the total current from the emf.
Now that we have found the equivalent resistance of the whole circuit, we can find the total current \( I_{total} \) using the total voltage:

\[
I_{total} = \frac{V_{total}}{R_{eq}} = \frac{144 \, \text{V}}{8 \, \Omega} = 18 \, \text{A}.
\]

### Conclusion
We now have:
- Total impressed emf (voltage) across the system: **144 volts**
- Total current drawn from the source: **18 A**