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