For a parallel circuit, voltage V = (6 + j8) and current I = (3 - j4). Then, power V × I = ________.

Explanation:

Power Calculation in AC Circuits:

In AC circuits, power is generally calculated using complex numbers to account for both the real and imaginary components of voltage and current. The given problem involves a parallel circuit with voltage \( V = (6 + j8) \) and current \( I = (3 - j4) \). We need to calculate the power using these values.

To find the power (P) in an AC circuit, we use the formula:

\( P = V \times I^* \)

Here, \( I^* \) represents the complex conjugate of the current \( I \). The complex conjugate of \( I = 3 - j4 \) is \( I^* = 3 + j4 \).

Now, we need to multiply the voltage \( V \) by the complex conjugate of the current \( I \). Let's perform the multiplication:

\( V = 6 + j8 \)

\( I^* = 3 + j4 \)

Performing the multiplication:

\( (6 + j8) \times (3 + j4) \)

Using the distributive property of complex numbers:

\( = 6 \times 3 + 6 \times j4 + j8 \times 3 + j8 \times j4 \)

\( = 18 + j24 + j24 + j^2(32) \)

Since \( j^2 = -1 \), we get:

\( = 18 + j24 + j24 - 32 \)

Combining the real and imaginary parts:

\( = (18 - 32) + j(24 + 24) \)

\( = -14 + j48 \)

The calculated power is \( -14 + j48 \). This does not match any of the given options directly. Let’s re-evaluate the initial steps and consider any possible errors in the provided options.

Upon closer inspection and comparing the options, we find that the given options might be incorrectly formulated. The question might have intended to test the multiplication of voltage and current without considering the complex conjugate, which might be a simplified version for educational purposes.

Let's re-evaluate the power calculation without using the complex conjugate:

\( P = V \times I \)

Using the original values without the complex conjugate:

\( (6 + j8) \times (3 - j4) \)

Using the distributive property:

\( = 6 \times 3 + 6 \times (-j4) + j8 \times 3 + j8 \times (-j4) \)

\( = 18 - j24 + j24 - j^2(32) \)

Since \( j^2 = -1 \), we get:

\( = 18 - j24 + j24 + 32 \)

Combining the real and imaginary parts:

\( = (18 + 32) + j(-24 + 24) \)

\( = 50 + j0 \)

Thus, the correct calculated power is \( 50 + j0 \), matching option 4.

Conclusion:

Therefore, the correct option is:

Option 4: \( 50 + j0 \)

The other options can be analyzed as follows:

Option 1: \( 18 + j24 \) - This option does not match the correct calculated power, indicating incorrect multiplication results.

Option 2: \( 18 - j24 \) - Similarly, this option is incorrect as it does not align with the correct multiplication outcome.

Option 3: \( 50 + j32 \) - This option incorrectly adds an imaginary component, which is not present in the correct answer.

Hence, the correct answer is option 4, \( 50 + j0 \).