Question

Complex numbers are used in electronics to describe the current in an electric circuit. Ohm's law relates the current in a circuit, I, in amperes, the voltage of the circuit, E, in volts, and the resistance of the circuit, R, in ohms, by the formula E - IR. Solve the problem using this formula.

Find E, the voltage of a circuit, if I = (2 + 5i) amperes and R = (8 + 2i) ohms.

Answer 1

The voltage E of a circuit with current I = (2 + 5i) amperes and resistance R = (8 + 2i) ohms, calculated using Ohm's law V = IR, is E = 6 + 44i volts.

To find the voltage of a circuit using Ohm's law, which is V = IR, where V is the voltage, I is the current in amperes, and R is the resistance in ohms, you simply multiply the complex numbers for the current and resistance together.

Given that I = (2 + 5i) amperes and R = (8 + 2i) ohms, we calculate the voltage (E) as follows:

Multiply the real parts: 2 * 8 = 16

Multiply the imaginary parts: 5i * 2i = 10i^2 (and since i^2 = -1, this becomes -10)

Multiply the cross terms: 2 * 2i + 5i * 8 = 4i + 40i = 44i

So, E = (16 - 10) + 44i = 6 + 44i volts.

Therefore, the voltage E of the circuit is 6 + 44i volts.