Complex numbers are often used when dealing with alternating current (AC) circuits. In the equation V = IZ, V is voltage, I is current, and Z is a value known as impedance. If V = 1+i and Z=2-$, find I.
I=1/5+3/5i
I=V/Z=(1+j)/(2-j)
multipy num and denom by 2+j
(1+j)(2+j)/3=(2+j+2j-1)/3=(1+3j)/3
To find the current (I), we need to rearrange the equation V = IZ and solve for I.
Given:
V = 1 + i (Complex number representing voltage)
Z = 2 - $ (Complex number representing impedance)
Let's substitute the values into the equation V = IZ:
1 + i = I(2 - $)
Now, let's distribute I to each term inside the parentheses:
1 + i = 2I - I$
To isolate I, let's move the terms involving I to the left side:
2I - I$ = 1 + i
Next, let's group the terms involving I on the left side:
(2 - $)I = 1 + i
To solve for I, we can divide both sides by (2 - $):
I = (1 + i) / (2 - $)
Thus, the current (I) is equal to (1 + i) divided by (2 - $).
To find the value of I in the equation V = IZ, where V is the voltage, I is the current, and Z is the impedance, you can substitute the given values of V and Z into the equation and solve for I.
Given: V = 1 + i and Z = 2 - $
Substituting the values into the equation V = IZ:
1 + i = I(2 - $)
Expanding the equation:
1 + i = 2I - $I
Rearranging the equation to isolate the imaginary terms:
i + $I = 2I - 1
Now, separate the real and imaginary parts of the equation:
$I = 2I - 1 (for the real parts)
i = 0 (for the imaginary parts)
Solving the real part of the equation:
$I = 2I - 1
Rearranging the equation:
$I - 2I = -1
Consolidating like terms:
-$I = -1
Dividing both sides by -$:
I = -1 / -$
Note that -$ is the negative of the imaginary unit i, so -$ = -i.
Substituting this value:
I = -1 / -i
Now, to simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part:
I = -1 * i / (-i * i)
Remember that i * i equals -1:
I = -i / (-(-1))
Simplifying further:
I = -i / 1
Which simplifies to:
I = -i
Therefore, the value of I is -i.