i) You are testing the voltage across a capacitor in an AC circuit. The instrument you are using indicates this voltage to have a magnitude of 100 V and a phase angle of 45 degrees. Convert this voltage into a complex number.

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1. The given voltage magnitude is $$ 100 $$ V and the phase angle is $$ 45^\circ $$.

2. In polar form, the voltage is written as:
   $$
   100 \angle 45^\circ
   $$

3. To convert to rectangular form, we use the formulas:
   $$
   V = V_m (\cos \theta + j \sin \theta)
   $$
   where $$ V_m = 100 $$ V and $$ \theta = 45^\circ $$.

4. Evaluate the trigonometric functions:
   $$
   \cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}
   $$

5. Substitute the values:
   $$
   V = 100 \left(\frac{\sqrt{2}}{2} + j \frac{\sqrt{2}}{2}\right)
   $$

6. Simplify the expression:
   $$
   V = 50\sqrt{2} + j50\sqrt{2}
   $$

Thus, the complex representation of the voltage is:
$$
50\sqrt{2} + j50\sqrt{2}
$$
The voltage can be expressed as $$ 50\sqrt{2} + j50\sqrt{2} $$.

i) You are testing the voltage across a capacitor in an AC circuit. The instrument you are using indicates this voltage to have a magnitude of 100 V and a phase angle of 45 degrees. Convert this voltage into a complex number.

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