(a) Reactance is the opposition in ohms to th alternating current by a capacitor or inductor. It is defined by
\(X_{C} = \frac{V}{I}\) or \(X_{L} = \frac{V}{I}\)
\(X_{C} = \frac{1}{2 \pi f c}\) or \(X_{L} = 2 \pi fl\)
V - voltage (V), c - current, f - frequency (Hz), I - inductance (H), c - capacitance (F).
Impedance is the opposition in ohms to the alternating current by combination of resistor, inductor or capacitor. It is defined as:
\(Z = \sqrt{R^{2} + X_{L} ^{2}}\)
\(Z = \sqrt{R^{2} + X^{2}}\)
\(Z = \sqrt{R^{2} + (X_{C} - X_{L})^{2}}\)
(b)
Hence using ,
\(V = \sqrt{V_{R} ^{2} + (V_{L} - V_{C})^{2}}\)
\(240 = \sqrt{140^{2} + (50 - V_{C})^{2}}\)
Square both sides,
\(57600 = 19600 + (50 - V_{C})^{2} \implies 57600 - 19600 = (50 - V_{C})^{2}\)
\(38000 = (50 - V_{C})^{2}\)
\(\sqrt{38000} = 50 - V_{C}\)
\(\pm 194.936 = 50 - V_{C} \implies V_{C} = \pm 194.936 + 50\)
\(V_{C} = -144.936 ; V_{C} = 244.936v\)
The voltage cannot be negative, hence, \(V_{C} = 244.936v\)
(ii) Using \(X_{C} = \frac{1}{2 \pi fc} = \frac{V}{I}\)
\(\frac{244.936}{10} = \frac{1}{2\pi \times 50 \times c}\)
\(24.4936 = \frac{1}{2\pi \times 50 \times c} \implies c = \frac{1}{2\pi \times 50 \times 24.4936}\)
\(c = \frac{1}{7694.89} = 0.00013F = 130\mu F\)