(a)Explain resonance frequency as applied in RLC series Circuit. (ii) Sketch a diagram to illustrate the variation of frequency, f, with the resistance, R, ...

(a)(i) Resonance in a series RLC circuit

Resonance frequency, \(f_0\), is the frequency at which the inductive reactance equals the capacitive reactance:

\[X_L=X_C\]

At this frequency the resultant reactance is zero, so the impedance is minimum and equal to the resistance, \(Z=R\). The current is therefore maximum.

(a)(ii) Variation of \(R\), \(X_C\) and \(X_L\) with frequency

At the resonant frequency \(f_0\), the \(X_L\) and \(X_C\) curves intersect. The resistance \(R\) remains constant as frequency changes.

(a)(iii) Phase relationship between current and supply voltage

1. When \(f=f_0\), \(X_L=X_C\); the current is in phase with the supply voltage.
2. When \(f<f_0\), \(X_C>X_L\); the circuit is capacitive and the current leads the supply voltage.
3. When \(f>f_0\), \(X_L>X_C\); the circuit is inductive and the current lags the supply voltage.

(b)(i) Mutual inductance

Mutual inductance is the induced e.m.f. in one coil per unit rate of change of current in a nearby magnetically linked coil:

\[M=\frac{E}{\mathrm{d}I/\mathrm{d}t}\]

(b)(ii) Angular speed of the generator coil

For a rotating coil generator,

\[E_0=NBA\omega\]

Radius of coil:

\[r=\frac{8.0\times10^{-2}}{2}=4.0\times10^{-2}\ \text{m}\]

Area of coil:

\[A=\pi r^2=3.142(4.0\times10^{-2})^2=5.0272\times10^{-3}\ \text{m}^2\]

Hence,

\[\omega=\frac{E_0}{NBA}=\frac{480}{500\times0.25\times5.0272\times10^{-3}}\]

\[\omega=7.64\times10^2\ \text{rad s}^{-1}\]

Angular speed = \(7.64\times10^2\ \text{rad s}^{-1}\) (approximately \(763.8\ \text{rad s}^{-1}\)).