Question

L, C and R are connected in series with an alternating current source. When there will be resonance in the circuit? What will be the nature of impedance during resonance?
OR
State the Faraday's laws of electromagnetic induction. Define self induction, state the Lenz's law regarding direction of self induced current. In a coil, due to change of current at the rate of 1 A/s an e.m.f. of 1 V is produced. What will be the coefficient of self inductance of the coil?

Hint:

Series resonance occurs when \(X_L=X_C\), i.e. \(\omega_0=\dfrac{1}{\sqrt{LC}}\), where the impedance falls to its minimum value \(Z=R\) (purely resistive). For the coil, use \(\varepsilon=L\dfrac{dI}{dt}\) to find \(L\).

Answer 1

Option 1: Series LCR Resonance

Step 1: Impedance of a series LCR circuit. For a resistance \(R\), inductor \(L\) and capacitor \(C\) in series across an AC source of angular frequency \(\omega\), the inductive reactance is \(X_L=\omega L\) and the capacitive reactance is \(X_C=\dfrac{1}{\omega C}\). The net impedance is \[ Z=\sqrt{R^{2}+\left(X_L-X_C\right)^{2}}=\sqrt{R^{2}+\left(\omega L-\dfrac{1}{\omega C}\right)^{2}}. \] Step 2: Condition for resonance. Resonance is the state in which the current \(I=\dfrac{V}{Z}\) is maximum, which happens when the impedance \(Z\) is minimum. From the formula, \(Z\) is least when the reactive part becomes zero: \[ X_L-X_C=0\ \Rightarrow\ \omega L=\dfrac{1}{\omega C}. \] Step 3: Resonant frequency. Solving the above, \[ \omega^{2}=\dfrac{1}{LC}\ \Rightarrow\ \omega_0=\dfrac{1}{\sqrt{LC}},\qquad f_0=\dfrac{1}{2\pi\sqrt{LC}}. \] So the circuit resonates at frequency \(f_0=\dfrac{1}{2\pi\sqrt{LC}}\). Step 4: Nature of impedance at resonance. Putting \(X_L=X_C\) in the impedance formula, \[ Z_{res}=\sqrt{R^{2}+0}=R. \] Hence at resonance the impedance is minimum and purely resistive (ohmic), equal to \(R\); the two reactances cancel. The current is therefore maximum, \(I_{max}=\dfrac{V}{R}\), the source voltage and current are in phase, and the power factor \(\cos\phi=\dfrac{R}{Z}=1\). \[\boxed{\omega_0=\dfrac{1}{\sqrt{LC}},\quad Z_{res}=R\ \text{(minimum, purely resistive)}}\]

Option 2: Faraday's Laws, Self Induction, Lenz's Law and Numerical

Faraday's laws of electromagnetic induction:
First law: Whenever the magnetic flux linked with a closed circuit changes, an e.m.f. (and hence an induced current in a closed loop) is set up in the circuit; it lasts only as long as the flux keeps changing.
Second law: The magnitude of the induced e.m.f. equals the rate of change of magnetic flux linkage: \[ \varepsilon=-\dfrac{d\Phi}{dt}\qquad(\text{for }N\text{ turns},\ \varepsilon=-N\dfrac{d\Phi}{dt}). \] The negative sign is Lenz's law.

Self induction: When the current in a coil changes, the flux linked with the coil itself changes, so an e.m.f. is induced in the same coil which opposes the change of current. This property is called self induction. Since flux linkage is proportional to current, \(N\Phi=L\,I\), giving \[ \varepsilon=-L\dfrac{dI}{dt}. \] The constant \(L\) is the coefficient of self induction (self inductance).

Lenz's law (direction of self induced current): The self induced current always flows in a direction that opposes the very change of current (increase or decrease) that produces it. It is a consequence of conservation of energy.

Numerical:
Step 1: Given \(\dfrac{dI}{dt}=1\ \text{A/s}\) and \(|\varepsilon|=1\ \text{V}\).
Step 2: Formula \(|\varepsilon|=L\dfrac{dI}{dt}\Rightarrow L=\dfrac{|\varepsilon|}{dI/dt}\).
Step 3: Substituting, \(L=\dfrac{1\ \text{V}}{1\ \text{A/s}}=1\ \text{H}\).
\[\boxed{L=1\ \text{henry}}\]