Question

A coil of inductance L, a capacitor of capacitance C and a resistor of resistance R are connected in series with an alternating source of emf E = E₀ sin ωt . Write expressions for
(i) Total impedance of circuit
(ii) Frequency of source emf for which circuit will show resonance.

Answer 1

this is only the first part of the question

Answer 2

Answer:

The impedance of the circuit is  $\sqrt{R^{2} +(w_{L}-\frac{1}{w} _{c}) ^{2} }$. The frequency at which resonance takes place is  $f=(\frac{1}{2\pi } )*\sqrt{(\frac{1}{LC} )}$

Explanation:

• In a series RLC circuit, resistor, inductor, and capacitor are connected in a serial way.
• For resistance, the current and voltage are in phase.
• For the inductor, the current and voltage are out of phase. Voltage leads the current.
• For capacitors, the current and voltage are out of phase. Voltage lags the current.

Step 1:

In a series RLC circuit,

For resistance, $V_{R} =I*R$

For inductor, $V_{L} =I*X_{L}$

For capacitor, $V_{c} =I*X_{c}$

where, $X_{L}$ and $X_{c}$ are reactance of inductor and capacitor respectively.

As voltages for inductor and capacitor are $180$ degrees opposite to each other.

So, net voltage due to both $= V_{L} -V_{c}$

                                              $= I*(X_{L}-X_{c})$

The voltage across the resistor is $90$ degrees to this net voltage.

So, total voltage for combination, V $=\sqrt{R^{2} +(X_{L}-X_{c}) ^{2} } *(I)$

As impedance $=\frac{V}{I}$

Net impedance $= \sqrt{R^{2} +(w_{L}-\frac{1}{w} _{c}) ^{2} }$

Step 2:

During resonance, the inductive reactance and capacitive reactance are the same in magnitude but $180$ out of phase.

So, $X_{L} =X_{c}$

ω$L$ $=$ ($1$/ω) $*$ $C$

$(2\pi f)*L=\frac{1}{2\pi fC}$

$f=(\frac{1}{2\pi } )*\sqrt{(\frac{1}{LC} )}$

Thus, the resonance frequency is  $f=(\frac{1}{2\pi } )*\sqrt{(\frac{1}{LC} )}$ .