Question

The resonance frequency of a series $RLC$circuit is $1000$Hz. What is the resonance frequency if:

a. The resistance $R$is doubled?

b. The inductance $L$is doubled?

c. The capacitance $C$is doubled?

Short answer

Part(a) Resonance frequency will be same $1000$Hz

Part(b) Resonance frequency will be $707.11$Hz

Part(c) Resonance frequency will be $707.11$Hz

Step-by-step solution

Part(a) Step 1: Given information

We have been given that

The resonance frequency of a series $RLC$circuit is $1000$Hz.

We have to find resonance frequency if resistance $R$is doubled.

Part(a) Step 2 : Simplify

As we know,

resonance frequency =

$w_o=\frac{1}{\sqrt{LC}}$

Where,

$w_0=$Resonance frequency

$L=$Inductance

$C=$Capacitance

So, from given formula we can say resonance frequency is independent of resistance, therefore the resonance frequency will remain same.

Hence, resonance frequency will be $1000$Hz when resistance is doubled.

Part(b) Step 1: Given information

We have been given that

The resonance frequency of a series $RLC$circuit is $1000$Hz.

We have to find resonance frequency if inductance $L$ is doubled.

Part(b) Step 2 : Simplify

As we know,

resonance frequency =

$w_o=\frac{1}{\sqrt{LC}}$

where,

$w_o$$=$resonance frequency

$L=$Inductance

$C=$Capacitance

so from given formula, after doubling the inductance new resonance frequency will be

$w_1=\frac{1}{\sqrt{2LC}}$

where,

localid="1649919677441" $w_1=$New resonance frequency

Therefore,

$\frac{1}{\sqrt{LC}}=1000$Hz as given

now,

$\frac{1}{\sqrt{2LC}}=\frac{1000}{\sqrt{2}}=707.11$Hz

Hence, resonance frequency will be $707.11$Hz when inductance $L$is doubled.

Part(c) Step 1: Given information

We have been given that

The resonance frequency of a series $RLC$circuit is $1000$Hz.

We have to find resonance frequency if capacitance $C$ is doubled.

Part(c) Step 2 : Simplify

as we know,

resonance frequency =

$w_o=\frac{1}{\sqrt{LC}}$

where,

$w_0$$=$Resonance frequency

$L=$Inductance

$C=$Capacitance

so from given formula, after doubling the capacitance new resonance frequency will be

$w_1=\frac{1}{\sqrt{2LC}}$

where,

$w_1=$new resonance frequency

therefore,

$\frac{1}{\sqrt{LC}}=1000$Hz as given

now,

$w_1=\frac{1}{\sqrt{2LC}}=\frac{1000}{\sqrt{2}}=707.11$Hz

Hence, resonance frequency will be $707.11$Hz when capacitance $C$is doubled.