The fractional half-width Δ ω d of a resonance curve is the width of the curve at half the maximum value of I . (a) Show that Δ ω d ω = R ( 3 C L ) 1 / 2 , where ω is the angular frequency at resonance. (b) What happens to the ratio Δ ω d ω with an increase in R ?

In a R L C circuit, the current amplitude is maximum at resonance, where the driven angular frequency is equal to the natural angular frequency. It the falls off rapidly away resonance. Case is shown in figure, (a) We use the expressions of sidebands of the curve as, ω 1 = + 3 C R + 3 C 2 R 2 + 4 L C 2 L C , ω 2 = - 3 C R + 3 C 2 R 2 + 4 L C 2 L C . The resonance angular frequency is ω = 1 L C . Thus, fractional band width is Δ ω d ω = ω 1 - ω 2 ω = 2 3 C R L C 2 L C = R 3 C L (b) Note that the value of Δ ω d ω increase linearly with R that is, the larger the resistance, the broader the peak.

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IMPORTANT

Earn 100

In the given figure, $R = 25.0 Ω, C = 4.70 μ F,$ $L = 25.0 mH .$ The generator provides an emf with RMS voltage $75.0 V$ and frequency $550 Hz$ .

(a) What is the RMS current?

What is the RMS voltage across
(b) $R, (c) C, (d) L, (e) C$ $L$ and together, and
(f) $R, C,$ $L$ and together? At what average rate is energy dissipated by $(g) R, (h) C,$ and
(i) $L ?$

Important Questions on Electromagnetic Oscillations and Alternating Current

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IMPORTANT

Principles Of Physics International Student Version>Chapter 31 - Electromagnetic Oscillations and Alternating Current>Problems>Q 40

A series circuit containing inductance

L_{1}

and capacitance

C_{1}

oscillates at angular frequency

ω .

A second series circuit, containing inductance

C

and capacitance

C_{2}

, oscillates at the same angular frequency. In terms of

ω,

what is the angular frequency of oscillation of a series circuit containing all four of these elements? Neglect resistance. (Hint: Use the formulas for equivalent capacitance and equivalent inductance)

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IMPORTANT

Principles Of Physics International Student Version>Chapter 31 - Electromagnetic Oscillations and Alternating Current>Problems>Q 41

A variable capacitor with a range from $10$ to $410 pF$ is used with a coil to form a variable-frequency $L C$ circuit to tune the input to a radio.

(a) What is the ratio of maximum frequency to minimum frequency that can be obtained with such a capacitor? If this circuit is to obtain frequencies from $0.54 MHz$ to $1.60 MHz,$ the ratio computed in (a) is too large. By adding a capacitor in parallel to the variable capacitor, this range can be adjusted. To obtain the desired frequency range,

(b) what capacitance should be added and

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IMPORTANT

Principles Of Physics International Student Version>Chapter 31 - Electromagnetic Oscillations and Alternating Current>Problems>Q 42

What is the maximum value of an ac voltage whose rms value is

220 V ?

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IMPORTANT

Principles Of Physics International Student Version>Chapter 31 - Electromagnetic Oscillations and Alternating Current>Problems>Q 43

In an oscillating

L C

circuit,

L = 25.0 mH

and

C = 2.89 μ F

. At time

t = 0

the current is

9.20 mA

, the charge on the capacitor is

3.80 μ C,

and the capacitor is charging. What are

(a)

the total energy in the circuit, (b) the maximum charge on the capacitor, and
(c) the maximum current? (d) If the charge on the capacitor is given by

q = Q \cos (ω t + ϕ),

what is the phase angle

ϕ ?

(e) Suppose the data are the same, except that the capacitor is discharging at

t = 0 .

What then is

ϕ ?

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IMPORTANT

Principles Of Physics International Student Version>Chapter 31 - Electromagnetic Oscillations and Alternating Current>Problems>Q 44

An AC generator has emf

E = E_{m}

sin

ω_{d} t,

E_{m} = 30.0 V

with and

ω_{d} = 377 rad s^{- 1}

. It is connected to a

12.7 H

inductor.

(a)

What is the maximum value of the current? (b) When the current is a maximum, what is the emf of the generator?
(c) When the emf of the generator is

- 15.0 V

and increasing in magnitude, what is the current?

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IMPORTANT

Principles Of Physics International Student Version>Chapter 31 - Electromagnetic Oscillations and Alternating Current>Problems>Q 45

An electric motor has an effective resistance of

61.0 Ω

and an inductive reactance of

52.0 Ω

when working under load. The RMS voltage across the alternating source is

420 V

. Calculate the RMS current.

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IMPORTANT

Principles Of Physics International Student Version>Chapter 31 - Electromagnetic Oscillations and Alternating Current>Problems>Q 46

0.25 kg

body oscillates in

SHM

on a spring that, when extended

2.0 mm

from its equilibrium position, has

8.0 N

an restoring force. What are (a) the angular frequency of oscillation, (b) the period of oscillation, and (c) the capacitance of

L C

an circuit with the same period if

L

5.0 H

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IMPORTANT

Principles Of Physics International Student Version>Chapter 31 - Electromagnetic Oscillations and Alternating Current>Problems>Q 47

The fractional half-width

Δ ω_{d}

of a resonance curve is the width of the curve at half the maximum value of

I .

(a) Show that

\frac{Δ ω_{d}}{ω} = R (\frac{3 C}{L})^{1 / 2},

where

ω

is the angular frequency at resonance. (b) What happens to the ratio

\frac{Δ ω_{d}}{ω}

with an increase in

R ?

Important Questions on Electromagnetic Oscillations and Alternating Current

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