Advertisements
Advertisements
प्रश्न
A solenoid having inductance 4.0 H and resistance 10 Ω is connected to a 4.0 V battery at t = 0. Find (a) the time constant, (b) the time elapsed before the current reaches 0.63 of its steady-state value, (c) the power delivered by the battery at this instant and (d) the power dissipated in Joule heating at this instant.
Advertisements
उत्तर
Given:-
Inductance, L = 4.0 H
Resistance, R = 10 Ω
Emf of the battery, E = 4 V
(a) Time constant
\[\tau = \frac{L}{R} = \frac{4}{10} = 0 . 4 s\]
(b) As the current reaches 0.63 of its steady-state value, i = 0.63 i0.
Now,
0.63 i0 = i0(1 − e−t/τ)
⇒ e−t/τ = 1 − 0.063 = 0.37
⇒ ln e−t/τ = ln 0.37
`rArr -t/tau=-0.9942`
⇒ t = 0.942 × 0.4
= 0.3977 = 0.4 s
(c) The current in the LR circuit at an instant is given by
i = i0(1 − e−t/τ)
\[= \frac{4}{10}(1 - e^{- 0 . 4/0 . 4} )\]
= 0.4 × 0.6321
= 0.2528 A
Power delivered, P = Vi
⇒ P = 4 × 0.2528
= 1.01 = 1 W
(d) Power dissipated in Joule heating, P' = i2R
⇒ P' = (0.2258)2 × 10
= 0.639 = 0.64 W
APPEARS IN
संबंधित प्रश्न
Why does current in a steady state not flow in a capacitor connected across a battery? However momentary current does flow during charging or discharging of the capacitor. Explain.
A source of ac voltage v = v0 sin ωt, is connected across a pure inductor of inductance L. Derive the expressions for the instantaneous current in the circuit. Show that average power dissipated in the circuit is zero.
An inductor-coil of inductance 17 mH is constructed from a copper wire of length 100 m and cross-sectional area 1 mm2. Calculate the time constant of the circuit if this inductor is joined across an ideal battery. The resistivity of copper = 1.7 × 10−8 Ω-m.
An LR circuit having a time constant of 50 ms is connected with an ideal battery of emf ε. find the time elapsed before (a) the current reaches half its maximum value, (b) the power dissipated in heat reaches half its maximum value and (c) the magnetic field energy stored in the circuit reaches half its maximum value.
Answer the following question.
Draw the diagram of a device that is used to decrease high ac voltage into a low ac voltage and state its working principle. Write four sources of energy loss in this device.
Choose the correct answer from given options
The phase difference between the current and the voltage in series LCR circuit at resonance is
A series LCR circuit with L = 0.12 H, C = 480 nF, R = 23 Ω is connected to a 230 V variable frequency supply.
(a) What is the source frequency for which current amplitude is maximum. Obtain this maximum value.
(b) What is the source frequency for which average power absorbed by the circuit is maximum. Obtain the value of this maximum power.
(c) For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these frequencies?
(d) What is the Q-factor of the given circuit?
In a series LCR circuit supplied with AC, ______.
Assertion: When the frequency of the AC source in an LCR circuit equals the resonant frequency, the reactance of the circuit is zero, and so there is no current through the inductor or the capacitor.
Reason: The net current in the inductor and capacitor is zero.
A series LCR circuit containing 5.0 H inductor, 80 µF capacitor and 40 Ω resistor is connected to 230 V variable frequency ac source. The angular frequencies of the source at which power transferred to the circuit is half the power at the resonant angular frequency are likely to be ______.
Which of the following combinations should be selected for better tuning of an LCR circuit used for communication?
As the frequency of an ac circuit increases, the current first increases and then decreases. What combination of circuit elements is most likely to comprise the circuit?
- Inductor and capacitor.
- Resistor and inductor.
- Resistor and capacitor.
- Resistor, inductor and capacitor.
For an LCR circuit driven at frequency ω, the equation reads
`L (di)/(dt) + Ri + q/C = v_i = v_m` sin ωt
- Multiply the equation by i and simplify where possible.
- Interpret each term physically.
- Cast the equation in the form of a conservation of energy statement.
- Integrate the equation over one cycle to find that the phase difference between v and i must be acute.
A series LCR circuit driven by 300 V at a frequency of 50 Hz contains a resistance R = 3 kΩ, an inductor of inductive reactance XL = 250 πΩ, and an unknown capacitor. The value of capacitance to maximize the average power should be ______.
Draw the phasor diagram for a series LRC circuit connected to an AC source.
Three students, X, Y and Z performed an experiment for studying the variation of ac with frequency in a series LCR circuit and obtained the graphs as shown below. They all used
- an AC source of the same emf and
- inductance of the same value.
- Who used minimum resistance?
- In which case will the quality Q factor be maximum?
- What did the students conclude about the nature of impedance at resonant frequency (f0)?
- An ideal capacitor is connected across 220 V, 50 Hz, and 220 V, 100 Hz supplies. Find the ratio of current flowing through it in the two cases.
Which of the following combinations should be selected for better tuning of an L-C-R circuit used for communication?
A 20Ω resistance, 10 mH inductance coil and 15µF capacitor are joined in series. When a suitable frequency alternating current source is joined to this combination, the circuit resonates. If the resistance is made \[\frac {1}{3}\] rd, the resonant frequency ______.
A resistance of 200Ω and an inductor of \[\frac {1}{2π}\]Н are connected in series to a.c. voltage of 40 V and 100 Hz frequency. The phase angle between the voltage and current is ______.
