Advertisements
Advertisements
Question
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.
Solution
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
RELATED QUESTIONS
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.
In a series LCR circuit, obtain the condition under which the impedance of the circuit is minimum ?
A series LCR circuit is connected to an ac source. Using the phasor diagram, derive the expression for the impedance of the circuit. Plot a graph to show the variation of current with frequency of the source, explaining the nature of its variation.
A series LCR circuit is connected to a source having voltage v = vm sin ωt. Derive the expression for the instantaneous current I and its phase relationship to the applied voltage.
Obtain the condition for resonance to occur. Define ‘power factor’. State the conditions under which it is (i) maximum and (ii) minimum.
Find the value of t/τ for which the current in an LR circuit builds up to (a) 90%, (b) 99% and (c) 99.9% of the steady-state value.
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.
Draw a labelled graph showing a variation of impedance of a series LCR circuit with frequency of the a.c. supply.
Answer the following question.
In a series LCR circuit connected across an ac source of variable frequency, obtain the expression for its impedance and draw a plot showing its variation with frequency of the ac source.
Use the expression for Lorentz force acting on the charge carriers of a conductor to obtain the expression for the induced emf across the conductor of length l moving with velocity v through a magnetic field B acting perpendicular to its length.
A series LCR circuit with R = 20 Ω, L = 1.5 H and C = 35 µF is connected to a variable-frequency 200 V ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?
Keeping the source frequency equal to the resonating frequency of the series LCR circuit, if the three elements, L, C and R are arranged in parallel, show that the total current in the parallel LCR circuit is minimum at this frequency. Obtain the current rms value in each branch of the circuit for the elements and source specified for this frequency.
In an L.C.R. series a.c. circuit, the current ______.
In a series LCR circuit the voltage across an inductor, capacitor and resistor are 20 V, 20 V and 40 V respectively. The phase difference between the applied voltage and the current in the circuit is ______.
At resonant frequency the current amplitude in series LCR circuit is ______.
The resonant frequency of a RF oscillator is 1 MHz and its bandwidth is 10 kHz. The quality factor will be :
In series LCR AC-circuit, the phase angle between current and voltage is
A series LCR circuit containing a 5.0 H inductor, 80 µF capacitors, and 40 Ω resistor is connected to a 230 V variable frequency ac source. The angular frequencies of the source at which power is transferred to the circuit are half the power at the resonant angular frequency are likely to be ______.
Three students, X, Y and Z performed an experiment for studying the variation of a.c. 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 220V, 50Hz, and 220V, 100Hz supplies. Find the ratio of current flowing through it in the two cases.
