Advertisements
Advertisements
प्रश्न
How many time constants will elapse before the energy stored in the capacitor reaches half of its equilibrium value in a charging RC circuit?
Advertisements
उत्तर
The equilibrium value of energy in a capacitor,
\[U = \frac{1}{2}\frac{Q^2}{C},\] where Q is the steady state charge.
Let q be the charge for which energy reaches half its equilibrium. Then,
\[\frac{1}{2}\frac{q^2}{C} = \frac{1}{2}U\]
\[ \Rightarrow \frac{1}{2}\frac{q^2}{C} = \frac{1}{2}\left( \frac{1}{2}\frac{Q^2}{C} \right)\]
\[ \Rightarrow q = \sqrt{\frac{Q^2}{2}}\]
The growth of charge in a capacitor,
\[q = Q\left( 1 - e^{- \frac{t}{RC}} \right)\]
\[ \because q = \sqrt{\frac{Q^2}{2} ,}\]
\[ \sqrt{\frac{Q^2}{2}} = Q\left( 1 - e^{- \frac{t}{RC}} \right)\]
\[ \Rightarrow \frac{Q}{\sqrt{2}} = Q\left( 1 - e^{- \frac{t}{RC}} \right)\]
\[ \Rightarrow e^{- \frac{t}{RC}} = \left( 1 - \frac{1}{\sqrt{2}} \right)\]
\[ \Rightarrow - \frac{t}{RC} = \ln\left( 1 - \frac{1}{\sqrt{2}} \right)\]
Let t = nRC
\[So, - \frac{nRC}{RC} = \ln\left( 1 - \frac{1}{\sqrt{2}} \right)\]
\[ \Rightarrow n = 1 . 23\]
APPEARS IN
संबंधित प्रश्न
Explain what would happen if the capacitor given in previous question a 3 mm thick mica sheet (of dielectric constant = 6) were inserted between the plates,
- While the voltage supply remained connected.
- After the supply was disconnected.
A 600 pF capacitor is charged by a 200 V supply. It is then disconnected from the supply and is connected to another uncharged 600 pF capacitor. How much electrostatic energy is lost in the process?
A capacitor C1 of capacitance 1 μF and a capacitor C2 of capacitance 2 μF are separately charged by a common battery for a long time. The two capacitors are then separately discharged through equal resistors. Both the discharge circuits are connected at t = 0.
(a) The current in each of the two discharging circuits is zero at t = 0.
(b) The currents in the two discharging circuits at t = 0 are equal but not zero.
(c) The currents in the two discharging circuits at t = 0 are unequal.
(d) C1 loses 50% of its initial charge sooner than C2 loses 50% of its initial charge.
A capacitance C, a resistance R and an emf ε are connected in series at t = 0. What is the maximum value of (a) the potential difference across the resistor (b) the current in the circuit (c) the potential difference across the capacitor (d) the energy stored in the capacitor (e) the power delivered by the battery and (f) the power converted into heat?
Two capacitors of capacitances 4⋅0 µF and 6⋅0 µF are connected in series with a battery of 20 V. Find the energy supplied by the battery.
By evaluating ∫i2Rdt, show that when a capacitor is charged by connecting it to a battery through a resistor, the energy dissipated as heat equals the energy stored in the capacitor.
Find the charge on each of the capacitors 0.20 ms after the switch S is closed in the figure.
A capacitor with stored energy 4⋅0 J is connected with an identical capacitor with no electric field in between. Find the total energy stored in the two capacitors.
A capacitor of capacitance 100 μF is connected across a battery of emf 6 V through a resistance of 20 kΩ for 4 s. The battery is then replaced by a thick wire. What will be the charge on the capacitor 4 s after the battery is disconnected?
Consider the situation shown in figure. The switch is closed at t = 0 when the capacitors are uncharged. Find the charge on the capacitor C1 as a function of time t.
A metal sphere of radius R is charged to a potential V.
- Find the electrostatic energy stored in the electric field within a concentric sphere of radius 2 R.
- Show that the electrostatic field energy stored outside the sphere of radius 2 R equals that stored within it.
Figure shows two identical parallel plate capacitors connected to a battery through a switch S. Initially, the switch is closed so that the capacitors are completely charged. The switch is now opened and the free space between the plates of the capacitors is filled with a dielectric of dielectric constant 3. Find the ratio of the initial total energy stored in the capacitors to the final total energy stored.
Obtain the expression for the energy stored in a capacitor connected across a dc battery.
If the p. d. across a capacitor is increased from 10 V to 30 V, then the energy stored with the capacitor ____________.
An air-filled parallel plate capacitor has a uniform electric field `overset(->)("E")` in the space between the plates. If the distance between the plates is 'd' and the area of each plate is 'A', the energy stored in the capacitor is ______
(∈0 = permittivity of free space)
Prove that, if an insulated, uncharged conductor is placed near a charged conductor and no other conductors are present, the uncharged body must be intermediate in potential between that of the charged body and that of infinity.
A fully charged capacitor C with initial charge q0 is connected to a coil of self-inductance L at t = 0. The time at which the energy is stored equally between the electric and magnetic fields is ______.
Derive an expression for energy stored in a capacitor.
