Advertisements
Advertisements
प्रश्न
A capacitor of capacitance C is connected to a battery of emf ε at t = 0 through a resistance R. Find the maximum rate at which energy is stored in the capacitor. When does the rate have this maximum value?
Advertisements
उत्तर
The rate of growth of charge for the capacitor,
\[q = \epsilon C \left(1 − e^\frac{- t}{RC}\right)\]
Let E be the energy stored inside the capacitor. Then,
\[E = \frac{q^2}{2C} = \frac{\epsilon^2 C^2}{2C} \left( 1 - e^{- \frac{t}{RC}} \right)^2 \]
\[ \Rightarrow E = \frac{\epsilon^2 C}{2} \left( 1 - e^{- \frac{t}{RC}} \right)^2\]
Let r be the rate of energy stored inside the capacitor. Then,
\[r = \frac{dE}{dt} = \frac{2 \epsilon^2 C}{2}\left( 1 - e^{- \frac{t}{RC}} \right)\left( - e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right)\]
\[ \Rightarrow r = \frac{\epsilon^2}{R}\left( 1 - e^{- \frac{t}{RC}} \right)\left( e^{- \frac{t}{RC}} \right)\]
\[\frac{dr}{dt} = \frac{\epsilon^2}{R}\left[ \left( - e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right)\left( e^{- \frac{t}{RC}} \right) + \left( 1 - e^{- \frac{t}{RC}} \right)\left( e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right) \right]\]
For r to be maximum,
\[\frac{dr}{dt} = 0\]
\[\Rightarrow \frac{\epsilon^2}{R}\left[ \left( - e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right)\left( e^{- \frac{t}{RC}} \right) + \left( 1 - e^{- \frac{t}{RC}} \right)\left( e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right) \right] = 0\]
\[ \Rightarrow \left[ \frac{e^{- \frac{2t}{RC}}}{RC} + \frac{e^{- \frac{2t}{RC}}}{RC} - \frac{e^\frac{- t}{RC}}{RC} \right] = 0\]
\[ \Rightarrow 2 e^{- \frac{2t}{RC}} = e^{- \frac{t}{RC}} \]
\[ \Rightarrow e^{- \frac{t}{RC}} = \frac{1}{2}\]
\[ \Rightarrow - \frac{t}{RC} = - \ln2\]
\[ \Rightarrow t = RC\ln2\]
APPEARS IN
संबंधित प्रश्न
Obtain the expression for the energy stored per unit volume in a charged parallel plate capacitor.
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) Find the current in the 20 Ω resistor shown in the figure. (b) If a capacitor of capacitance 4 μF is joined between the points A and B, what would be the electrostatic energy stored in it in steady state?
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?
A 100 μF capacitor is joined to a 24 V battery through a 1.0 MΩ resistor. Plot qualitative graphs (a) between current and time for the first 10 minutes and (b) between charge and time for the same period.
How many time constants will elapse before the current in a charging RC circuit drops to half of its initial value? Answer the same question for a discharging RC circuit.
A capacitance C charged to a potential difference V is discharged by connecting its plates through a resistance R. Find the heat dissipated in one time constant after the connections are made. Do this by calculating ∫ i2R dt and also by finding the decrease in the energy stored in the capacitor.
Each capacitor in figure has a capacitance of 10 µF. The emf of the battery is 100 V. Find the energy stored in each of the four capacitors.
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 point charge Q is placed at the origin. Find the electrostatic energy stored outside the sphere of radius R centred at the origin.
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.
A large conducting plane has a surface charge density `1.0 xx 10^-4 "Cm"^-2` . Find the electrostatic energy stored in a cubical volume of edge 1⋅0 cm in front of the plane.
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. Hence define energy density of 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)
A 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 ______
(ε0 = permittivity of free space)
Do free electrons travel to region of higher potential or lower potential?
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 ______.
A parallel plate capacitor (A) of capacitance C is charged by a battery to voltage V. The battery is disconnected and an uncharged capacitor (B) of capacitance 2C is connected across A. Find the ratio of total electrostatic energy stored in A and B finally and that stored in A initially.
