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
Find the charge on the capacitor as shown in the circuit.
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.
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 capacitor of capacitance 12.0 μF is connected to a battery of emf 6.00 V and internal resistance 1.00 Ω through resistanceless leads. 12.0 μs after the connections are made, what will be (a) the current in the circuit (b) the power delivered by the battery (c) the power dissipated in heat and (d) the rate at which the energy stored in the capacitor is increasing?
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?
A capacitor of capacitance C is given a charge Q. At t = 0, it is connected to an ideal battery of emf ε through a resistance R. Find the charge on the capacitor at time t.
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.
A 2µF capacitor is charge to 100 volt and then its plate are connected by a conducting wire. The heat produced is:-
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)
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.
In a capacitor of capacitance 20 µF, the distance between the plates is 2 mm. If a dielectric slab of width 1 mm and dielectric constant 2 is inserted between the plates, what is the new capacitance?
If the charge on the parallel plate capacitor is increased by 3 C, the energy stored in it increases by 44%. The original charge on the capacitor is ______.
Where is the electrical potential energy stored when a capacitor is charged?
Which set gives the equivalent expressions for the energy stored in a capacitor?
What is the SI unit of energy stored in a capacitor?
If an additional small charge \(dq\) is supplied to a capacitor when its potential difference is \(V\), what is the small work done?
Using \(Q=CV\), which expression is an equivalent form of \(U=\frac{Q^2}{2C}\)?
A capacitor with capacitance \(3\times10^{-9}\,\mathrm{F}\) is connected to \(400\,\mathrm{V}\). A dielectric slab with \(K=3\) completely fills the space, and the voltage is kept constant. What is the energy change when the slab is removed?
