Advertisements
Advertisements
Question
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.
Advertisements
Solution
The growth of charge on the capacitor at time t ,
\[Q = Q_0 \left( 1 - e^{- \frac{t}{RC}} \right)\]
\[i = \frac{dQ}{dt} = \left( \frac{Q_0}{RC} \right) e^{- \frac{t}{RC}}\]
Heat dissipated during time t1 to t2,
\[U = \int_{t_1}^{t_2} i^2 Rdt\]
\[ = \frac{{Q_0}^2}{2C}\left( e^{- \frac{2 t_1}{RC}} - e^{- \frac{2 t_2}{RC}} \right)\]
\[ \because Q_0 = C V_{0,} \]
\[ U = \frac{1}{2}C {V_0}^2 \left( e^{- \frac{2 t_1}{RC}} - e^{- \frac{2 t_2}{RC}} \right)\]
The potential difference across a capacitor at any time t,
\[V = V_0 \left( 1 - e^{- \frac{t}{RC}} \right)\]
The energy stored in the capacitor at any time t,
\[E = \frac{1}{2}C V^2 = \frac{1}{2}C {V_0}^2 \left( 1 - e^{- \frac{2t}{RC}} \right)^2\]
∴ The energy stored in the capacitor from t1 to t2,
\[E = \frac{1}{2}C {V_0}^2 \left( e^{- \frac{2 t_1}{RC}} \right) - \frac{1}{2}C {V_0}^2 \left( 1 - e^{- \frac{2 t_2}{RC}} \right)\]
\[ \Rightarrow E = \frac{1}{2}C {V_0}^2 \left( e^{- \frac{2 t_1}{RC}} - e^{- \frac{2 t_2}{RC}} \right)\]
APPEARS IN
RELATED QUESTIONS
Obtain the expression for the energy stored per unit volume in a charged parallel plate capacitor.
Find the ratio of energy stored in the two configurations if they are both connected to the same source.
The energy density in the electric field created by a point charge falls off with the distance from the point charge as
A capacitor of capacitance 500 μF is connected to a battery through a 10 kΩ resistor. The charge stored in the capacitor in the first 5 s is larger than the charge stored in the next.
(a) 5 s
(b) 50 s
(c) 500 s
(d) 500 s
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.
The plates of a capacitor of capacitance 10 μF, charged to 60 μC, are joined together by a wire of resistance 10 Ω at t = 0. Find the charge on the capacitor in the circuit at (a) t = 0 (b) t = 30 μs (c) t = 120 μs and (d) t = 1.0 ms.
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 charge on a capacitors falls to 0.1% of its maximum value in 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 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.
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 C is given a charge Q. At t = 0, it is connected to an uncharged capacitor of equal capacitance through a resistance R. Find the charge on the second capacitor as a function of time.
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 point charge Q is placed at the origin. Find the electrostatic energy stored outside the sphere of radius R centred at the origin.
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.
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 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 ‘`vec "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)
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?
