Advertisements
Advertisements
Question
A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 µC. The space between the concentric spheres is filled with a liquid of dielectric constant 32.
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere?
(c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller.
Advertisements
Solution
Radius of the inner sphere, r2 = 12 cm = 0.12 m
Radius of the outer sphere, r1 = 13 cm = 0.13 m
Charge on the inner sphere, q = 2.5 µC = `2.5 xx 10^-6 "C"`
Dielectric constant of a liquid, ` in_"r" = 32`
(a) Capacitance of the capacitor is given by the relation
C = `(4piin_0in_1"r"_1"r"_2)/("r"_1 - "r"_2)`
Where,
`in_0` = Permittivity of free space = `8.85 xx 10^-12 "C"^2 "N"^-1 "m"^-2`
`1/(4piin_0) = 9 xx 10^9 "Nm"^2 "C"^-2`
∴ C = `(32 xx 0.12 xx 0.13)/(9 xx 10^9 xx (0.13 - 0.12))`
`≈ 5.5 xx 10^-9 "F"`
Hence, the capacitance of the capacitor is approximate `5.5 xx 10^-9 "F"`.
(b) Potential of the inner sphere is given by,
`"V" = "q"/"C"`
= `(2.5 xx 10^-6)/(5.5 xx 10^-9) = 4.5 xx 10^2 "V"`
Hence, the potential of the inner sphere is `4.5 xx 10^2 "V"`.
(c) Radius of an isolated sphere, r = 12 × 10−2 m
Capacitance of the sphere is given by the relation,
`"C'" = 4piin_0"r"`
= `4pi xx 8.85 xx 10^-12 xx 12 xx 10^-12`
= `1.33 xx 10^-11 "F"`
The capacitance of the isolated sphere is less in comparison to the concentric spheres. This is because the outer sphere of the concentric spheres is earthed. Hence, the potential difference is less and the capacitance is more than the isolated sphere.
APPEARS IN
RELATED QUESTIONS
A capacitor of capacitance C is charged fully by connecting it to a battery of emf E. It is then disconnected from the battery. If the separation between the plates of the capacitor is now doubled, how will the following change?
(i) charge stored by the capacitor.
(ii) Field strength between the plates.
(iii) Energy stored by the capacitor.
Justify your answer in each case.
A capacitor of capacitance ‘C’ is charged to ‘V’ volts by a battery. After some time the battery is disconnected and the distance between the plates is doubled. Now a slab of dielectric constant, 1 < k < 2, is introduced to fill the space between the plates. How will the following be affected? (b) The energy stored in the capacitor Justify your answer by writing the necessary expressions
Find the equivalent capacitance of the network shown in the figure, when each capacitor is of 1 μF. When the ends X and Y are connected to a 6 V battery, find out (i) the charge and (ii) the energy stored in the network.
As `C = (1/V) Q` , can you say that the capacitance C is proportional to the charge Q?
Suppose, one wishes to construct a 1⋅0 farad capacitor using circular discs. If the separation between the discs be kept at 1⋅0 mm, what would be the radius of the discs?
Find the charge appearing on each of the three capacitors shown in figure .
Find the charge supplied by the battery in the arrangement shown in figure.
Find the equivalent capacitance of the system shown in figure between the points a and b.
A cylindrical capacitor is constructed using two coaxial cylinders of the same length 10 cm and of radii 2 mm and 4 mm. (a) Calculate the capacitance. (b) Another capacitor of the same length is constructed with cylinders of radii 4 mm and 8 mm. Calculate the capacitance.
Consider the situation shown in the figure. The switch S is open for a long time and then closed. (a) Find the charge flown through the battery when the switch S is closed. (b) Find the work done by the battery.(c) Find the change in energy stored in the capacitors.(d) Find the heat developed in the system.
A sphercial capacitor is made of two conducting spherical shells of radii a and b. The space between the shells is filled with a dielectric of dielectric constant K up to a radius c as shown in figure . Calculate the capacitance.
Define ‘capacitance’. Give its unit.
Obtain the expression for capacitance for a parallel plate capacitor.
For the given capacitor configuration
- Find the charges on each capacitor
- potential difference across them
- energy stored in each capacitor.
Capacitors P and Q have identical cross-sectional areas A and separation d. The space between the capacitors is filled with a dielectric of dielectric constant Er as shown in the figure. Calculate the capacitance of capacitors P and Q.
The work done in placing a charge of 8 × 10–18 coulomb on a condenser of capacity 100 micro-farad is ______.
The displacement current of 4.425 µA is developed in the space between the plates of the parallel plate capacitor when voltage is changing at a rate of 106 Vs-1. The area of each plate of the capacitor is 40 cm2. The distance between each plate of the capacitor is x × 10-3 m. The value of x is ______.
(Permittivity of free space, ε0 = 8.85 × 10-12C2N-1m-2).
Two identical capacitors are connected as shown and have an initial charge of Q0. The separation between the plates of each capacitor is d0. Suddenly the left plate of the upper capacitor and right plate of the lower capacitor start moving with speed v towards the left while the other plate of each capacitor remains fixed. `("given" (Q_0V)/(2d_0) = 10 A)`. The value of current in the circuit is ______ A.
Two spherical conductors of capacities 4 µF and 3 µF are charged to same potential having radii 4 cm and 3 cm respectively. If ‘σ1’ and ‘σ2’ represent surface density of charge on respective conductors then ______.
