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
Obtain the equivalent capacitance of the network in Figure. For a 300 V supply, determine the charge and voltage across each capacitor.
A capacitor of capacitance C is charged to a potential V. The flux of the electric field through a closed surface enclosing the capacitor is
Three capacitors of capacitances 6 µF each are available. The minimum and maximum capacitances, which may be obtained are
A parallel-plate capacitor has plate area 25⋅0 cm2 and a separation of 2⋅00 mm between the plates. The capacitor is connected to a battery of 12⋅0 V. (a) Find the charge on the capacitor. (b) The plate separation is decreased to 1⋅00 mm. Find the extra charge given by the battery to the positive plate.
Find the capacitance of the combination shown in figure between A and B.
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.
A parallel-plate capacitor with the plate area 100 cm2 and the separation between the plates 1⋅0 cm is connected across a battery of emf 24 volts. Find the force of attraction between the plates.
Consider the situation shown in figure. The plates of the capacitor have plate area A and are clamped in the laboratory. The dielectric slab is released from rest with a length a inside the capacitor. Neglecting any effect of friction or gravity, show that the slab will execute periodic motion and find its time period.
You are provided with 8 μF capacitors. Show with the help of a diagram how you will arrange minimum number of them to get a resultant capacitance of 20 μF.
Three circuits, each consisting of a switch 'S' and two capacitors, are initially charged, as shown in the figure. After the switch has been closed, in which circuit will the charge on the left-hand capacitor
(i) increase,
(ii) decrease, and
(iii) remains the same? Give reasons.
Derive the expression for resultant capacitance, when the capacitor is connected in parallel.
The positive terminal of 12 V battery is connected to the ground. Then the negative terminal will be at ______.
Dielectric constant for a metal is ______.
Three capacitors 2µF, 3µF, and 6µF are joined in series with each other. The equivalent capacitance is ____________.
The radius of a sphere of capacity 1 microfarad in the air is ______
A parallel plate capacitor is filled by a dielectric whose relative permittivity varies with the applied voltage (U) as ε = αU where α = 2V–1. A similar capacitor with no dielectric is charged to U0 = 78V. It is then connected to the uncharged capacitor with the dielectric. Find the final voltage on the capacitors.
A leaky parallel plate capacitor is filled completely with a material having dielectric constant K = 5 and electric conductivity σ = 7.4 × 10-12 Ω-1 m-1. If the charge on the plate at the instant t = 0 is q = 8.85 µC, then the leakage current at the instant t = 12 s is ______ × 10-1 µA.
Read the following paragraph and answer the questions.
| A capacitor is a system of two conductors separated by an insulator. The two conductors have equal and opposite charges with a potential difference between them. The capacitance of a capacitor depends on the geometrical configuration (shape, size and separation) of the system and also on the nature of the insulator separating the two conductors. They are used to store charges. Like resistors, capacitors can be arranged in series or parallel or a combination of both to obtain the desired value of capacitance. |
- Find the equivalent capacitance between points A and B in the given diagram.
- A dielectric slab is inserted between the plates of the parallel plate capacitor. The electric field between the plates decreases. Explain.
- A capacitor A of capacitance C, having charge Q is connected across another uncharged capacitor B of capacitance 2C. Find an expression for (a) the potential difference across the combination and (b) the charge lost by capacitor A.
OR
Two slabs of dielectric constants 2K and K fill the space between the plates of a parallel plate capacitor of plate area A and plate separation d as shown in the figure. Find an expression for the capacitance of the system.
