###### Advertisements

###### Advertisements

Figure shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 cm. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to 0.15 A.

**(a) **Calculate the capacitance and the rate of charge of potential difference between the plates.

**(b) **Obtain the displacement current across the plates.

**(c)** Is Kirchhoff’s first rule (junction rule) valid at each plate of the capacitor? Explain.

###### Advertisements

#### Solution

Radius of each circular plate, r = 12 cm = 0.12 m

Distance between the plates, d = 5 cm = 0.05 m

Charging current, I = 0.15 A

Permittivity of free space, ε_{0} = 8.85 × 10^{−12 }C^{2 }N^{−1 }m^{−2}

**(a)** Capacitance between the two plates is given by the relation,

C =`(ε_0"A")/"d"`

Where,

A = Area of each plate = πr^{2}

`"C" = (ε_0pi"r"^2)/"d"`

= `(8.85 xx 10^-12 xx pi xx (0.12)^2)/0.05`

= 8.0032 × 10^{−12 }F

= 80.032 pF

Charge on each plate, q = CV

Where,

V = Potential difference across the plates

Differentiation on both sides with respect to time (t) gives:

`("dq")/("dt") = "C"("dV")/("dt")`

But, `("dq")/("dt")` = current (I)

∴ `("dV")/("dt") = "I"/"C"`

= `(0.15)/(80.032 xx 10^-12)`

= 1.87 × 10^{9 }V/s

Therefore, the change in the potential difference between the plates is 1.87 ×10^{9} V/s.

**(b)** The displacement current across the plates is the same as the conduction current. Hence, the displacement current, i_{d} is 0.15 A.

**(c) **Yes

Kirchhoff’s first rule is valid at each plate of the capacitor provided that we take the sum of conduction and displacement for current.

#### APPEARS IN

#### RELATED QUESTIONS

A parallel plate capacitor (Figure) made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to a 230 V ac supply with a (angular) frequency of 300 rad s^{−1}.

**(a) **What is the rms value of the conduction current?

**(b)** Is the conduction current equal to the displacement current?

**(c) **Determine the amplitude of B** **at a point 3.0 cm from the axis between the plates.

When an ideal capacitor is charged by a dc battery, no current flows. However, when an ac source is used, the current flows continuously. How does one explain this, based on the concept of displacement current?

A parallel-plate capacitor of plate-area A and plate separation d is joined to a battery of emf ε and internal resistance R at t = 0. Consider a plane surface of area A/2, parallel to the plates and situated symmetrically between them. Find the displacement current through this surface as a function of time.

Without the concept of displacement current it is not possible to correctly apply Ampere’s law on a path parallel to the plates of parallel plate capacitor having capacitance C in ______.

If the total energy of a particle executing SHM is E, then the potential energy V and the kinetic energy K of the particle in terms of E when its displacement is half of its amplitude will be ______.

A spring balance has a scale that reads 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this spring, when displaced and released, oscillates with a period of 0.60 s. What is the weight of the body?

A cylinder of radius R, length Land density p floats upright in a fluid of density p_{0}. The cylinder is given a gentle downward push as a result of which there is a vertical displacement of size x; it is then released; the time period of resulting (undampe (D) oscillations is ______.

The displacement of a particle from its mean position is given by x = 4 sin (10πt + 1.5π) cos (10πt + 1.5π). The motion of the particle is

Displacement current goes through the gap between the plantes of a capacitors. When the charge of the capacitor:-

Which of the following is the unit of displacement current?

A parallel plate capacitor of plate separation 2 mm is connected in an electric circuit having source voltage 400 V. What is the value of the displacement current for 10^{-6} second if the plate area is 60 cm^{2}?

According to Maxwell's hypothesis, a changing electric field gives rise to ______.

A capacitor of capacitance ‘C’, is connected across an ac source of voltage V, given by V = V_{0} sinωt The displacement current between the plates of the capacitor would then be given by ______.

A capacitor of capacitance ‘C’, is connected across an ac source of voltage V, given by V = V_{0} sinωt The displacement current between the plates of the capacitor would then be given by ______

A capacitor of capacitance ‘C’, is connected across an ac source of voltage V, given by

V = V_{0}sinωt

The displacement current between the plates of the capacitor would then be given by:

The charge on a parallel plate capacitor varies as q = q_{0} cos 2πνt. The plates are very large and close together (area = A, separation = d). Neglecting the edge effects, find the displacement current through the capacitor?

Show that the magnetic field B at a point in between the plates of a parallel-plate capacitor during charging is `(ε_0mu_r)/2 (dE)/(dt)` (symbols having usual meaning).

Show that average value of radiant flux density ‘S’ over a single period ‘T’ is given by S = `1/(2cmu_0) E_0^2`.

You are given a 2 µF parallel plate capacitor. How would you establish an instantaneous displacement current of 1 mA in the space between its plates?

Sea water at frequency ν = 4 × 10^{8} Hz has permittivity ε ≈ 80 εo, permeability µ ≈ µo and resistivity ρ = 0.25 Ω–m. Imagine a parallel plate capacitor immersed in seawater and driven by an alternating voltage source V(t) = V_{o} sin (2πνt). What fraction of the conduction current density is the displacement current density?

A long straight cable of length `l` is placed symmetrically along z-axis and has radius a(<< l). The cable consists of a thin wire and a co-axial conducting tube. An alternating current I(t) = I_{0} sin (2πνt) flows down the central thin wire and returns along the co-axial conducting tube. The induced electric field at a distance s from the wire inside the cable is E(s,t) = µ_{0}I_{0}ν cos (2πνt) In `(s/a)hatk`.

- Calculate the displacement current density inside the cable.
- Integrate the displacement current density across the cross-section of the cable to find the total displacement current I
^{d}. - Compare the conduction current I0 with the displacement current `I_0^d`.

AC voltage V(t) = 20 sinωt of frequency 50 Hz is applied to a parallel plate capacitor. The separation between the plates is 2 mm and the area is 1 m^{2}. The amplitude of the oscillating displacement current for the applied AC voltage is ______.

[take ε_{0} = 8.85 × 10^{-12} F/m]

A particle is moving with speed v = b`sqrtx` along positive x-axis. Calculate the speed of the particle at time t = τ (assume that the particle is at origin at t = 0).

A parallel plate capacitor is charged to 100 × 10^{-6} C. Due to radiations, falling from a radiating source, the plate loses charge at the rate of 2 × 10^{-7} Cs^{-1}. The magnitude of displacement current is ______.

Draw a neat labelled diagram of displacement current in the space between the plates of the capacitor.