Advertisements
Advertisements
Question
Show that average value of radiant flux density ‘S’ over a single period ‘T’ is given by S = `1/(2cmu_0) E_0^2`.
Advertisements
Solution
Radiant flux density S = `1/mu_0 (vecE xx vecB)`
c = `1/sqrt(mu_0 ε_0)`
or c2 = `1/(mu_0ε_0)`
`1/mu_0 = ε_0c^2`
∴ S = `ε_0c^2 (vecE xx vecB)` ......(I)
Let electromagnetic waves be propagated along X-axis so its electric and magnetic field vectors are along Y and Z axis.
∴ `vecE = E_0 cos(kx - ωt)hatj`
`vecB = B_0 cos(kx - ωt)hatk`
`vecE xx vecB = (E_0B_0) cos^2 (kx - ωt)(hatj xx hatk)`
Put E × B in I
∴ S = `ε_0c^2 E_0B_0 cos^2 (kx - ωt)hati`
So average value of the magnitude of radiant flux density over complete cycle is
`S_(av) = c^2 ε_0E_0B_0 [1/T] int_0^T cos^2 (kx - ωt)dt hati`
= `(c^2ε_0E_0B_0)/T [T/2] = c^2/2 ε_0E_0 (E_0/c)` .....`[because c = E_0/B_0 or B_0 = E_0/c_0]`
= `(cε_0E_0^2)/2 [c = 1/sqrt(mu_0ε_0) or ε_0 = 1/(mu_0c^2)]`
= `c/2 * 1/(mu_0c^2) E_0^2`
`S_(av) = E_0^2/(2 mu_0c)` Hence proved.
APPEARS IN
RELATED QUESTIONS
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.
- Calculate the capacitance and the rate of charge of the potential difference between the plates.
- Obtain the displacement current across the plates.
- Is Kirchhoff’s first rule (junction rule) valid at each plate of the capacitor? Explain.
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.
- What is the rms value of the conduction current?
- Is the conduction current equal to the displacement current?
- 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 ______.
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 cm2?
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 = V0 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 = V0 sinωt The displacement current between the plates of the capacitor would then be given by ______
An electromagnetic wave travelling along z-axis is given as: E = E0 cos (kz – ωt.). Choose the correct options from the following;
- The associated magnetic field is given as `B = 1/c hatk xx E = 1/ω (hatk xx E)`.
- The electromagnetic field can be written in terms of the associated magnetic field as `E = c(B xx hatk)`.
- `hatk.E = 0, hatk.B` = 0.
- `hatk xx E = 0, hatk xx B` = 0.
A variable frequency a.c source is connected to a capacitor. How will the displacement current change with decrease in frequency?
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).
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 × 108 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) = Vo sin (2πνt). What fraction of the conduction current density is the displacement current density?
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 m2. The amplitude of the oscillating displacement current for the applied AC voltage is ______.
[take ε0 = 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).
Draw a neat labelled diagram of displacement current in the space between the plates of the capacitor.
