###### Advertisements

###### Advertisements

A plane electromagnetic wave is passing through a region. Consider (a) electric field (b) magnetic field (c) electrical energy in a small volume and (d) magnetic energy in a small volume. Construct the pairs of the quantities that oscillate with equal frequencies.

###### Advertisements

#### Solution

Let the electromagnetic wave be propagating in the z-direction. The vibrations of the electric and magnetic fields are given by:

E_{x}= E_{0} sin (kz – ωt)

B_{y}= B_{0} sin (kz – ωt)

Let the volume of the region be V.

The angular frequency of the vibrations of the electric and magnetic fields are same and are equal to ω. Therefore, their frequency, `f = ω/(2pi)` , is same .

The electrical energy in the region,

`U_E = (1/2 ∈_0 E^2) xx V`

It can be written as :

`U_E = (1/2 ∈_0 (E_0^2 sin^2(kz - ωt))xx V`

`U_E = (1/2 ∈_0 E_0^2 xx ((1-cos2(kz-ωt)))/2) xx V`

`U_E = (1/4 ∈_0 E_0^2 xx (1-cos2(kz-ωt))) xx V`

The magnetic energy in the region ,

`U_B = (B^2/(2u_0)) xx V`

`U_B = ((B_0^2 sin^2(kz-ωt))/(2u_0)) xx V`

⇒ `U_B = ((B_0^2(1-cos(2kz-2ωt)))/(4u_0)) xx V`

The angular frequency of the electric and magnetic energies is same and is equal to 2ω.Therefore, their frequency,

`f^' = (2ω)/(2pi) = 2f`

, will be same.

Thus, the electric and magnetic fields have same frequencies and the electrical and magnetic energies will have same frequencies.

#### APPEARS IN

#### RELATED QUESTIONS

How are electromagnetic waves produced?

In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 × 10^{10} Hz and amplitude 48 V m^{−1}.

**(a)** What is the wavelength of the wave?

**(b) **What is the amplitude of the oscillating magnetic field?

**(c) **Show that the average energy density of the E field equals the average energy density of the B** **field. [c = 3 × 10^{8} m s^{−1}]

How is the speed of em-waves in vacuum determined by the electric and magnetic field?

In the study of a photoelectric effect the graph between the stopping potential V and frequency *v* of the incident radiation on two different metals P and Q is shown below:

(i) Which one of the two metals has higher threshold frequency?

(ii) Determine the work function of the metal which has greater value.

(iii) Find the maximum kinetic energy of electron emitted by light of frequency 8 × 10^{14} Hz for this metal.

Do electromagnetic waves carry energy and momentum ?

Identify the electromagnetic waves whose wavelengths vary as:

(a) 10^{–12} m < λ < 10^{–8} m

(b) 10^{–3} m < λ < 10^{–1} m

Write one use for each.

What is the ratio of the speed of gamma rays to that of radio waves in a vacuum?

How are electric vector `(vec "E")`, magnetic vector `(vec "B")` and velocity vector `(vec "C")` oriented in an electromagnetic wave?

How does a charge q oscillating at certain frequency produce electromagnetic waves?

Sketch a schematic diagram depicting electric and magnetic fields for an electromagnetic wave propagating along the Z-direction.

A wire carries an alternating current i = i_{0} sin ωt. Is there an electric field in the vicinity of the wire?

A capacitor is connected to an alternating-current source. Is there a magnetic field between the plates?

Can an electromagnetic wave be polarised?

An electromagnetic wave going through vacuum is described by

E = E_{0} sin (kx − ωt); B = B_{0} sin (kx − ωt).

Which of the following equations is true?

Displacement current goes through the gap between the plates of a capacitor when the charge of the capacitor

(a) increases

(b) decreases

(c) does not change

(d) is zero

Speed of electromagnetic waves is the same

The energy contained in a small volume through which an electromagnetic wave is passing oscillates with

A laser beam has intensity 2.5 × 10^{14} W m^{−2}. Find amplitudes of electric and magnetic fields in the beam.

The energy associated with light of which of the following colours is minimum :

This is an example of step-up transformer .

Define frequency modulation and state any one advantage of frequency modulation (FM) over amplitude modulation (AM).

State any one property which is common to all electromagnetic waves.

The energy levels of an atom of a certain element are shown in the given figure. Which one of the transitions A, B, C, D or E will result in the emission of photons of electromagnetic radiation of wavelength 618.75 nm? Support your answer with mathematical calculations.

The dimension of `1/(mu_0 ∈_0)` is

If the amplitude of the magnetic field is 3 x 10^{-6} T, then amplitude of the electric field for a electromagnetic waves is ______.

The electric and magnetic fields, associated with an electromagnetic wave, propagating along negative X-axis can be represented by ______.

In an electromagnetic wave in free space the rms value of the electric field is 3 V m^{-1}. The peak value of the magnetic field is ______.

If the magnetic monopole exists, then which of the Maxwell’s equation to be modified?

Let E = E_{0} sin[10^{6} x -ωt] be the electric field of plane electromagnetic wave, the value of ω is ______.

Give two uses of Microwaves.

Write notes on Ampere-Maxwell law.

Explain the Maxwell’s modification of Ampere’s circuital law.

Explain the importance of Maxwell’s correction.

Maxwell’s displacement current is a response to inconsistency in the ______.

The direction of propagation of an electromagnetic plane wave is ______.

The speed of plane electromagnetic waves is maximum in ______.

A man standing on the road has to hold his umbrella at 30° with the vertical to keep the rain away. He throws away the umbrella and starts, running at 10 km/h and finds raindrops hitting his head vertically. The speed of the raindrops with respect to the road is ______.

Which one of the following does not represent simple harmonic motion? Here y denotes the instantaneous displacement. Here, A and B are constants and co is the angular frequency.

Dimensions of 1/(µOE_{0}) is

An accelerate electron would produce.

Maxwell's equation describe the fundamental law of

If a source is transmitting electromagnetic waves of frequency 8.2 × 10^{6} Hz. then wavelength of electromagnetic waves transmitted from the source will be.

The velocity of electromagnetic wave is parallel to

Which of the following are not electromagnetic waves?

The sun delivers 10^{3}w/m^{2} of electromagnetic flux to the earth's surface. The total power that is incident on a roof of dimension 8m/10m will be

Dimensions of ε_{0} `(d phi_ε)/(dt)` are of

Electromagnetic waves are produced by ______.

Which of the following type of radiations are radiated by an oscillating electric charge?

For a plane electromagnetic wave propagating in x-direction, which one of the following combinations gives the correct possible directions for electric field (E) and magnetic field (B) respectively?

For a plane electromagnetic wave propagating in the x-direction, which one of the following combinations gives the correct possible directions for the electric field (E) and magnetic field (B) respectively?

A plane EM wave travelling in vacuum along z direction is given by `E = E_0 sin(kz - ωt)hati` and `B = B_0 sin(kz - ωt)hatj`

- Evaluate `oint E.dl` over the rectangular loop 1234 shown in figure.
- Evaluate `int B.ds` over the surface bounded by loop 1234.
- Use equation `oint E.dl = (-dphi_B)/(dt)` to prove `E_0/B_0` = c.
- By using similar process and the equation `ointB.dl = mu_0I + ε_0 (dphi_E)/(dt)`, prove that c = `1/sqrt(mu_0ε_0)`

A plane electromagnetic wave of frequency 500 MHz is travelling in a vacuum along a y-direction.

At a particular point in space and time, `vec"B"` = 8.0 × 10^{-8} `hat"Z"`T. The value of the electric field at this point is ______.

(speed of light = 3 × 10^{8} ms^{-1})

`hat x, hat y, hat z` are unit vectors along x, y, and Z directions.

For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric (U_{e}) and magnetic (U_{m}) fields is ______.

A plane electromagnetic wave, has frequency of 2.0 × 10^{10} Hz and its energy density is 1.02 × 10^{-8} J/m^{3} in vacuum. The amplitude of the magnetic field of the wave is close to `(1/(4piepsilon_0) = 9xx10^9"Nm"^2/"C"^2 "and speed of light" = 3 xx 10^8 "m" "s"^-1)`:

The electric field in a plane electromagnetic wave is given by `vecE = 200cos[((0.5 xx 10^3)/m)x - (1.5 xx 10^11 "rad"/s xx t)]V/mhatj`. If the wave falls normally on a perfectly reflecting surface having an area of 100 cm^{2}. If the radiation pressure exerted by the E.M. wave on the surface during a 10-minute exposure is `x/10^9 N/m^2`. Find the value of x.

A plane electromagnetic wave travels in free space along the x-direction. The electric field component of the wave at a particular point of space and time is E = 6 Vm^{-1} along the y-direction. Its corresponding magnetic field component, B would be ______.

Sunlight falls normally on a surface of area 36 cm^{2} and exerts an average force of 7.2 × 10^{-9} N within a time period of 20 minutes. Considering a case of complete absorption the energy flux of incident light is ______.

The electric field in an electromagnetic wave is given by E = 56.5 sin ω(t - x/c)Nc^{-1}. Find the intensity of the wave if it is propagating along x-axis in the free space.

(Given ε_{0} = 8.85 × 10^{-12} C^{2} N^{-1} m^{-2})

An electromagnetic wave of frequency 3 GHz enters a dielectric medium of relative electric permittivity 2.25 from vacuum. The wavelength of this wave in that medium will be ______ × 10^{-2} cm.

In a plane electromagnetic wave, the direction of electric field and magnetic field are represented by `hat"k"` and 2`hat"i" - 2hat"j"`, respectively. What is the unit vector along direction of propagation of the wave.

A 27 mW laser beam has a cross-sectional area of 10 mm^{2}. The magnitude of the maximum electric field in this electromagnetic wave is given by:

[Given permittivity of space ∈_{0} = 9 × 10^{-12} SI units, Speed of light c = 3 10^{8} m/s]

Name the electromagnetic wave/radiation which is used to study crystal structure.