Ncert Exercises & Solutions

An electromagnetic wave travelling along z-axis is given as E=E₀ cos (kz - \(\omega\)t). Choose the correct options from the following.

(a) The associated magnetic field is given as \(B=\frac1c\widehat k\times E=\frac{1}{\omega}(\widehat k\times E)\)
(b) The electromagnetic field can be written in terms of the associated magnetic field as \(E=c(B\times\widehat k)\)
(c) \(\widehat k.E=0,\widehat k.B=0\)
(d) \(\widehat k\times E=0,\widehat k\times B=0\)

1. (a, b, c)

2. (a, c, d)

3. (b, c, d)

4. (b, d)

(1) Hint: The direction of the electromagnetic wave is perpendicular to both the electric field and the magnetic field.
Step 1: Find the relation between the magnetic field and the electric field.

Suppose an electromagnetic wave is traveling along negative z-direction. Its electric field is given by;

E = E_{0} \cos (kz - ωt)

which is perpendicular to z-axis. It acts along negative y-direction.

The associated magnetic field

\vec{B}

in an electromagnetic wave is along x-axis i.e., along

\hat{k} \times \vec{E} .

As,

B_{0} = \frac{E_{0}}{c}

\vec{B} = \frac{1}{c} (\hat{k} \times \vec{E})

The associated electric field can be written in terms of magnetic field as:

\vec{E} = c (\vec{B} \times \hat{k})

Step 2:

Angle between \hat{k} and \vec{E} is 90 ° and between \hat{k} and \vec{B} is 90 °,

therefore, \hat{k} . \vec{E} = E \times 1 \times \cos 90 ° = 0 and \hat{k} . \vec{B} = E \times 1 \times \cos 90 ° = 0

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