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Electromagnetic Waves question

2020 · 2 Sep · Shift 2 · Q62
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  5. /2020 · 2 Sep · Shift 2 · Q62

Electromagnetic Waves question

2020 · 2 Sep · Shift 2 · Q62

JEE MainPhysicsElectromagnetic WavesMCQ+4 / −1
In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by k^\widehat kk and 2i^−2j^2\widehat i - 2\widehat j2i−2j​, respectively. What is the unit vector along direction of propagation of the wave?
  1. A
    15(i^+2j^){1 \over {\sqrt 5 }}\left( {\widehat i + 2\widehat j} \right)5​1​(i+2j​)
  2. B
    15(2i^+j^){1 \over {\sqrt 5 }}\left( {2\widehat i + \widehat j} \right)5​1​(2i+j​)
  3. C
    12(i^+j^){1 \over {\sqrt 2 }}\left( {\widehat i + \widehat j} \right)2​1​(i+j​)
  4. D
    12(j^+k^){1 \over {\sqrt 2 }}\left( {\widehat j + \widehat k} \right)2​1​(j​+k)
View written solutionFree

Correct answer: C

  1. In an electromagnetic wave, the direction of propagation is along the Poynting vector:
S⃗∝E⃗×B⃗\vec S \propto \vec E \times \vec BS∝E×B

So we need the unit vector along:

n^=E⃗×B⃗\hat n = \vec E \times \vec Bn^=E×B
  1. Given directions:
  • Electric field along k^\hat kk^
  • Magnetic field along 2i^−2j^2\hat i - 2\hat j2i^−2j^​

Thus,

E⃗=k^,B⃗=2i^−2j^\vec E = \hat k, \qquad \vec B = 2\hat i - 2\hat jE=k^,B=2i^−2j^​
  1. Compute the cross product:
E⃗×B⃗=k^×(2i^−2j^)\vec E \times \vec B = \hat k \times (2\hat i - 2\hat j)E×B=k^×(2i^−2j^​)

Using distributive property,

=2(k^×i^)−2(k^×j^)= 2(\hat k \times \hat i) - 2(\hat k \times \hat j)=2(k^×i^)−2(k^×j^​)

Now use the standard vector products:

k^×i^=j^,k^×j^=−i^\hat k \times \hat i = \hat j, \qquad \hat k \times \hat j = -\hat ik^×i^=j^​,k^×j^​=−i^

Therefore,

E⃗×B⃗=2j^−2(−i^)=2i^+2j^\vec E \times \vec B = 2\hat j - 2(-\hat i) = 2\hat i + 2\hat jE×B=2j^​−2(−i^)=2i^+2j^​
  1. So the propagation direction is along:
2i^+2j^=2(i^+j^)2\hat i + 2\hat j = 2(\hat i + \hat j)2i^+2j^​=2(i^+j^​)

Its unit vector is:

i^+j^12+12=12(i^+j^)\frac{\hat i + \hat j}{\sqrt{1^2+1^2}} = \frac{1}{\sqrt 2}(\hat i + \hat j)12+12​i^+j^​​=2​1​(i^+j^​)
  1. Compare with the options:
12(i^+j^)\frac{1}{\sqrt 2}(\hat i + \hat j)2​1​(i^+j^​)

This is Option C.

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