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The dispersion relation for electromagnetic waves travelling in a plasma is given as w2 = c2k2 + wp2, where c and wp are constants. In this plasma, the group velocity is :
- a)proportional to but not equal to the phase velocity
- b)inversely proportional to the phase velocity
- c)equal to the phase velocity
- d)a constant
Correct answer is option 'B'. Can you explain this answer?
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Dispersion relation:
The dispersion relation for electromagnetic waves traveling in a plasma is given by:
w^2 = c^2k^2 - wp^2
where w is the angular frequency, c is the speed of light, k is the wave number, and wp is the plasma frequency.
Group velocity and phase velocity:
In wave propagation, the phase velocity is the velocity at which the phase of a wave propagates in space. Mathematically, it is given by:
vp = w/k
On the other hand, the group velocity is the velocity at which the overall shape or envelope of the wave propagates. Mathematically, it is given by:
vg = dω/dk
where ω is the angular frequency.
Relationship between group velocity and phase velocity:
To determine the relationship between the group velocity and phase velocity, we need to find the derivative of the dispersion relation with respect to the wave number k.
Differentiating both sides of the dispersion relation with respect to k, we get:
2w(dw/dk) = 2ck^2 - 0
dw/dk = ck^2/wp^2
Substituting this expression into the group velocity equation, we get:
vg = dω/dk = c - ck^2/wp^2
Explanation of the answer:
From the derived expression for the group velocity, we can observe that it is inversely proportional to the phase velocity (vp = w/k).
This means that as the phase velocity increases, the group velocity decreases and vice versa. The group velocity is not equal to the phase velocity, but it is inversely proportional to it.
Therefore, the correct answer is option 'B': The group velocity is inversely proportional to the phase velocity.
The dispersion relation for electromagnetic waves traveling in a plasma is given by:
w^2 = c^2k^2 - wp^2
where w is the angular frequency, c is the speed of light, k is the wave number, and wp is the plasma frequency.
Group velocity and phase velocity:
In wave propagation, the phase velocity is the velocity at which the phase of a wave propagates in space. Mathematically, it is given by:
vp = w/k
On the other hand, the group velocity is the velocity at which the overall shape or envelope of the wave propagates. Mathematically, it is given by:
vg = dω/dk
where ω is the angular frequency.
Relationship between group velocity and phase velocity:
To determine the relationship between the group velocity and phase velocity, we need to find the derivative of the dispersion relation with respect to the wave number k.
Differentiating both sides of the dispersion relation with respect to k, we get:
2w(dw/dk) = 2ck^2 - 0
dw/dk = ck^2/wp^2
Substituting this expression into the group velocity equation, we get:
vg = dω/dk = c - ck^2/wp^2
Explanation of the answer:
From the derived expression for the group velocity, we can observe that it is inversely proportional to the phase velocity (vp = w/k).
This means that as the phase velocity increases, the group velocity decreases and vice versa. The group velocity is not equal to the phase velocity, but it is inversely proportional to it.
Therefore, the correct answer is option 'B': The group velocity is inversely proportional to the phase velocity.
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The dispersion relation for electromagnetic waves travelling in a plasma is given as w2 = c2k2 + wp2, where c and wp are constants. In this plasma, the group velocity is :a)proportional to but not equal to the phase velocityb)inversely proportional to the phase velocityc)equal to the phase velocityd)a constantCorrect answer is option 'B'. Can you explain this answer?
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