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Waves question

2009 · Shift 2 · Q44
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Waves question

2009 · Shift 2 · Q44

JEE AdvancedPhysicsWavesMultiple correct+4 / −2
Under the influence of the Coulomb field of charge +Q, a charge −-− q is moving around it in an elliptical orbit. Find out the correct statement(s):
  1. A
    The angular momentum of the charge −-− q is constant.
  2. B
    The linear momentum of the charge −-− q is constant.
  3. C
    The angular velocity of the charge −-− q is constant.
  4. D
    The linear speed of the charge −-− q is constant.
View written solutionFree

Correct answer: A

  1. Nature of force

The charge −q-q−q moves under the Coulomb attraction due to charge +Q+Q+Q. The force is

F⃗=−14πε0Qqr2r^\vec F = -\frac{1}{4\pi\varepsilon_0}\frac{Qq}{r^2}\hat rF=−4πε0​1​r2Qq​r^

which is a central force, i.e. it is always directed along the line joining the two charges.

  1. Check angular momentum

For a central force,

τ⃗=r⃗×F⃗=0\vec \tau = \vec r \times \vec F = 0τ=r×F=0

because F⃗∥r⃗\vec F \parallel \vec rF∥r.

Hence,

dL⃗dt=τ⃗=0\frac{d\vec L}{dt} = \vec \tau = 0dtdL​=τ=0

so angular momentum is conserved.

Therefore, Option A is correct.

  1. Check linear momentum

Linear momentum remains constant only if net force is zero. Here the Coulomb force is nonzero and continuously changes direction and magnitude along the orbit. Thus,

dp⃗dt=F⃗≠0\frac{d\vec p}{dt} = \vec F \neq 0dtdp​​=F=0

So linear momentum is not constant.

Therefore, Option B is incorrect.

  1. Check angular velocity

Angular momentum is constant:

L=mr2θ˙L = mr^2\dot\thetaL=mr2θ˙

Thus,

θ˙=Lmr2\dot\theta = \frac{L}{mr^2}θ˙=mr2L​

In an elliptical orbit, rrr changes continuously, so θ˙\dot\thetaθ˙ also changes. Hence angular velocity is not constant.

Therefore, Option C is incorrect.

  1. Check linear speed

In an elliptical orbit under an inverse-square central force, speed changes from point to point. The particle moves faster near the centre and slower when farther away. So linear speed is not constant.

Therefore, Option D is incorrect.

  1. Final conclusion

Only the angular momentum remains constant.

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