
A charged particle of mass and charge moving under the influence of uniform electric field and a uniform magnetic field follows a trajectory from point P to Q as shown in figure. The velocities at P and Q are respectively, and . Then which of the following statements (A, B, C, D) are the correct? (Trajectory shown in schematic and not to scale)
(A)
(B) Rate of work done by the electric field at P is
(C) Rate of work done by both the field at Q is zero
(D) The difference between the magnitude of angular momentum of the particle at P and Q is .
A. (A), (B), (C), (D)
B. (A), (C), (D)
C. (B), (C), (D)
D. (A), (B), (C)

Answer
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Hint : To solve this question, we need to use the work energy theorem between the points P and Q. The rate of work done can be found out by using the formula for the power. Also, the difference between the magnitudes of the angular momentum can be calculated by using its formula.
Formula used: The formulae which are used to solve this question are given by
, here is the force applied by the electric field on a charge .
, here is the total work done by a force in displacing a particle from to , is the displacement, and is the angle between and .
, here is the power supplied by a force on a particle moving with a velocity , and is the angle between the force and the velocity.
Complete step by step answer
We know that both the electric and the magnetic forces are conservative forces. Since the particle moves in a region of the electric and the magnetic fields, so the energy of the particle remains conserved between the points P and Q. So from the work energy theorem we have
………………………..(1)
The work done is the total work done by the electric and the magnetic force on the charged particle.
………………………..(2)
Now, work done by the electric field is
We know that . So we get
………………………..(3)
As the electric field is parallel to the -axis, along which the displacement of the particle is , so we have
So from (3) we have
………………………..(4)
Also, the work done by magnetic field is
As we know that the magnetic force is always perpendicular to the motion of the charged particle, so the work done by the magnetic force is always equal to zero.
………………………..(5)
Substituting (4) and (5) in (2) we get
………………………..(6)
Now, the kinetic energy at point P is
………………………..(7)
Also, the kinetic energy at at Q is
………………………..(8)
Substituting (6) (7) and (8) in (1) we get
Dividing by we get
………………………..(9)
Hence, the statement (A) is correct.
Now, we know that the rate of work done is equal to the power which is given by
………………………..(10)
At P, the velocity of the charged particle is parallel to the electric field. So the power supplied by the electric field is given by
From (9)
Hence, the statement (B) is also correct.
As we know that the magnetic force is always perpendicular to the motion of the charged particle. So we have . So from (10) we get
Hence, the statement (C) is also correct.
Now, the angular momentum about O at P is given by
Also, the angular momentum at Q is given by
So the difference is given by
Hence the statement (D) is incorrect.
Thus the correct statements are (A), (B) and (C).
Hence, the correct answer is option D.
Note
In this question we were not given the point about which the angular momentum was supposed to be calculated. So we assumed that point to be the origin while doing the calculation.
Formula used: The formulae which are used to solve this question are given by
Complete step by step answer
We know that both the electric and the magnetic forces are conservative forces. Since the particle moves in a region of the electric and the magnetic fields, so the energy of the particle remains conserved between the points P and Q. So from the work energy theorem we have
The work done
Now, work done by the electric field is
We know that
As the electric field is parallel to the
So from (3) we have
Also, the work done by magnetic field is
As we know that the magnetic force is always perpendicular to the motion of the charged particle, so the work done by the magnetic force is always equal to zero.
Substituting (4) and (5) in (2) we get
Now, the kinetic energy at point P is
Also, the kinetic energy at at Q is
Substituting (6) (7) and (8) in (1) we get
Dividing by
Hence, the statement (A) is correct.
Now, we know that the rate of work done is equal to the power which is given by
At P, the velocity of the charged particle is
From (9)
Hence, the statement (B) is also correct.
As we know that the magnetic force is always perpendicular to the motion of the charged particle. So we have
Hence, the statement (C) is also correct.
Now, the angular momentum about O at P is given by
Also, the angular momentum at Q is given by
So the difference is given by
Hence the statement (D) is incorrect.
Thus the correct statements are (A), (B) and (C).
Hence, the correct answer is option D.
Note
In this question we were not given the point about which the angular momentum was supposed to be calculated. So we assumed that point to be the origin while doing the calculation.
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