In this section each equation has some statements (A, B, C, D ……. .) given in column-I and some

In this section each equation has some statements (A, B, C,...

In this section each equation has some statements $(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D} \ldots \ldots . .)$ given in column-I and some statements $(\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}, \mathrm{T}, \ldots \ldots)$ in column-II. Any given statement in column-I can have correct matching with ONE OR MORE statements (s) in column-II.\begin{tabular}{l|r} Column-I & Column-II \\ \hline \end{tabular} (A) A charged particle moving (P) un deflected parallel to direction of mag. field (B) A charged particle moving (Q) circular path perpendicular to the direction of magnetic field (C) A charged particle moving (R) Helical path at an angle in a region of strong mag. field (D) A charged particle moving (S) parabolic path in a strong and uniform electric field of large region

In this section each equation has some statements $(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D} \ldots \ldots . .)$ given in column-I and some statements
$(\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}, \mathrm{T}, \ldots \ldots)$ in column-II. Any given statement in
column-I can have correct matching with ONE OR MORE statements (s) in column-II.\begin{tabular}{l|r} Column-I & Column-II \\ \hline \end{tabular}
(A) A charged particle moving
(P) un deflected parallel to direction of mag. field
(B) A charged particle moving
(Q) circular path perpendicular to the direction of magnetic field
(C) A charged particle moving
(R) Helical path at an angle in a region of strong mag. field
(D) A charged particle moving
(S) parabolic path in a strong and uniform electric field of large region

Key Concepts

Motion of Charged Particles in Electric Fields

In an electric field, a charged particle experiences a force directly proportional to the electric field strength and the charge, acting in the direction of the field for a positive charge and opposite for a negative charge. This force imparts a linear acceleration, leading to parabolic trajectories if the particle is also under the influence of other forces, or altering its speed and direction in a uniform field. This principle is fundamental in understanding particle acceleration and deflection in devices such as cathode ray tubes.

Helical Motion

When a charged particle's velocity has both perpendicular and parallel components with respect to a magnetic field, the perpendicular component leads to circular motion while the parallel component allows the particle to move in the direction of the magnetic field. The resulting trajectory is helical. The pitch of the helix depends on the magnitude of the parallel component of the velocity relative to the perpendicular one.

Motion of Charged Particles in Magnetic Fields

When a charged particle moves in a magnetic field, the magnetic part of the Lorentz force acts perpendicular to both the velocity of the particle and the magnetic field. This results in the particle executing circular or helical motion, depending on the angle between the velocity vector and the magnetic field. The magnitude of the force and the ensuing acceleration are determined by the charge, the velocity, and the magnetic field strength.

Lorentz Force

The Lorentz force is the combined force experienced by a charged particle when it is in the presence of electric and magnetic fields. This force is given by the equation F = q(E + v × B), where q is the charge, E is the electric field, v is the velocity of the particle, and B is the magnetic field. The force's direction and magnitude determine the resulting motion of the particle, making it a fundamental concept in electromagnetism.

Circular Motion in Magnetic Fields

If the velocity of a charged particle is entirely perpendicular to a uniform magnetic field, the particle experiences a centripetal force that causes it to move in a uniform circular path. The radius of this path depends on the balance between the magnetic force and the centripetal force required for circular motion. This concept is key in devices like cyclotrons, where particles are accelerated along circular paths.

Transcript

00:01 Using this equation and the definition of the period, find the correct expression for the period.

00:08 So for part a, we know that the orbital radius classically is mv over qb.

00:24 So that makes the period 2 pi r not over v, which is 2 pi m, the v.

00:40 Will cancel over qb.

00:45 So in the relativistic limit, r is equal to the momentum over qb, which is the momentum is gamma mv over qb, and this is gamma r0, which gives us a period of gamma 2 pi m over qb.

01:22 Which is gamma t knot, so this is t knot.

01:29 Part b is t independent of v.

01:37 It's not.

01:39 So it depends on, it's not independent of v.

01:55 Part c, if a 10 mega electron moves in a circular path in a uniform magnetic field of magnitude 2 .2 tesla, what are the radius, according to chapter 28, the correct radius, the period, according to 28, and the real period.

02:17 Okay, so the 10 mega electron volts is the kinetic energy.

02:23 So we know that the rest mass energy of an electron is 0 .511 mega electron volts.

02:34 So the classical kinetic energy is 1 .5 mb squared, which is 1⁄2 mc squared, b squared over 6.

02:43 Squared, which is one -half -m -c -squared, beta -squared.

02:54 So if the classical kinetic energy is 10 mega -electron volts, solving for beta, and we get 6 .256, since we know the rest -mass energy is 0 .511 mega -electron volts.

03:15 But a beta factor of 6 .256 is impossible because it is higher than the speed of light.

03:24 But to find the classical radius, let's use this energy...

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