Magnetic Force, Helical Motion & Lorentz Force

Magnetic Force

The force due to Magnetic Field acts on electric charges in a particular manner. If you simply place an electric charge in a magnetic field nothing happens, i.e., when an electric charge is at rest in a constant magnetic field it experiences no magnetic force. Electric charges have to move to experience a force from a magnetic field.

IMPORTANT

1. The force on a charge q is proportional to the magnitude of the charge and its speed v. When the charge is not moving, there is no magnetic force on it.
2. The direction of the force is perpendicular to the direction of both the velocity v and the magnetic induction field B.
3. When the velocity of the charge is parallel to the magnetic field the force is zero.
F = |F| = qvB sin0 = 0

The above points may be summarised in a mathematical expression as

F = q( v ×  B)                                     (10)

The magnitude of the force is given by     F = qvB sinθ

where θ is the angle between the vectors v and B.

Work done by a Magnetic Force is zero.

Since F is always perpendicular to v there is no component of acceleration in the direction of velocity vector, therefore, magnitude of the velocity does not change, thus kinetic energy of the charge does not change, hence no work is done by the magnetic force.

(a)  What is the magnetic field produced by a point charge q moving at velocity v ?Illustration–4

(b)  What is the force between two equal charges moving parallel to each other with the same velocity ?

Solution

(a)  The Biot-Savart law refers the magnetic field produced by a current element, Id.

Since I = dq/dt we rewrite

This force is attractive.

The net forced between the charge is

Motion of a charged particle in a Uniform Magnetic Field

Note that time period is independent of velocity. All particles with same charge to mass ratio (q/m), have the same period.

Illustration–6

An electron with a kinetic energy of 103 eV moves perpendicular to the direction of uniform field B = 104 T.

(a)  What is the period of its orbit ?                      (b)        What is the radius of the body ?

Solution

(a)  Using equation (12)

(b)  Using equation (11), the radius of the orbit is given by

Illustration–7

A particle of mass m and charge q is projected into a region having a perpendicular uniform magnetic field B. Find the angle of  deviation q of the particle as it comes out of the magnetic field if width d of the region is equal to

Helical Motion

The parallel component v|| is unaffected by the magnetic field B, therefore, the particle moves with constant velocity parallel to the field. The resultant motion is a uniform circular motion perpendicular to field lines and a constant linear motion along the field lines. This is called a circular helical path.

The pitch of a helix is defined as the linear distance moved by the particle in one revolution.

Illustration–8

An electron moving at 4 ´ 106 m/s enters a uniform field B = 0.04 T at 30° to the lines. What is the pitch of the helical path.

Lorentz Force

When a particle is subjected to both electric and magnetic fields in the same region, the total force on it is called the Lorentz force.

We know that the path of particle in a uniform magnetic field is a helix and the path in a uniform electric field is parabola. When a particle is subject to both electric and magnetic fields, the motion is in general quite complex. However, the special cases in which the fields are either parallel to perpendicular to each other are simple to analyze.

Illustration–9

 Uniform electric  E and magnetic B  field are directed along the y – axis as shown in the figure (13) . A particle with specific change (q/m) leaves the origin O along the x-axis with velocity vo.       (a)  Find the coordinate yn of the particle when it crosses the y-axis for the nth time.       (b) Find the angle between the particle vector and the y-axis at that moment.
Solution

(a)  Since magnetic field does not accelerate a particle in its own direction, therefore, the distance moved along the y-axis is only due to electric field.

DRILL EXERCISE–2

1. (a) What is the minimum magnetic field needed to exert a 5.4 × 10–15 N force on an electron moving at 2.1 × 107 m/s? (b) What magnetic field strength would be required if the field were at 45º to the electron’s velocity?
1. What is the magnitude of the magnetic force on a proton moving at 2.5 × 105 m/s (a) at right angles; (b) at 30º; (c) parallel to a magnetic field of 0.50 T?
1. A velocity selector uses a 60-mT magnetic field perpendicular to a 24-kN/C electric field. At what speed will charged particles pass through the selector undeflected?
1. How long does it take an electron to complete a circular orbit at right angles to a 1.0-G magnetic field?
1. Two protons, moving in a plane perpendicular to a uniform magnetic field of 500 G, undergo an elastic head-on collision. How much time elapses before they collide again? Hint: Draw a picture.

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Magnetic Force, Helical Motion & Lorentz Force

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