Electromagnetic Waves, Antennas, and RF: Magnetic Fields

EM08 introduced electric current and current density: charge in motion. The next step is to study a new vector field whose most immediate effect is also tied to moving charge: the magnetic field.

In this article the magnetic field is introduced operationally through the force it exerts on a moving charged particle. The sources of magnetic fields are postponed to EM10, where currents will be used to generate magnetic fields through the Biot–Savart law and related results.

The central magnetic-force law is

(1)

Here q is the particle charge, v is its velocity, and B is the magnetic field, often called the magnetic flux density. The cross product makes magnetic force fundamentally directional: the force depends not only on the field magnitude but also on the direction of particle motion [1, 2, 3, 5].

1 The magnetic field as a vector field

Like the electric field introduced in EM05, the magnetic field is a vector field:

(2)

At each point in space and time, B has both magnitude and direction.

Its SI unit is the tesla:

(3)

The unit follows directly from the magnetic-force law. For motion perpendicular to the field,

(4)

so

(5)

Because 1 A = 1 C/s, this may also be written as

(6)

2 The magnetic part of the Lorentz force

The complete electromagnetic force on a point charge is the Lorentz force

(7)

The electric part is

(8)

while the magnetic part is

(9)

This equation immediately reveals an important difference between electric and magnetic forces. A stationary charge can experience an electric force, but if

(10)

then

(11)

A magnetic field therefore does not exert a magnetic force on a stationary point charge.

3 Cross-product geometry

For two vectors v and B, the vector

(12)

is perpendicular to both. Its magnitude is

(13)

where 𝜃 is the angle between v and B.

Therefore the magnetic-force magnitude is

(14)

For a positive charge, the force direction is the direction of v × B. For a negative charge, the force direction is reversed.

3.1 Three important angles

If v is parallel to B,

(15)

and

(16)

If v is perpendicular to B,

(17)

and the magnetic force has its maximum magnitude:

(18)

If v is antiparallel to B,

(19)

and again

(20)

4 Only perpendicular velocity contributes

Decompose the velocity into components parallel and perpendicular to the field:

(21)

Then

(22)

Because v_∥ is parallel to B,

(23)

Thus

(24)

The parallel velocity passes through the magnetic field without being changed by the magnetic force. The perpendicular velocity is continuously deflected.

5 A magnetic field does no work on a point charge

Instantaneous mechanical power delivered by a force is

(25)

For the magnetic force,

PB	= FB ⋅ v	(26)
	= q(v × B) ⋅ v.	(27)

The vector v × B is perpendicular to v, so their dot product is zero:

(28)

Therefore an ideal magnetic field acting alone does not change the kinetic energy of a point charge:

(29)

in the nonrelativistic case.

The magnetic field can change the direction of velocity, but not its speed. This is why magnetic forces naturally produce curved particle trajectories.

6 Circular motion for velocity perpendicular to the field

Suppose a charged particle enters a uniform magnetic field with

(30)

The magnetic force is always perpendicular to the velocity. Because the speed remains constant while the direction changes, the force acts as a centripetal force.

Equating magnetic and centripetal force magnitudes gives

(31)

Cancel one factor of v:

(32)

Therefore the orbit radius is

(33)

This result is nonrelativistic. It shows that a larger momentum produces a larger orbit radius, while a stronger magnetic field bends the trajectory more tightly.

6.1 Cyclotron angular frequency

The angular speed for circular motion is

(34)

Using

(35)

we obtain

(36)

The corresponding period is

(37)

In this ideal nonrelativistic model, the cyclotron angular frequency does not depend on speed.

7 Helical motion when velocity has two components

Now suppose

(38)

The perpendicular component produces circular motion, while the parallel component remains unchanged. The result is a helix around the magnetic-field direction.

The helix radius is

(39)

During one cyclotron period, the particle moves a parallel distance

(40)

where p is the helix pitch. Therefore

(41)

This decomposition is widely useful in plasma physics, charged-particle instrumentation, and space environments.

8 Magnetic force on a current-carrying conductor

EM08 described electric current as moving charge. Since moving charges experience magnetic force, a current-carrying Conductor placed in a magnetic field can experience a net force.

For a short straight conductor segment with vector length L carrying conventional current I in a uniform magnetic field,

(42)

Its magnitude is

(43)

The vector L points in the conventional-current direction.

For a differential conductor element,

(44)

This result will become especially useful once EM10 develops the magnetic fields generated by currents themselves.

9 A note about B and H

In this article the magnetic field is described using B, measured in tesla. Electromagnetic-wave and materials discussions also commonly use the magnetic field intensity H, measured in A/m.

In vacuum they are related by

(45)

The distinction becomes more important in material media. The present article keeps the focus on B because it appears directly in the Lorentz-force law. Later wave articles will return to H when electromagnetic wave impedance and power flow are developed.

10 Worked Example 1: force on a positive charge

A proton moves with

(46)

through a uniform field

(47)

For a proton,

(48)

The direction is

(49)

The magnitude is

FB	= qvB	(50)
	= (1.602 × 10−19)(3.0 × 106)(0.20)	(51)
	≈ 9.61 × 10−14 N.	(52)

Thus

(53)

11 Worked Example 2: the same motion for an electron

Suppose an electron has the same velocity and moves through the same field as in Example 1.

The vector v × B is still directed toward −y, but the electron charge is negative:

(54)

Therefore

(55)

The positive and negative particles bend in opposite directions.

12 Worked Example 3: force at an oblique angle

A particle has

(56)

and the angle between v and B is 30^∘.

Then

FB	= |q|vB sin 𝜃	(57)
	= (2.0 × 10−6)(400)(0.50) sin 30∘	(58)
	= 2.0 × 10−4 N.	(59)

Thus

(60)

13 Worked Example 4: proving that speed stays constant

For a particle moving only under magnetic force,

(61)

Dot both sides with v:

(62)

The right side is zero. The left side is

(63)

Therefore

(64)

The magnetic field changes velocity direction but not speed.

14 Worked Example 5: proton orbit radius

A proton moves perpendicular to a uniform magnetic field with

(65)

Using

(66)

we find

r	=	(67)
	=	(68)
	≈ 6.96 × 10−2 m.	(69)

Thus

(70)

The cyclotron period is

Tc	=	(71)
	≈ 2.19 × 10−7 s.	(72)

So

(73)

15 Worked Example 6: helical motion

A particle has perpendicular and parallel velocity components

(74)

Let

(75)

The helix radius is

r	=	(76)
	=	(77)
	≈ 2.49 × 10−2 m.	(78)

Thus

(79)

The period is

(80)

The pitch is

p	= v∥Tc	(81)
	= (4.0 × 105)(5.22 × 10−7)	(82)
	≈ 0.209 m.	(83)

Therefore

(84)

16 Worked Example 7: force on a straight current segment

A straight conductor segment has length

(85)

and carries

(86)

It lies perpendicular to a uniform magnetic field

(87)

The force magnitude is

F	= ILB	(88)
	= (3.0)(0.40)(0.25)	(89)
	= 0.30 N.	(90)

Thus

(91)

Its direction is determined by L × B.

17 Common misconceptions

• Magnetic field does not mean magnetic force is always present. A stationary point charge has zero magnetic force.
• The force is not generally along B. Magnetic force is perpendicular to both v and B.
• The force is not generally along v. Its perpendicularity to velocity is exactly why magnetic force does no work on a point charge.
• A negative charge reverses the right-hand-rule force direction.
• Parallel motion is not bent. Only v_⊥ contributes to magnetic force.
• A circular orbit does not imply changing speed. The velocity vector changes direction while its magnitude remains constant.
• B and H are related but are not the same quantity.

18 Why this matters for radio waves and antennas

The present article has introduced the magnetic field through its mechanical effect on moving charge. That is only one part of the electromagnetic story.

EM08 established that electric current is organized charge motion. EM10 will show that currents generate magnetic fields. Later, Faraday’s law and Maxwell’s correction to Ampere’s law will show that changing electric and magnetic fields are coupled.

That sequence eventually leads to electromagnetic waves in which electric and magnetic fields propagate together. In a simple plane wave the geometry will become

(92)

with electromagnetic power flow connected to the vector product of the electric and magnetic fields.

The magnetic-field foundations developed here therefore become part of the mathematical language used later for radio waves, antennas, GPS links, and RF power flow.

19 Summary

The magnetic field is a vector field

(93)

whose force on a moving point charge is

(94)

The force magnitude is

(95)

Because magnetic force is perpendicular to velocity,

(96)

so a magnetic field acting alone changes velocity direction without changing particle speed.

For perpendicular motion in a uniform field,

(97)

A current-carrying conductor experiences

(98)

EM10 next reverses the viewpoint: instead of asking how a magnetic field acts on moving charge, it asks how moving charge and electric current produce magnetic fields.

References