Electromagnetic Waves, Antennas, and RF: Magnetic Forces on Charges and Currents

EM09 introduced the magnetic field through the Lorentz force on a moving charge,

(1)

It also stated the corresponding force on a straight current-carrying segment. EM10 now connects those two descriptions carefully. The main question is:

(2)

That bridge is important for motors, coils, transmission structures, and antennas. Later articles will describe how currents generate magnetic fields; here the magnetic field is treated as externally specified, and the focus is the mechanical force that field exerts on charges and currents [1, 2, 3, 5].

1 Review: magnetic force on one moving charge

For a particle of charge q and velocity v in a magnetic field B,

(3)

The magnitude is

(4)

where 𝜃 is the angle between v and B.

For a positive charge, the force direction follows the right-hand rule for v × B. A negative charge reverses that direction.

Because the cross product is perpendicular to v,

(5)

so the magnetic field does no instantaneous work on an isolated point charge:

(6)

The force can nevertheless redirect the motion.

2 From many moving charges to electric current

Consider a straight Conductor with cross-sectional area A. Suppose its mobile charge carriers have number density n, charge q, and drift velocity v_d.

The current density is

(7)

For a uniform conductor carrying conventional current I along a unit direction ℓ,

(8)

and

(9)

The sign of q is already contained in J = nqv_d. This is why conventional current points opposite to electron drift in an ordinary metal.

3 Deriving the force on a current element

Take a small conductor element of vector length

(10)

Its volume is

(11)

The number of mobile carriers in this volume is

(12)

Each carrier experiences magnetic force

(13)

Therefore the total force on the carriers in the small element is

dF	= dN qvd × B	(14)
	= nAdℓqvd × B.	(15)

Since

(16)

and, for a uniform straight conductor,

(17)

we obtain

(18)

This equation is the differential force law for a thin current-carrying wire.

4 Force on a finite wire

For a wire following a path C,

(19)

If B is uniform, it can be taken outside the integral:

(20)

The line integral of dℓ is simply the displacement from the wire’s starting point to its ending point:

(21)

Therefore, for a wire segment in a uniform field,

(22)

For a straight segment, define

(23)

and obtain the familiar result

(24)

Its magnitude is

(25)

5 Force density in a continuous current distribution

The current-density description is even more general.

For a small volume dV carrying current density J, the force is

(26)

This identifies the magnetic force per unit volume:

(27)

Its SI units are

(28)

The total magnetic force on a volume V is therefore

(29)

This form is especially important later when currents are distributed through conductors, coils, and antenna structures rather than confined to an ideal filament.

6 A closed current loop in a uniform field

For a closed loop in a uniform magnetic field,

(30)

Because B is uniform,

(31)

But a closed path returns to its starting point, so

(32)

Hence

(33)

for a closed current loop in a uniform magnetic field.

Zero net force does not imply zero mechanical effect. Different parts of the loop can experience opposite forces that form a couple and produce torque.

7 Torque on a rectangular current loop

Consider a rectangular loop carrying current I in a uniform magnetic field. Let the loop have area A and unit normal n. Let 𝜃 be the angle between n and B.

Opposite sides of the loop experience equal and opposite magnetic forces. The net force is zero, but the separated forces produce a torque.

For one turn, the torque magnitude is

(34)

For N identical turns,

(35)

8 Magnetic dipole moment

The magnetic dipole moment of a planar current loop is defined as

(36)

For N turns,

(37)

The unit normal is set by a right-hand rule: curl the fingers of the right hand in the direction of conventional current; the thumb gives n and therefore the direction of μ.

Using μ, the torque law becomes

(38)

Its magnitude is

(39)

The magnetic moment therefore packages the current, loop area, number of turns, and loop orientation into one vector.

9 Magnetic dipole potential energy

A magnetic dipole in an external magnetic field has orientation-dependent potential energy

(40)

Therefore

(41)

The minimum energy occurs at

(42)

when μ is aligned with B.

The maximum energy occurs at

(43)

when they are antiparallel.

10 Does this contradict the statement that magnetic force does no work?

At first glance, rotational motion of a current loop may seem to contradict the result from EM09 that magnetic force does no work on an individual point charge.

There is no contradiction.

For an isolated charge,

(44)

In a conductor, however, mobile charges are constrained by the material, and maintaining a current generally involves electric fields, lattice forces, and possibly an external source. Mechanical energy can be exchanged among the field, the conductor, and the source while the magnetic force on each individual carrier remains perpendicular to that carrier’s instantaneous velocity contribution associated with the Lorentz force.

For introductory calculations, the potential-energy formula

(45)

is therefore used for the mechanical orientation of a current loop in an externally imposed field.

11 Worked Example 1: vector magnetic force on a charge

A positive charge

(46)

moves with

(47)

through

(48)

Then

FB	= qv × B	(49)
	= (2.0 × 10−6)(3.0 × 104)(0.20)(x ×z).	(50)

Since

(51)

we obtain

(52)

12 Worked Example 2: force on a straight conductor

A wire of length

(53)

carries

(54)

along +x through a field

(55)

The force is

F	= IL × B	(56)
	= (4.0)(0.50)(0.30)(x ×z)	(57)
	= −0.60y N.	(58)

Thus

(59)

13 Worked Example 3: force density

Suppose

(60)

and

(61)

Then

fB	= J × B	(62)
	= (2.5 × 106)(0.040)(x ×z)	(63)
	= −1.0 × 105y N/m3.	(64)

Hence

(65)

14 Worked Example 4: curved wire in a uniform field

A wire carries current I = 2.0 A from

(66)

to

(67)

along an arbitrary curved path. A uniform field is

(68)

The net force depends only on the endpoint displacement:

F	= I(r2 − r1) × B	(69)
	= 2.0(0.30x + 0.40y) × (0.50z).	(70)

Using

(71)

we get

(72)

15 Worked Example 5: torque on a single current loop

A one-turn loop has

(73)

and its normal makes an angle

(74)

with a uniform field

(75)

The magnetic moment magnitude is

(76)

The torque magnitude is

τ	= μB sin 𝜃	(77)
	= (0.060)(0.40) sin 30∘	(78)
	= 1.2 × 10−2 N m.	(79)

Thus

(80)

16 Worked Example 6: multiturn coil

A coil has

(81)

Its normal is perpendicular to a field

(82)

The magnetic moment is

μ	= NIA	(83)
	= 100(0.50)(4.0 × 10−4)	(84)
	= 2.0 × 10−2 A m2.	(85)

Since 𝜃 = 90^∘,

(86)

Therefore

(87)

17 Worked Example 7: dipole energy change

A magnetic dipole has

(88)

in a field

(89)

Compare the potential energy at 𝜃 = 90^∘ with the energy at alignment, 𝜃 = 0.

At 90^∘,

(90)

At 0,

U0	= −μB	(91)
	= −(0.080)(0.25)	(92)
	= −2.0 × 10−2 J.	(93)

Therefore the change from 90^∘ to alignment is

(94)

The aligned state is lower in energy.

18 Common misconceptions

• Current direction is conventional current direction. In a metal, electron drift is opposite to the direction used for I dℓ.
• The force is not generally parallel to the current. It is proportional to dℓ× B.
• A closed loop can have zero net force and still have nonzero torque.
• The magnetic dipole moment is not the magnetic field. μ characterizes the loop; B characterizes the applied field.
• A uniform magnetic field produces zero net force on a closed loop, but a nonuniform field need not.
• J × B is a force density. It must be integrated over volume to obtain total force.
• Magnetic-force zero-work on a point charge does not forbid mechanical torque on a constrained current loop.

19 Why this matters for antennas and RF structures

An antenna is not just a geometric object. It supports time-varying charge and current distributions. Later articles will use those currents as sources of electromagnetic fields.

The present article establishes the mechanical side of that current-field interaction. In distributed form,

(95)

shows that a magnetic field acts locally on current density.

This language becomes useful in antenna conductors, inductive structures, motors, coils, and electromagnetic stress analysis. It also prepares the notation needed in EM11, where the direction of reasoning is reversed: currents become the source of magnetic fields through the Biot–Savart law.

20 Summary

The microscopic magnetic-force law is

(96)

For a thin current element,

(97)

For a current density,

(98)

and

(99)

A closed loop in a uniform field has zero net force but can experience torque. Defining

(100)

we obtain

(101)

and

(102)

EM11 next asks how moving charge and current generate magnetic fields.

References