CHAPTER 3
The Magnetic Field
3.1. Force due to the Magnetic Field
3.1.1 The Lorentz force law
F IGURE 3.1.1: Direction of magnetic field produced F IGURE 3.1.2: Direction of magnetic force by Lorentz
by a constant current law
Magnetism was a phenomena known well before the discovery of electricity through the discovery of
lodestones which are iron-rich rocks that are naturally occurring permanent magnets. Hans Christian
Oersted conducted thirteen years of careful experimentation to show that electric currents can create
magnetic fields. Andre-Marie Ampere then postulated that the only source of magnetic fields have
to be electrical currents and similar electrical effects. Ampere also postulated that there were no such
things as magnetic monopoles1 , i.e. there is no such thing as an isolated magnetic point charge. The
magnetic effects in a magnet were due to microscopic electrical loops that existed within the magnet.
In our discussion of electrostatics, we had assumed that the charges are not moving. However mag-
netic fields can only be generated by charges in motion. Therefore we will now be discussing moving
charges and their magnetic effects. If a charge moves with constant velocity, it will produce a constant mag-
1
Interestingly, many high energy theories in physics predict that magnetic monopoles should have been created during
the early moments after the Big Bang. Since we have never had a confirmed detection of a magnetic monopole in our
universe till date, a major open question in physics is the fate of these early universe monopoles.
71
,CHAPTER 3. THE MAGNETIC FIELD 72
netic field. Therefore constant electric currents, which is a series of point charges moving at constant
velocity, produce relatively static magnetic fields.
Figure2 3.1.1 gives us the direction of the magnetic field produced by a constant current (or a charge
moving with constant velocity). Take our right hand: if the thumb is the direction of the current/moving
charge, then our curled fingers gives us the direction of the magnetic field. Therefore we can see that
magnetic fields do not necessarily travel in straight lines. This is important to keep in mind, especially
because it can complicate the vector nature of the problem.
When it comes to force due to a magnetic field, we have to understand that since the magnetic field
is due to an electric charge, we will also have to consider the force created by the electric field also.
Therefore the complete expression for the force is given by
⃗F = q ⃗E + ⃗v × B
⃗ (3.1.1)
This is called the Lorentz force law. Here q is the charge, ⃗E is the electric field, ⃗v is the velocity, and
⃗ is the magnetic field.
B
If we have no electric fields to worry about, i.e. ⃗E = 0 then the force is purely due to the magnetic
field
⃗F = q ⃗v × B⃗
Figure3 3.1.2 shows us the direction of the force due to the magnetic field only. We can again use the
right hand rule to work out the direction of the force: if the direction of current/moving charge is the
index finger and the direction of the magnetic field is the middle finger, then the thumb gives us the
direction of the resulting force.
An interesting consequence of the nature of this
vector force is shown visually in Figure4 3.1.3.
In this setup, two wires are initially kept paral-
lel to each other and hooked to the same battery.
Thus when the switch is flipped, both of them
will have the same current flowing through the
wires. In figure (a), the current is flowing in
the opposite directions, and the wires will repel
each other via the Lorentz force. Meanwhile, in
figure (b), the wires have current flowing in the
same direction and we can see that they will at-
tract each other via the Lorentz force5 .
From equation 3.1.1 we can also define what
a magnetic field is. Fundamentally, the force
on any charge is described by two vector fields. F IGURE 3.1.3: Magnetic forces between current carry-
The first field generates the component of force ing wires
that is independent of motion (the electrostatic
force), while the second field generates the com-
ponent of force that is proportional to both the speed and direction of the moving charge. This second
⃗ whose value can be computed from the Lorentz force law.
vector field is called the magnetic field (B)
Alternate names for B ⃗ are the magnetic flux density and the magnetic induction.
2
Image taken from Griffiths, D.J. (2013). Introduction to electrodynamics (4th ed.). Pearson. Page 211.
3
Ibid.
4
Ibid.
5
I invite you all to use the right hand rule and work out the direction of the Lorentz force due to each wire, while taking
the direction of the current into account.
, CHAPTER 3. THE MAGNETIC FIELD 73
The SI unit of the magnetic field is Tesla and the cgs unit is gauss. The relation between the two is
1 T = 104 gauss .
Example A point charge with q = −3.64 nC is moving with a velocity of 2.75 × 103 î m/s. What is the force
on the charge if it is moving in a magnetic field given by 0.75 î + 0.75 k̂ T.
Here there is no electric field, so we have ⃗E = 0. Thus the total force is
⃗F = q ⃗v × B ⃗
= −3.64 × 10−9 2.75 × 103 î × 0.75 î + 0.75 k̂
= −3.64 × 10−9 × 2.75 × 103 × 0.75 î × î + î × k̂
| {z } | {z }
=0 =−ĵ
−6
= 9.26 × 10 ĵ N
3.1.2 Magnetic forces do no work
Another important consequence of the Lorentz force law is that magnetic forces do no work. Proving
this is relatively simple. Let us temporarily define C⃗ = ⃗v × B.
⃗ Now let us take some charge moving
with a velocity ⃗v. Therefore, its total displacement during some small time instant dt is given by
d⃗l = ⃗v dt. The small amount of work done is then given by
dW = ⃗F • d⃗l = q ⃗v × B
⃗ • (⃗v dt)
⃗ • ⃗v dt
= qC
Let us now look more closely at C ⃗ • ⃗v. Since C
⃗ is a cross product of ⃗v and another vector, it is always
perpendicular in direction to ⃗v for any possible case. Thus C ⃗ • ⃗v = 0, since the dot product between any
two perpendicular vectors is always 0. This implies that
dW = 0 (3.1.2)
Thus magnetic fields will alter the direction of the charged particle but will never increase or decrease its
velocity (i.e. cause an acceleration).
3.1.3 Magnetic forces due to currents
Let us consider a wire with a line charge density of λ that is travelling along the wire with a velocity
⃗v. This we can describe the current in the wire as
⃗I = λ⃗v (3.1.3)
Here ⃗I is defined as a vector whose direction is the same as the direction of the wire. Usually we only
truly care about the magnitude of the current, so the full vector formulation is ignored. The magnetic
force due to the current carrying wire in then
ˆ
⃗Fmag = ⃗v × B⃗ dq
ˆ
= ⃗ λ dl
⃗v × B
ˆL
= ⃗I × B
⃗ dl
L
The Magnetic Field
3.1. Force due to the Magnetic Field
3.1.1 The Lorentz force law
F IGURE 3.1.1: Direction of magnetic field produced F IGURE 3.1.2: Direction of magnetic force by Lorentz
by a constant current law
Magnetism was a phenomena known well before the discovery of electricity through the discovery of
lodestones which are iron-rich rocks that are naturally occurring permanent magnets. Hans Christian
Oersted conducted thirteen years of careful experimentation to show that electric currents can create
magnetic fields. Andre-Marie Ampere then postulated that the only source of magnetic fields have
to be electrical currents and similar electrical effects. Ampere also postulated that there were no such
things as magnetic monopoles1 , i.e. there is no such thing as an isolated magnetic point charge. The
magnetic effects in a magnet were due to microscopic electrical loops that existed within the magnet.
In our discussion of electrostatics, we had assumed that the charges are not moving. However mag-
netic fields can only be generated by charges in motion. Therefore we will now be discussing moving
charges and their magnetic effects. If a charge moves with constant velocity, it will produce a constant mag-
1
Interestingly, many high energy theories in physics predict that magnetic monopoles should have been created during
the early moments after the Big Bang. Since we have never had a confirmed detection of a magnetic monopole in our
universe till date, a major open question in physics is the fate of these early universe monopoles.
71
,CHAPTER 3. THE MAGNETIC FIELD 72
netic field. Therefore constant electric currents, which is a series of point charges moving at constant
velocity, produce relatively static magnetic fields.
Figure2 3.1.1 gives us the direction of the magnetic field produced by a constant current (or a charge
moving with constant velocity). Take our right hand: if the thumb is the direction of the current/moving
charge, then our curled fingers gives us the direction of the magnetic field. Therefore we can see that
magnetic fields do not necessarily travel in straight lines. This is important to keep in mind, especially
because it can complicate the vector nature of the problem.
When it comes to force due to a magnetic field, we have to understand that since the magnetic field
is due to an electric charge, we will also have to consider the force created by the electric field also.
Therefore the complete expression for the force is given by
⃗F = q ⃗E + ⃗v × B
⃗ (3.1.1)
This is called the Lorentz force law. Here q is the charge, ⃗E is the electric field, ⃗v is the velocity, and
⃗ is the magnetic field.
B
If we have no electric fields to worry about, i.e. ⃗E = 0 then the force is purely due to the magnetic
field
⃗F = q ⃗v × B⃗
Figure3 3.1.2 shows us the direction of the force due to the magnetic field only. We can again use the
right hand rule to work out the direction of the force: if the direction of current/moving charge is the
index finger and the direction of the magnetic field is the middle finger, then the thumb gives us the
direction of the resulting force.
An interesting consequence of the nature of this
vector force is shown visually in Figure4 3.1.3.
In this setup, two wires are initially kept paral-
lel to each other and hooked to the same battery.
Thus when the switch is flipped, both of them
will have the same current flowing through the
wires. In figure (a), the current is flowing in
the opposite directions, and the wires will repel
each other via the Lorentz force. Meanwhile, in
figure (b), the wires have current flowing in the
same direction and we can see that they will at-
tract each other via the Lorentz force5 .
From equation 3.1.1 we can also define what
a magnetic field is. Fundamentally, the force
on any charge is described by two vector fields. F IGURE 3.1.3: Magnetic forces between current carry-
The first field generates the component of force ing wires
that is independent of motion (the electrostatic
force), while the second field generates the com-
ponent of force that is proportional to both the speed and direction of the moving charge. This second
⃗ whose value can be computed from the Lorentz force law.
vector field is called the magnetic field (B)
Alternate names for B ⃗ are the magnetic flux density and the magnetic induction.
2
Image taken from Griffiths, D.J. (2013). Introduction to electrodynamics (4th ed.). Pearson. Page 211.
3
Ibid.
4
Ibid.
5
I invite you all to use the right hand rule and work out the direction of the Lorentz force due to each wire, while taking
the direction of the current into account.
, CHAPTER 3. THE MAGNETIC FIELD 73
The SI unit of the magnetic field is Tesla and the cgs unit is gauss. The relation between the two is
1 T = 104 gauss .
Example A point charge with q = −3.64 nC is moving with a velocity of 2.75 × 103 î m/s. What is the force
on the charge if it is moving in a magnetic field given by 0.75 î + 0.75 k̂ T.
Here there is no electric field, so we have ⃗E = 0. Thus the total force is
⃗F = q ⃗v × B ⃗
= −3.64 × 10−9 2.75 × 103 î × 0.75 î + 0.75 k̂
= −3.64 × 10−9 × 2.75 × 103 × 0.75 î × î + î × k̂
| {z } | {z }
=0 =−ĵ
−6
= 9.26 × 10 ĵ N
3.1.2 Magnetic forces do no work
Another important consequence of the Lorentz force law is that magnetic forces do no work. Proving
this is relatively simple. Let us temporarily define C⃗ = ⃗v × B.
⃗ Now let us take some charge moving
with a velocity ⃗v. Therefore, its total displacement during some small time instant dt is given by
d⃗l = ⃗v dt. The small amount of work done is then given by
dW = ⃗F • d⃗l = q ⃗v × B
⃗ • (⃗v dt)
⃗ • ⃗v dt
= qC
Let us now look more closely at C ⃗ • ⃗v. Since C
⃗ is a cross product of ⃗v and another vector, it is always
perpendicular in direction to ⃗v for any possible case. Thus C ⃗ • ⃗v = 0, since the dot product between any
two perpendicular vectors is always 0. This implies that
dW = 0 (3.1.2)
Thus magnetic fields will alter the direction of the charged particle but will never increase or decrease its
velocity (i.e. cause an acceleration).
3.1.3 Magnetic forces due to currents
Let us consider a wire with a line charge density of λ that is travelling along the wire with a velocity
⃗v. This we can describe the current in the wire as
⃗I = λ⃗v (3.1.3)
Here ⃗I is defined as a vector whose direction is the same as the direction of the wire. Usually we only
truly care about the magnitude of the current, so the full vector formulation is ignored. The magnetic
force due to the current carrying wire in then
ˆ
⃗Fmag = ⃗v × B⃗ dq
ˆ
= ⃗ λ dl
⃗v × B
ˆL
= ⃗I × B
⃗ dl
L
