BE1-HEM1 Electromagnetics 1
Notes 4:Magnetic force and Magnetic Field
1. Introduction
In addition to the force described by Coulomb’s Law, another force exists between charges, but only when
they move. This force can be demonstrated when current carrying wires of two different circuits I1 and
I2 are placed parallel to each other as shown in the diagram below.
I1
I2
- - - - - -
+ + + + + +
v
F
F
- - - - - -
+ + + + + +
v
Suppose the current in each wire is the same I1 = I2 , and is described by positive charges moving
at velocity v , and that the corresponding negative charges in the wires are stationary (this describes
conventional current, although in reality it is the electrons that are moving, and the positive ions are
stationary). If the positive and negative charges are equally spaced, the overall charge density is zero,
and so no Coulomb force exists between the charges. However, an attractive force between the wires is
experienced - each wire feels the same magnitude of force - which becomes an equal but repulsive force
if the currents are travelling in opposite directions to each other. If the wires are perpendicular to each
other, this force drops to zero, and in general the magnitude of the force is proportional to the parallel
component of the direction of the wires relative to each other (ie proportional to cos θ , where θ is the
angle between them). This force is known as the magnetic force.
In general, we have to consider the force between two complete circuits carrying currents I1 and I2 as
shown:-
I1
I2
2. Magnetic field
When analyzing the electric force between two charges, we started with the equation describing the force
- Coulomb’s Law - and then, to make calculations more simple, we introduced a two step process - first
calculating the electric field E due to the first charge, and then find the force on the second elemental
charge within that electric field dF = Edq .
Magnetic force is slightly more involved, and the best way to introduce it is the other way around -
to assume and describe the existence of a magnetic field B being generated by the first wire, and intro-
duce the description of force on a current element in the second wire due to its presence in that magnetic
field. We can then formulate an equation for magnetic force between the wires.
However, the two stage principle is the same for electric and magnetic field, ie that the magnetic field is
a mathematical concept introduced to make the force easier to analyze, but we must always remember
that the force is the more fundamental quantity.
1
, BE1-HEM1 Electromagnetics 1:Notes 4:Magnetic force and Magnetic Field
3. Magnetic field around a long wire - qualitative
The magnetic field B is a vector field. It is often referred to as the magnetic flux density and has the
N
units Tesla ( Am ).
The B-field vectors around a straight wire are tangential to perpendicular concentric circles, with the
direction of the vectors given by the right hand grip rule - if the thumb is pointing in the direction of
the current, the direction of the fingers gives the direction of the vectors. Their magnitude decreases
inversely proportionally to the radius of the circles - ie the perpendicular distance from the wire. In the
same way that electric field lines were drawn for electric field, magnetic field lines are drawn tangential to
the vectors of the magnetic field, and the distance between the field lines should indicate its magnitude.
I
I
Vectors Field lines
We will see shortly a more general method of calculating B.
4. Magnetic force on a current element
When considering the electric field, we found the force on an element of charge dq . For magnetic field, we
will consider a current element dI. This is a vector in the direction of current flow d̂l whose magnitude
is the size of the current I × the small length of the element dl.
Current element dI = Idl
I
Vector dl
In the diagram below, the current I1 flowing in the first wire is generating the magnetic field B1 , and
we will consider a small element of current dI2 in the second wire that lies within B1 . (The wires are
not touching, but separated by a certain distance)
I I2 B
dI2
dI 2 θ
dF
dI2 experiences a force dF21 due to its presence in the magnetic field B1 that is perpendicular to both
B1 and its own direction. In the diagram, this would be perpendicularly into the page. The magnitude
of this force is dF21 = B1 dI2 sin θ . Taking the direction into account, this force can be expressed as a
vector (cross) product:-
dF21 = dI2 × B1
2
Notes 4:Magnetic force and Magnetic Field
1. Introduction
In addition to the force described by Coulomb’s Law, another force exists between charges, but only when
they move. This force can be demonstrated when current carrying wires of two different circuits I1 and
I2 are placed parallel to each other as shown in the diagram below.
I1
I2
- - - - - -
+ + + + + +
v
F
F
- - - - - -
+ + + + + +
v
Suppose the current in each wire is the same I1 = I2 , and is described by positive charges moving
at velocity v , and that the corresponding negative charges in the wires are stationary (this describes
conventional current, although in reality it is the electrons that are moving, and the positive ions are
stationary). If the positive and negative charges are equally spaced, the overall charge density is zero,
and so no Coulomb force exists between the charges. However, an attractive force between the wires is
experienced - each wire feels the same magnitude of force - which becomes an equal but repulsive force
if the currents are travelling in opposite directions to each other. If the wires are perpendicular to each
other, this force drops to zero, and in general the magnitude of the force is proportional to the parallel
component of the direction of the wires relative to each other (ie proportional to cos θ , where θ is the
angle between them). This force is known as the magnetic force.
In general, we have to consider the force between two complete circuits carrying currents I1 and I2 as
shown:-
I1
I2
2. Magnetic field
When analyzing the electric force between two charges, we started with the equation describing the force
- Coulomb’s Law - and then, to make calculations more simple, we introduced a two step process - first
calculating the electric field E due to the first charge, and then find the force on the second elemental
charge within that electric field dF = Edq .
Magnetic force is slightly more involved, and the best way to introduce it is the other way around -
to assume and describe the existence of a magnetic field B being generated by the first wire, and intro-
duce the description of force on a current element in the second wire due to its presence in that magnetic
field. We can then formulate an equation for magnetic force between the wires.
However, the two stage principle is the same for electric and magnetic field, ie that the magnetic field is
a mathematical concept introduced to make the force easier to analyze, but we must always remember
that the force is the more fundamental quantity.
1
, BE1-HEM1 Electromagnetics 1:Notes 4:Magnetic force and Magnetic Field
3. Magnetic field around a long wire - qualitative
The magnetic field B is a vector field. It is often referred to as the magnetic flux density and has the
N
units Tesla ( Am ).
The B-field vectors around a straight wire are tangential to perpendicular concentric circles, with the
direction of the vectors given by the right hand grip rule - if the thumb is pointing in the direction of
the current, the direction of the fingers gives the direction of the vectors. Their magnitude decreases
inversely proportionally to the radius of the circles - ie the perpendicular distance from the wire. In the
same way that electric field lines were drawn for electric field, magnetic field lines are drawn tangential to
the vectors of the magnetic field, and the distance between the field lines should indicate its magnitude.
I
I
Vectors Field lines
We will see shortly a more general method of calculating B.
4. Magnetic force on a current element
When considering the electric field, we found the force on an element of charge dq . For magnetic field, we
will consider a current element dI. This is a vector in the direction of current flow d̂l whose magnitude
is the size of the current I × the small length of the element dl.
Current element dI = Idl
I
Vector dl
In the diagram below, the current I1 flowing in the first wire is generating the magnetic field B1 , and
we will consider a small element of current dI2 in the second wire that lies within B1 . (The wires are
not touching, but separated by a certain distance)
I I2 B
dI2
dI 2 θ
dF
dI2 experiences a force dF21 due to its presence in the magnetic field B1 that is perpendicular to both
B1 and its own direction. In the diagram, this would be perpendicularly into the page. The magnitude
of this force is dF21 = B1 dI2 sin θ . Taking the direction into account, this force can be expressed as a
vector (cross) product:-
dF21 = dI2 × B1
2
