Introduction
There is a strong connection between the electric and magnetic fields. In this lesson, we will discuss magnetic materials, magnetic forces, motion of a charge in magnetic field, and magnetic field created by an electrical current.
Magnetic Materials, Magnetic Field, and Magnetic Forces
What properties do refrigerator magnets have in common with Earth? First, the medium around each of them is characterized by a certain property called a magnetic field. A magnetic field manifests itself by the interaction between the magnet producing the field and other magnets or iron-based objects that might be in the vicinity. Any magnet, regardless of shape, has two poles: a north pole and a south pole. If you divide the magnet into two pieces, each of the pieces will also display the properties of north and south poles. Further reducing the size of the magnet does not separate the poles but keeps a north and a south pole in each of the pieces.
Some materials are naturally magnetic (for example, magnetite discovered to be magnetic for the first time about 2,000 years ago), and others can be magnetized (for example, iron-based objects), and still others will never display any magnetic properties. The existence of these categories is determined, as before with electrical charges, by the structure of the substance and the specific bonds between the atoms and molecules that are connected with each other.
Around a magnetic field, the magnetic properties can be described by the magnetic field lines, which are similar to the electric field lines. The direction of the magnetic field lines is from the N to the S pole outside the magnet and the other way inside the magnet (as shown in Figure 15.1 for a bar magnet).
The magnetic interaction, like that in the case of electrical charge interaction, is of two kinds: attraction and repulsion. The attraction force is established between two magnets facing each other with the different poles (N and S). The repulsion force is established between two poles of the same kind (N and N, or Sand S).
Magnetic Force
The magnetic force is proportional to the strength of the magnetic field, to the velocity of the moving charge, and to the value of the charge.
F = q · v 
B
Motion of a Point Charge in a Magnetic Field
Consider a point -like charge in a magnetic field. If the charge is in motion in the field, it can be shown that a force is acting on the charge that will curve the straight trajectory of the charge in the absence of any other interaction.
B is the value of the magnetic field at a point. This quantity is a vector because the magnetic field, similar to the electric field, has a definite direction. The unit for the magnetic field can be determined from the equation for magnetic force if we take only the scalar part of the magnetic force. Then you can see that magnetic field is measured in
This is a unit called a tesla (1 T) for Nicola Tesla (1856– 1943). A field of 1 T is a very large field (compare to the field of Earth, which is less than 10 –4 T). Therefore, the usual unit for magnetic fields is gauss (1 G).
1 G = 10–4T
The cross-product between the velocity and the magnetic field B can be interpreted in the following manner. To determine the magnetic force, you should consider only the speed perpendicular to the magnetic field, hence the magnetic force value is:
F = q · v · B · sin α
where α is the angle between the velocity v and magnetic field B.
If you construct a plane that contains both the velocity and the magnetic field, the resultant magnetic force is perpendicular to the field, whereas the direction (up or down) is given to the sign of the charge. See Figure 15.2.
The rule to determine the direction of the force is called the right-hand rule or the corkscrew rule.
In the case of the corkscrew for a positive charge, as shown in Figure 15.2, if you rotate the screw so that the velocity vector becomes superimposed on the magnetic field, the direction of advance of the screw is the direction of the force.
If the magnetic force is the net force on the particle, then the charge will be accelerated in the direction of the net force and the trajectory will be curved.
Right-Hand Rule
Using the right hand, palm up, point the fingers in the direction of the vector magnetic field B and the thumb in the direction of the vector velocity v. The force will be perpendicular to the palm, and, if the charge is positive, the force will point upward. The force will be pointing downward if the charge, is negative.
Example
In a particle accelerator, an electron is accelerated in a field of 5,000 gauss. If the speed of the electron is 3 · 106 m/s, find the magnetic force on the electron if the electron enters the field at an angle of 30° with respect to the magnetic field.
Solution
Set the quantities we are given and the unknown in the equation of the magnetic field.
B = 5,000 G = 5,000 G · 1 T/104 G = 0.5 T
v = 3 · 106 m/s
α = 30°
F = ?
F = q · v · B · sin α
F = –1.6 · 10–19 C · 3 · 106 m/s · 0.5 T · sin 30°
F = –120 · 10–15 N
F = –1.2 · 10–13 N
The Magnetic Force on an Electrical Current in a Magnetic Field
As we have seen, a charge is acted upon by a magnetic field. But a charge in motion creates an electrical current, so some interaction must take place between a wire carrying current and an applied magnetic field.
We will consider a long wire of length L carrying a current I. If the free electrons are conducting electricity through the wire then once we have a magnetic field, each electron will sense the field and interact with it.
In a time Δ t, N electrons will be passing through the cross-sectional area with a charge of N · qe. Let's consider a wire with a circular cross-sectional area and electrons moving with an average velocity v (this is also called drift velocity). See Figure 15.4.
Then, by the definition of the electrical current:
I = N · qe/ Δt
where N is the density of charges moving through the conductor.
Solving for the total charge:
I · Δt = N · qe
Also, the volume that the N charges occupy:
v = A · L = A · (v · Δt)
And then the time can be found to be:
Δt = V/A · v
Introducing this is the current formula:
I · V/ A · v = N · qe
We also know that for a charge, the magnetic force is:
F = q · v · B · sin α
Then for N charges, the magnetic force will be calculated the same, but the charge is N · qe. Replacing the charge and simplifying, we get:
F = I · L · B · sin α
In this formula, the angle α is the angle between the electrical current and the applied magnetic field.
This result is very similar to the magnetic force on a moving charge that we worked with at the beginning of this lesson, although the place of the q · v product is taken by the I · L. The same rules, the right-hand rule or the corkscrew rule, can be employed to find the direction of the resultant force, and you can probably already draw the conclusion that the force will be perpendicular to the plane formed by the current and magnetic field directions, similar to the conclusion of the first part. However, determination of the magnetic force direction is made easier by the fact that we work directly with current and not charge: The right-hand thumb will take the direction of the electric current, the fingers point in the direction of the magnetic field, and the magnetic force will exit the palm perpendicularly.
Example
In Figure 15.5, the same wire is conducting the same current, but the direction of the field and of the magnetic field are different in the two images. Determine the direction of the magnetic force.
Solution
As shown by the figure below, the right hand rule is applied for each of the drawings:
- The current is in the direction of the y-axis, magnetic field is in the direction of the z-axis. The resultant magnetic force vector is in the direction of the direction of the x-axis.
- Both the current and the magnetic field are in the direction of the z-axis, and so the angle between them is α = 0°. The product is zero. So, no magnetic force is acting on the conductor in Figure 15.6.
- In Figure 15.7, the current is in the –x-axis direction, and the magnetic field is on z-axis direction. If you use the right-hand rule or the corkscrew rule, you determine a y direction for the magnetic force.
Magnetic Field Created by an Electric Current
As we learned in the previous section, an electric current is acted upon by a magnetic force. Our question now is: Does an electric current influence or create a magnetic field? The answer to this can be demonstrated with a battery, a wire, and a compass. In the presence of current through the wire, the compass is deflected from its normal position showing the magnetic field of Earth.
If we consider a long, current -carrying wire, the compass around the wire will show magnetic field lines of a circular pattern with direction influenced by the direction of the current through the wire. Again, the right-hand rule can be used to determine the magnetic field direction this time.
Experimentally, one can find the value of the magnetic field, its dependence on the current strength, and the distance relative to the wire.
In Figure 15.8, we show the magnetic field vector drawn as a vector tangent to the magnetic field line, and the size of the vector decreases with increased distance from the wire.
The magnetic field generated by a long, straight wire transporting an electrical current I at a point around the wire is proportional to the value of the current and inversely proportional to the relative distance to the wire.
The constant μ0 is called permeability of the vacuum, and its universal value is:
μ0 = 4 · π · 10– 7 T · m/ A
Magnetic field around the wire decreases as the larger the distance as shown in the equation. The pictorial interpretation in Figure 15.9 shows magnetic field lines at different distances from the current (concentric circles).
r2 – r1 < r3 – r2 < r4 – r3
In the presence of a current passing through a wire, a charge moving in proximity will be affected by the magnetic field produced by the wire, in the same manner as it would be affected by the same field created by a magnet.
Determining Direction of the Magnetic Field
To determine the direction, the extended thumb of the right hand points in the direction of the current, and the fingers are wrapped around the wire. The fingers will point in the direction of the magnetic field.
Example 1
Consider a positive charge moving in an upward direction and in close proximity to a magnetic field, such as the one in Figure 15.10. Find the direction of the magnetic force on the charge.
Solution 1
As seen in the figure, the current I is creating a magnetic field. The magnetic field line shows the direction of the field according to the application of the righthand rule. The vector describes a counterclockwise direction.
The magnetic force is seen to be oriented perpendicular to the plane of the velocity and the magnetic field and (for this case) toward the center of the circle. This means that the linear trajectory of the particle is going to be affected and curved as a result of the magnetic interaction.
If the wire in the previous figure affects a moving charge, then it could affect a number of charges, thus forming a current. So, the logical conclusion is that two currents in proximity to each other will affect each other. Because both currents determine a magnetic field around them, their interaction can be of attraction or of repulsion depending on the field, hence of the current.
Example 2
Find the type of interaction between two parallel wires in which electrical currents flow in the same direction.
Solution 2
Start by drawing the parallel wires and their corresponding flow (see Figure 15.11). Then figure the direction of the magnetic field, and, based on what we learned about the interaction between magnets, figure the type of interaction in this example.
The significance of the diagram is that the two wires behave as two magnets with opposite poles close to each other, and, as you may remember, this means that the two wires will attract each other.
There will also be a magnetic force acting on each of these wires because the magnetic field created by a current will interact with the electric current passing through the second wire as shown in Figure 15.12.
The general expression we learned in the previous sections was:
F = I · L · B · sin α
The angle between the current and the magnetic field is 90°; therefore, the sin of 90° = 1.
F = I · L · B
The force on wire 1 then is:
F1 = I1 · L · B2
Whereas the force on the second wire, considering the same length, is:
F2 = I2 · L · B1
If we also consider the expression of the magnetic field:
For each of the currents, the magnetic field is:
The distance between the two wires is the same. Then the forces become:
And on the second current:
The two expressions are identical, which means that each current will influence the other current with the same strength.
Practice problems of this concept can be found at: Magnetism Practice Questions
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