395
18.2Magnetic field and force
Effects of magnetic fields have been known since ancient times, and the magnetic compass has been used in navigation since the twelfth century and probably earlier. In modern times the use of magnetic fields is abundant in modern devices such as computers and mobile phones. Very powerful magnets are used to steer charged particles in circular paths in accelerators such as the Large Hadron Collider at CERN.
What produces magnetic fields?
Simple experiments reveal that bar magnets have two poles; these are called north and south. Two like poles repel and two unlike poles attract. This is very similar to positive and negative electric charges, but the poles of a magnet and electric charge are different things.
It is well known that the needle of a compass (the needle is a small bar magnet) aligns itself in an approximately northâsouth direction. This can be explained by assuming that the earth is itself a large magnet. The north pole of the compass needle is attracted by the earthâs magnetic south pole (which is in the geographic north). Just as an electric charge creates an electric field in the space around it, a magnet creates a similar (but distinct) field, a magnetic field. The magnetic needle of a compass can be used to investigate the presence of magnets. Since the compass needle aligns itself with a magnetic field (Figure 18.24), it follows that we can use the direction in which a compass needle is pointing to define the direction of the magnetic field at the location of the compass. In 1819 the Danish scientist H. C. Ãrsted (1777â1851) noticed a compass needle change direction when a current was turned on in a nearby wire. Although he could not explain why this happened, Ãrsted had demonstrated that electric currents produce magnetic fields. (The earthâs magnetic field is also thought to be created by currents in the earthâs molten iron core.)

Figure 18.24: A magnetic needle in an external magnetic field experiences forces that will align it with the direction of the magnetic field. The direction of the needle (from its south to the north pole) gives the direction of the external magnetic field.

396
A magnetic field is produced by electric currents. An electric field is produced by electric charges.
The vector representing a magnetic field is called magnetic flux density and its symbol is B. Its unit is the tesla, T, which will be defined later on. B is the analogue in magnetism of E in electricity.
In permanent magnets, such as bar magnets and horseshoe magnets, the magnetic field is created by microscopic currents due to the motion of electrons in the atoms of the material. These microscopic currents create microscopic magnetic fields; in materials called ferromagnetic these tiny fields are aligned so that a large observable macroscopic magnetic field is produced. Figure 18.25 shows the magnetic field lines for a bar magnet and a horseshoe magnet.

Figure 18.25: Magnetic field lines of a bar magnet and a horseshoe magnet. Notice that the arrows exit the north poles.

CHECK YOURSELF 6
Place the magnetic needle shown (red = North) at the two points marked and show how it is aligned. Ignore the magnetic field of the earth.

Figure 18.26: For check yourself question 6.

THEORY OF KNOWLEDGE
Magnetic poles appear similar to electric charges. We have north and south magnetic poles and positive and negative charges. The big difference is that we can isolate charges of the same sign but we cannot isolate, say, a north pole. If you cut a bar magnet in half you will only succeed in creating two smaller bar magnets, not isolated north and south poles. Yet theories of particle physics suggest that there may exist particles called magnetic monopoles which would be isolated north or south magnetic poles. Paul Dirac (1902â1984), one of the greatest but least well known physicists of the twentieth century, showed that the existence of magnetic monopoles would automatically explain why electric charge is quantised. Despite much search, no magnetic monopole has ever been found.
The field of a straight current carrying wire
How do we find the direction of the magnetic field created by a straight wire carrying electric current? Figure 18.27 shows small magnetic compasses around a long straight wire that carries current upwards. The compass needles align with the magnetic field. The direction of the needles at each point gives the direction of the magnetic field at that point.

Figure 18.27: Magnetic field around a straight wire.

Drawing a smooth curve through the compass needles gives a circle. The magnetic field is tangent to this circle (Figure 18.18). The imaginary curves whose tangents give the magnetic field are called magnetic field lines.
397

Figure 18.28: A three-dimensional view of the magnetic field pattern around a long straight wire. The magnetic field is symbolized by B. The cross in the wire indicates that the current is entering from left to right. The magnitude of the field decreases as we move away from the wire.

Drawing conventions in this chapter
A circle with a cross (â) indicates current or vector into the plane of the page.
A circle with a dot (â) indicates current or vector out of the plane of the page.
But which tangent do we take? The direction of the magnetic field around a straight wire carrying a current is given by the right-hand grip rule illustrated in Figure 18.29.
Grip the wire with the fingers of the right hand in such a way that the thumb points in the direction of the current. Then the direction in which the fingers curl is the direction of the âflowâ of the magnetic field vectors.

Figure 18.29: The right-hand grip rule for the magnetic field around a straight current-carrying wire. The thumb is in the direction of the current. The fingers curl in the direction of the magnetic field.

CHECK YOURSELF 7
Figure 18.30 shows a wire that goes vertically through a horizontal table. The current goes into the plane of the page.

Figure 18.30: For check yourself question 7.

Copy the diagram and draw arrows to show the direction and relative magnitude of the magnetic field at points P and Q.
The solenoid
Figure 18.31 shows the magnetic field lines for a solenoid (a coil). They are no longer circular as they were for the straight wire. The magnetic field lines within the solenoid are fairly uniform, indicating that the field is roughly constant in both magnitude and direction. Notice the similarity between the field outside the solenoid and that around the bar magnet. Notice also that magnetic field lines always exit from a north (N) pole and enter at a south (S) pole.

Figure 18.31: The magnetic field lines for a solenoid.

398
A different right-hand grip rule gives the direction of the magnetic field for a solenoid, illustrated in Figure 18.32. The fingers curl in the direction of the current. The thumb points in the direction of the magnetic field.

Figure 18.32: The right-hand grip rule for the magnetic field around a solenoid.

The field of a single loop of current
There is one last arrangement for which we need to know the magnetic field: a loop, i.e. a ring that carries a current. In Figure 18.33 the current flows clockwise if looked at from above. The magnetic field lines are closed loops around every point in the ring. Shown here are loops around just two points; you have to imagine similar loops around every point on the ring. For all practical purposes the field of a ring, just like that in the exterior of a coil, may be taken to be the same as that of a bar magnet.

Figure 18.33: The field of a ring. Outside the ring the field is very similar to that of a bar magnet.

CHECK YOURSELF 8
aA ring carrying current which is clockwise when looked at from above is brought near a bar magnet (Figure 18.34a). Do they attract or do they repel?
bTwo rings with counterclockwise currents when looked at from above have a common vertical axis (Figure 18.34b). Do they attract or do they repel?

Figure 18.34: For check yourself question 8.

WORKED EXAMPLE 18.6
Figure 18.35 shows two wires carrying equal currents into the page. State the direction of the magnetic field at point P which is equidistant from the two wires.

Figure 18.35: For worked example 18.6.

399
Answer
Using the right-hand grip rule for each wire, the magnetic fields are as shown in the following figure. The arrows representing the field are at right angles to the line joining P to each wire. Both fields have the same magnitude, as P is equidistant from both wires and the current is the same in both wires. The resultant field points to the left.

Figure 18.36: The magnetic fields from each wire are added as vectors to give the net magnetic field at P.

The magnetic force on a moving charge
Experiments show that an electric charge moving in a region of magnetic field experiences a new type of force called a magnetic force.
There is an exception to this rule: if the velocity of the charge is parallel to the direction of the field, the magnetic force is zero (Figure 18.37).

Figure 18.37: There is no magnetic force if the velocity is parallel to the magnetic field.

In any other direction there will be a force on the charge.A charge q moving with speed v in a region of magnetic field of magnetic flux density B will experience a magnetic force F given by:
F=qvBsinÎ¸
where Î¸ is the angle between the velocity direction and the B field direction.
In most of what we will do here Î¸ = 90Â° so that the magnetic force will be given by the simpler formula
F=qvB
We see that there is no magnetic force if the charge is not moving. This is different from the electric force on a charge, which is always non-zero whether the charge moves or not. The magnetic force on particles that are electrically neutral (q = 0) is, of course, zero.
The magnetic force is what we use to define the magnetic flux density.
If the magnetic force is F when a charge q moves with velocity v that makes an angle Î¸ with the direction of the field, then the magnitude of the magnetic field, B, also called the magnetic flux density, is defined to be:
B=FqvsinÎ¸
The unit of the magnetic flux density is the tesla (T). A magnetic flux density of 1 T produces a force of 1 N on a charge of 1 C moving at 1 m sâ1 at right angles to the direction of the field.
What about the direction of the magnetic force? An example is shown in Figure 18.38. The force is at right angles to both the velocity vector and the magnetic field.

Figure 18.38: The charge shown is positive. The direction of the force is perpendicular to both the velocity vector and the magnetic field vector.

There are a number of ârulesâ to help us find this direction. Three of these are shown in Figure 18.39.
400

Figure 18.39: The right-hand rule gives the direction of the force on a positive charge. The force on a negative charge is in the opposite direction.

Try the different versions and choose the one that you are comfortable with.
âFigure 18.39a. Hold your right hand as if you are going to shake hands. Place your hand so that the four fingers point in the direction of the field and the thumb in the direction of the velocity. The direction away from the palm is the direction of the force.
âFigure 18.39b. Hold your right hand as in Figure 18.39a, but then bend the middle finger at right angles to your palm. The middle finger now represents the force, the index finger the field and the thumb the velocity.
âFigure 18.39c. Curl the right-hand fingers so that they rotate from the vector v to the vector B (along the smallest of the two possible angles). The direction of the thumb is the direction of the force. (In this version you can also imagine you are rotating a screw in the direction from v to B. The direction the screw moves is the force direction.)
If you are familiar with the vector product of two vectors, you may recognise that FâÂ =qvâÂ ÃBâÂ .
These rules give the direction of the magnetic force on a positive charge. To find the force on a negative charge, pretend the charge is positive and then reverse the force direction you find.
CHECK YOURSELF 9
Figure 18.40 shows the magnetic force on a moving electron.

Figure 18.40: For check yourself question 9.

What is the direction of the magnetic field?
WORKED EXAMPLE 18.7
Express the tesla in terms of fundamental units.
Answer
From the definition B=FqvsinÎ¸ it follows that:
T=NCÃm sâ1=NAÃm=kgmsâ2Â AÃm=kgÂ sâ2Â Aâ1
The magnetic force on a current-carrying wire
A current in a wire consists of moving charges. So a current-carrying wire placed in a magnetic field will experience a magnetic force because there is a force on the moving charges in the wire.
Part of the wire in Figure 18.41 is in a region of magnetic field directed out of the page. In Figure 18.41a the current in the wire is zero and there is no force. Figure 18.41b and Figure 18.41c show the forces on the wire when there is current.

Figure 18.41: The magnetic force on a current-carrying wire.

The formula for the magnetic force on a length L (L is that length of the wire that finds itself in the region of the field) is:
F=BILsinÎ¸
where Î¸ is the angle between the current and the direction of the magnetic field.
401
To find the direction of the force, use the right-hand rules for the force on a charge (Figure 18.39) and replace velocity by current.
The force between two current-carrying wires
Consider two long, straight, parallel wires carrying currents I1 and I2 (Figure 18.42). The first wire (wire 1) creates a magnetic field in space. This field has magnitude B1 at the position of the second wire (wire 2). This means wire 2 experiences a magnetic force. Similarly, wire 2 creates a magnetic field of magnitude B2 at the position of wire 1, so that wire 1 also experiences a magnetic force. By Newtonâs third law, the force that wire 1 exerts on wire 2 must be accompanied by an equal and opposite force of wire 2 on wire 1. Therefore the forces experienced by the two wires are equal and opposite.
The currents are different in the two wires, so the magnetic fields are different, but the two forces are equal in magnitude.

Figure 18.42: The forces on two parallel currents are equal and opposite. a Force between parallel currents. b Force between anti-parallel currents.

We can use the right-hand rule to find the directions of these forces. Assume first that both currents are flowing into the page. Then the magnetic fields are as shown in Figure 18.42a and the forces are attractive. If wire 1 carries current into the page and wire 2 carries current out of the page, as shown in Figure 18.42b, the forces are repulsive. In both cases the forces are equal and opposite, consistent with Newtonâs third law. So we have found that if the currents are parallel, the forces are attractive, and if they are anti-parallel, the forces are repulsive.
It is useful to know that the magnitude of the magnetic flux density created at a distance r from a long straight wire carrying current I is given by the formula (which is not on the syllabus) B=Î¼0I2Ïr. Here Î¼0 is a constant of physics called magnetic permeability. Its value in vacuum is defined to be Î¼0 = 4Ï Ã 10â7 T m Aâ1. This constant is to magnetism what permittivity (Îµ0) is to electricity. Looking at Figure 18.43, the magnetic flux density created by the left wire at the position of the right wire is B1=Î¼0I12Ïr where r is the separation of the wires. Therefore the force experienced by a length L of the right wire is F=B1I2L=Î¼0I1I2L2Ïr. This gives the syllabus formula for the force per unit length:
FL=Î¼0I1I22Ïr
Notice that the magnetic field produced by the right wire at the position of the left is given by B2=Î¼0I22Ïr. The force on the left wire is then F=B2I1L=Î¼0I2I1L2Ïr which, as expected from Newtonâs third law, is equal to the force on the right wire.
CHECK YOURSELF 10
Calculate the unit and value of the quantity 1Îµ0Î¼0ââââââ.
WORKED EXAMPLE 18.8
Figure 18.43 shows three wires, X, Y and Z, carrying equal currents I. The directions of the currents are as shown. The separation of two adjacent wires is d.

Figure 18.43: For worked example 18.8.

402
Find the magnitude and direction of the force per unit length on wire Z when I = 4.0 A and d = 2.0 cm.
Answer
Parallel currents attract and anti-parallel repel. So X attracts Z and Y repels it. Y is closer to Z so the force it exerts is larger. Hence the force is to the right. The force per unit length from X is Î¼0I22Ï(2d) to the left and that from Y is Î¼0I22Ïd to the right. The net force per unit length is then Î¼0I22ÏdâÎ¼0I24Ïd=Î¼0I24Ïd to the right. For I = 4.0 A and d = 2.0 cm the force per unit length is 4ÏÃ10â7Ã4.024ÏÃ2.0Ã10â2=8.0Ã10â5Â NÂ mâ1.
EXAM TIP
It is simpler to remember that parallel currents attract rather than having to work out the direction of the magnetic field at Zâs location and then find the force.
NATURE OF SCIENCE
Introduced in the nineteenth century by Michael Faraday as âlines of forceâ, the concept of magnetic field lines allowed scientists to visualize the magnetic field around a magnet and the magnetic field around a moving charge. A few years later, in one of the greatest unifications in physics, James Clerk Maxwell showed that all magnetic and electric phenomena are different sides of the same general phenomenon, electromagnetism, and that light is a combination of electric and magnetic fields. In the early twentieth century, Albert Einstein showed that viewing electric and magnetic phenomena from different frames of reference leads naturally to the theory of relativity. At about the same time, trying to understand magnetism in different materials required the introduction of quantum theory, so magnetism played a crucial role in the development of physics.
WORKED EXAMPLE 18.9
An electron approaches a bar magnet, as shown in Figure 18.44. What is the direction of the force on the electron?

Figure 18.44: For worked example 18.9.

Answer
The magnetic flux density at the position of the electron is to the left. Placing the right hand so that the thumb points up the page (velocity direction) and the fingers to the left (field direction), the palm is pointing out of the page. But the charge is negative and so the force is into the page.
Links
We have now added two new forces to our list of forces which up to now really only included the gravitational force. All the other forces we met (normal forces, friction, tension etc.) are really macroscopic manifestations of electric or magnetic phenomena. Maxwell showed that electric and magnetic phenomena are closely linked and joined the electric and the magnetic force into one, the electromagnetic force, achieving the first unification in physics. Later on, in ChapterÂ 23, we will meet two new forces, the weak and strong nuclear forces. These are forces that are relevant in the subatomic world. In order of increasing strength of interaction the list is gravitational, weak nuclear, electromagnetic and strong nuclear. But this is a rather meaningless list since the comparison of interaction strength depends heavily on the context in which the comparison is made. Adding to the unification of Maxwell, the theory of Glashow, Salam and Weinberg that gave us the presently accepted standard model of particles implies that the electromagnetic force and the weak nuclear force are two aspects of the same force called the electroweak force. Are there other forces that we have not discovered yet?
403
TEST YOUR UNDERSTANDING
17Two wires carry equal currents into the plane of the page (Figure 18.45).

Figure 18.45: For question 17.

What is the direction of the magnetic field at point P halfway between the charges and at point Q?
18Two wires carry equal currents into the plane of the page (Figure 18.46).

Figure 18.46: For question 18.

What is the direction of the magnetic field at point P which is on the perpendicular bisector of the line joining the wires?
19Two wires carry equal currents in opposite directions at right angles to the plane of the page (Figure 18.47).

Figure 18.47: For question 19.

What is the direction of the magnetic field at point P which is on the perpendicular bisector of the line joining the wires?
20The current in a horizontal loop of wire is as shown in Figure 18.48.

Figure 18.48: For question 20.

What is the direction of the magnetic field at points P, Q and R?
21Determine the direction of the missing quantity from B, v and F in each of the cases shown in Figure 18.49. The circle represents a positive charge.

Figure 18.49: For question 21.

404
22The north pole of a bar magnet is brought close to a ring that carries current. For clarity only a cross section of the ring is shown (Figure 18.50).

Figure 18.50: For question 22.

Describe the force between the bar magnet and the ring.
23Currents are established in two parallel rings as shown in Figure 18.51. For clarity, only a cross section of each ring is shown.

Figure 18.51: For question 23.

Describe the force between the rings.
24A wire carrying current into the plane of the page is put in between two bar magnets as shown in Figure 18.52.

Figure 18.52: For question 24.

What is the direction of the force on the wire in each case?
25A current carrying wire is placed in between two bar magnets as shown in Figure 18.53.

Figure 18.53: For question 25.

What is the direction of the force on the wire?
26A bar magnet attracts a metallic ball hanging from a thread (Figure 18.54).

Figure 18.54: For question 26.

The magnet is reversed so that the south pole is facing the ball. What will happen now?
27Two wires carry equal currents into the plane of the page (Figure 18.55).

Figure 18.55: For question 27.

aOn the diagram draw:
ithe magnetic field created by wire X at the position of wire Y
iithe magnetic field created by wire Y at the position of wire X
iiithe magnetic forces on each wire.
bThe current in wire X is doubled. Make new drawings for ai, aii and aiii.
405
28Two wires carry equal currents in opposite directions at right angles to the plane of the page (Figure 18.56).

Figure 18.56: For question 28.

Wire X carries current into the page and is equidistant from the other two wires. What is the direction of the magnetic force on wire X?
29A ring carries current as shown in Figure 18.57. For clarity only a cross section of the ring is shown. A proton moves parallel to the plane of the ring.

Figure 18.57: For question 29.

What is the direction of the magnetic force on the proton?
30A square loop of wire is placed close to a straight wire (Figure 18.58).

Figure 18.58: For question 30.

Both wires have currents in them as shown. Carefully describe the force between the straight wire and the loop.
31Draw the magnetic field lines for two parallel wires carrying equal currents into the page. Repeat for anti-parallel currents.
32Copy Figure 18.59. Draw the magnetic field lines that result when the magnetic field of a long straight wire carrying current into the page is superimposed on a uniform magnetic field pointing to the right that lies on the page.

Figure 18.59: For question 32.

33A long straight wire carries current as shown in Figure 18.60. Two electrons move with velocities that are parallel and perpendicular to the current. Determine the direction of the magnetic force experienced by each electron.

Figure 18.60: For question 33.

34A proton moves past a bar magnet as shown in Figure 18.61. Find the direction of the force it experiences in each case.

Figure 18.61: For question 34.

406
35A bar magnet is placed in a uniform magnetic field as shown in Figure 18.62.

Figure 18.62: For question 35.

aSuggest whether there is a net force on the bar magnet.
bDescribe how it will move.
36A high-tension electricity wire running along a northâsouth line carries a current of 3000 A. The magnetic field of the earth at the position of the wire has a magnitude of 5.00 Ã 10â5 T and makes an angle of 30Â° below the horizontal.
Calculate the force experienced by a length of 30.0 m of the wire.
37A uniform magnetic field is established in the plane of the paper as shown in Figure 18.63. Two wires carry equal parallel currents normally to the plane of the paper at P and Q. Point R is on the line joining P to Q and closer to Q. The magnetic field at position R is zero.

Figure 18.63: For question 37.

aDetermine whether the currents are going into the paper or out of the paper.
bThe magnitude of the current is increased slightly. Determine whether the point where the magnetic field is zero moves to the right or to the left of R.
