1. University Of Zimbabwe HPH102: Electricity and Magnetism Magnetic Field Miss L G Thwala

2. HTH102: Course Coverage Introduction,
Magnetic field and Magnetic field lines,
Lorentz law,
Cyclotron and Hall effect,

3. Introduction
Many historians of science believe that the compass, which uses a magnetic needle, was used in China as early as the 13th century B. C., its invention being of Arabic or Indian origin.
The early Greeks knew about magnetism as early as 800 B.C. They discovered that the stone magnetite (Fe3O4) attracts pieces of iron.
Legend ascribes the name magnetite to the shepherd Magnes, the nails of whose shoes and the tip of whose staff stuck fast to chunks of magnetite while he pastured his flocks.
In 1269 a Frenchman Pierre de Maricourt mapped out the directions taken by a needle placed at various points on the surface of a spherical natural magnet. He found that the directions formed lines that encircled the sphere and passed through two points diametrically opposite each other, he called them poles of a magnet.
Subsequent experiments showed that every magnet, regardless of its shape, has two poles, called the north and south poles that exert forces on other magnetic poles just as electric charges exert forces on one another. That is, like poles repel each other and unlike poles attract each other.

4. Introduction …
The poles received their names because of the way a magnet behaves in the presence of the earth’s magnetic field.
If a bar magnet is suspended from its middle point and can swing freely in a horizontal plane, it will rotate until its north pole points to the earth’s geographic North Pole and its south pole points to the earth’s South Pole.
In 1600 William Gilbert extended the de Maricuourt’s experiments to a variety of materials. Using the fact that a compass needle orients in preferred directions, he suggested that the earth itself is a large permanent magnet.
In 1750 experimenters used a torsion balance to show that magnetic poles exert attractive and repulsive forces on each other and that these forces vary as the inverse square of the distance between interacting poles.
Although the force between two magnetic poles is similar to the force between two electric charges, there is an important difference.
Electric charges can be isolated whereas a single magnetic pole has never been isolated. Magnetic poles are always found in pairs.
All attempts to detect an isolated magnetic pole have been unsuccessful. No matter how many times a permanent magnet is cut in two, each piece always has a north and a south pole.

5. Introduction …
The relationship between magnetism and electricity was discovery in 1819 when during a lecture demonstration, the Danish physicist, Hans Christian Øersted found out that an electric current in a wire deflected a nearby compass needle.
Shortly afterwards, Andre Ampere formulated quantitative laws for calculating the magnetic force exerted by one current-carrying electrical conductor on another. He also suggested that on the atomic level, electric current loops are responsible for all magnetic phenomena.
In the 1820s, further connections between electricity and magnetism were demonstrated by Faraday and independently by Joseph Henry.
They showed that an electric current can be produced either by moving a magnet near a circuit or by changing the current in a nearby circuit.
This gave birth to the field of electromagnetic induction in 1831.
Maxwell put Faraday’s field ideas into mathematical form and predicted electromagnetic waves. Theoretical work by Maxwell showed that a changing electric field creates a magnetic field.

6. Introduction …
The dynamo, the motor and the transformer were the direct outcome of Faraday’s work.
Edison was responsible for opening the first power station in 1882, designed to supply electricity to domestic consumers in New York.
Shortly afterwards Tesla invented the induction motor which does not require brushes (to supply current to its rotor) and which is now the indispensible servant of industry.

7. Magnetic Field
As the forces exerted on charges by electric fields, they are additional magnetic forces experienced by moving charged particles.
Magnetic forces always act perpendicularly to the direction of motion of the charged particles.
E.g, if two parallel metal wires are brought near to each other and steady currents are set up in them, there is a force between the two as shown below.
If the currents are in the same direction the force is attractive; if the currents are in opposite direction the force is repulsive.
The force between the wires is not the same as the electrostatic force since the wires contain equal amounts of positive and negative charge and is electrically neutral. F I1 I2
Figure: The forces between parallel wires carrying currents

8. Magnetic Field …
Permanent magnets also cause forces to act on wires carrying currents.
If a current is passed through a wire suspended between the poles of a permanent magnet, the wire experiences a force perpendicular to the direction of the current and is deflected.
If the direction of the current is reversed the deflection of the wire is reversed.
A wire carrying a steady current and a permanent magnet both have the same effect on another wire carrying a current.
In order to describe these effects we introduce the magnetic field .
The current in one wire I1 gives rise to a magnetic field B1 which describes the forces on the moving charges constituting the current I2 in the other wire.
Experiments have shown that the force on the wire carrying the current I2 is proportional to I1, for a fixed current in the first wire.

9. Magnetic Field …
In a constant magnetic field, the magnitude of the force on a moving charge is proportional to the product of the charge q and the speed v of the charge.
In order to determine how the force on a moving charge depends on its velocity and the magnetic field, let us assume that the magnetic field in the middle of the gap between parallel pole faces of a permanent magnet is uniform and in a direction perpendicular to the pole faces.
Experiments with charged particles moving in the gap show that the particles always experience a force which is perpendicular to both the magnetic field and to the velocity of the particles.
The magnitude of the force is proportional to the speed of the particle v, its charge q, and to the sine of the angle between and i.e.
The units of the magnetic field are defined by choosing the constant of proportionality to be unity, leading to
This equation is known as the known as the Lorenz force law.

10. Magnetic Field …
The SI unit of magnetic field strength is the tesla (T).
The strengths of magnetic fields are often given in an alternative unit called the gauss (G), which is ten thousand times smaller than the tesla,
1 G = 10-4 T or 1 T = G
The magnetic properties of a magnet appear to originate at certain regions in the magnet which are called poles; in a bar magnet these are near the ends. Experiments show that:
Magnetic poles are of two kinds,
Like poles repel each other and unlike poles attract,
Poles always seem to occur in equal and opposite pairs, and
When no other magnet is near, a freely suspended magnet sets so that the line joining its poles is approximately parallel to the earth’s north-south axis, i.e. to the magnetic meridian.
The last fact suggests that the earth itself behaves like a permanent magnet and it makes it appropriate to call the pole of a magnet that points (more or less) towards the earth’s geographical North Pole, the north pole of the magnet and the other the South Pole.

11. Magnetic Field Lines
The space surrounding a magnet where a magnetic force is experienced is called a magnetic field.
The direction of the magnetic field at a point is taken as the direction of the force that acts on a north magnetic pole there. A magnetic field can be represented by magnetic field lines drawn so that:
The line ( or tangent to it if it is curved) gives the direction of the field at that point, and
The number of lines per unit cross-section area is an indication of the ‘strength’ of the field.
Arrows on the lines show the direction of the field and, since a north pole is repelled by the north pole of a magnet and attracted by the south, the arrows point away from the north poles and towards the south poles.
Field lines can be obtained quickly with iron filings or accurately plotted using a small compass.

12. Magnetic Field Lines …
Figure: Magnetic field pattern due to a bar magnet as displayed by iron filings.

13. Magnetic Field Lines …
Figure: Magnetic field patterns due to due to unlike poles of a bar magnet.

14. Magnetic Field Lines …
Figure: Magnetic field patterns due to due to like poles of a bar magnet.

15. Magnetic Field Lines …
Similar to electric field lines we can graphically represent the magnetic field by field lines.
The density of field lines indicates the strength of the magnetic field.
The tangent of the field lines indicates the direction of the magnetic field.
The magnetic field lines form closed loops, and differ from electric field lines in that they do not begin or end anywhere.
Figure: The magnetic field of a current in a long straight wire

16. Magnetic Field Lines … Consider a closed surface S around a volume V.
Since the lines of the field B are continuous, as many lines enter the volume V as they leave it.
This implies that the total outward flux of the magnetic field over the surface S is zero:
Figure: The magnetic field near a small circular loop of wire carrying a current

17. THE MOTION OF CHARGED PARTICLES IN ELECTRIC AND MAGNETIC FIELDS
The force on a particle of charge q moving with velocity v in electric and magnetic fields E and B is given by
If the particle is moving in a vacuum at the surface of the earth and there are no nearby molecules with which it collides or interact, and the only force acting on the particle, other than due to the electric and magnetic fields, is the force of gravity.
When considering atomic particles, it can be shown that gravitational forces become negligible. Neglecting the gravitation force, Newton’s law for the particle in a vacuum becomes
where is the momentum of the particle.

18. The motion of a charged particle in a uniform magnetic field…
Let us consider the motion of a charged particle of mass m in a uniform magnetic field.
Since the magnetic force always acts at right angles to the motion, the force cannot increase the energy of the particle and its speed will remain constant.
If the particle is moving in a plane perpendicular to the direction of the uniform magnetic field ,the force.
q x on the particle is constant in magnitude and is always at right angles to the direction of motion.
Figure: Motion of a positively charged particle in a uniform magnetic field

19. The motion of a charged particle in a uniform magnetic field …
The particle therefore moves in a circle of radius R (Fig.), where or
It can be seen from the last Eq. that particles of the same ratio of momentum to charge move in circles of the same radius in a uniform magnetic field.
Particles of a given value of (mv/q) are often selected by accepting only those that traverse a path of fixed radius when moving in a uniform field.
The angular frequency of the circular motion is given by**
The distance travelled by the particle in one revolution is equal to
The time T required to complete one revolution (period) is equal
The frequency f of this motion is equal to
The angular frequency ω given by Eq** does not depend on the radius of the orbit or the velocity of the particle. This frequency is called the cyclotron frequency.

20. The motion of a charged particle in a uniform magnetic field …
If a charged particle enters a region of uniform magnetic field with a velocity which is not perpendicular to B, the path of the particle is a helix.

21. The Cyclotron
One application of the effect of a magnetic field on the motion of a charged particle is the cyclotron.
A cyclotron consists of an evacuated cavity placed between the poles of a large electromagnet. The cavity is cut into two D-shaped pieces (called dees) with a gap between them. An oscillating high voltage is connected to the plates, generating an oscillating electric field in the region between the two dees.

22. The Cyclotron …
A charged particle, injected in the centre of the cyclotron, will carry out a uniform circular motion for the first half of one turn. The frequency of the motion of the particle depends on its mass, its charge and the magnetic field strength. The frequency of the oscillator is chosen such that each time the particle crosses the gap between the dees, it will be accelerated by the electric field.
As the energy of the ion increases, its radius of curvature will increase until it reaches the edge of the cyclotron and is extracted. During its motion in the cyclotron, the ion will cross the gap between the dees many times, and it will be accelerated to high energies.

23. The Cyclotron …
The cyclotron is used for accelerating particles such as protons and deuterons to high kinetic energy.
The high-energy particles are then used to bombard nuclei, causing nuclear reactions, which are studied to obtain information about the nucleus.
High energy protons or deuterons are also used to produce radioactive materials and for medical purposes.
The magnetic force on a charged particle moving in a uniform magnetic field can be balanced by an electrostatic field if the magnitudes and directions of the magnetic and electrostatic fields are properly chosen.
Since the electric force is in the direction of the electric field (for positive particles) and the magnetic force is perpendicular to the magnetic field, the electric and magnetic fields must be perpendicular to each other if the forces are to balance.

24. The motion of a charged particle in a uniform magnetic field…
Figure below shows a region of space between the plates of a capacitor where there is an electric field and a perpendicular magnetic field. Such an arrangement of perpendicular fields is called crossed fields.
Consider a particle of charge q entering this space from the left. If q is positive, the electric force q is down and the magnetic force q x is up. If the charge is negative the forces are reversed.
Since is perpendicular to the magnitude of the magnetic force is just qvB.
These two forces will balance if qE = qvB or
For given magnitudes of the electric and magnetic fields, the forces will balance only for particles with speed given by the above Eq . A particle of greater speed will be deflected in the direction of the magnetic force, while one of less speed will be deflected in the direction of the electric force. Such an arrangement of fields is called a velocity selector.
Figure: Crossed electric and magnetic fields

25. To Be Continued…