Physics 12 Unit 8 Magnetic Field and Electromagnetism Part I
1. Basics about magnets Magnets have been known by ancient people since long time ago, referring to the iron-rich rocks, called magnetite or lodestone, that were attracted strongly to each other. These materials have wide applications such as compasses. Unit 8 - Magnetic Field and Electromagnetism (Part I) 2
The effect of magnetism originates from the motion of electrons in a materials. An electron can produce a magnetic field by: 1. Orbiting around the nucleus of an atom 2. Spinning on its own axis Unit 8 - Magnetic Field and Electromagnetism (Part I) 3
Therefore, every atom of every element should produce some sort of magnetism. However, in reality, not every element is magnetic. Why? When an atom has all its electrons paired up, the magnetic field produced by one electron will be cancelled out by the field due to the other electron in a pair. As a result, the atom will generate no magnetic field overall. The elements having atoms with just paired electrons are called diamagnetic (or non-magnetic in simple terminology). Unit 8 - Magnetic Field and Electromagnetism (Part I) 4
When a materials consists of the atoms having one or more unpaired electrons, it is considered magnetic (also known as paramagnetic). These unpaired electrons behave like tiny magnets, and their induced magnetic fields point in random directions due to heat. Therefore, the total magnetic field is essentially zero without an externally applied magnetic field. Unit 8 - Magnetic Field and Electromagnetism (Part I) 5
When a magnetic field is applied, the electron spins will slowly align parallel to the applied field, causing a net attractive force. Note, however, that only a small portion of the electron spins in the materials will be oriented by the field; therefore the resulting force is not very strong. Also, when the external field is removed, the induced magnetic attraction dies off gradually. Unit 8 - Magnetic Field and Electromagnetism (Part I) 6
Example: Attraction of liquid oxygen by magnets Unit 8 - Magnetic Field and Electromagnetism (Part I) 7
There exists another type of magnetic substances called ferromagnetic materials. The atoms in a ferromagnetic substance are arranged in form of magnetic domain, a region in which the magnetic fields produced by the atoms are pointing in the same direction. Each domain acts like a miniature magnet. Their directions are randomly arranged in normal conditions (i.e., unmagnetized). Unit 8 - Magnetic Field and Electromagnetism (Part I) 8
In the presence of an external magnetic field, these domains rearrange themselves so that they line up with the applied field, making the materials temporarily magnetic: When the field is removed, these domains become disordered again. But some ferromagnetic materials remain magnetized even after the applied field has been removed. They are said to be transformed into permanent magnets. Unit 8 - Magnetic Field and Electromagnetism (Part I) 9
Typical ferromagnetic materials include iron, cobalt, nickel, their alloys, as well as some rare-earth elements (e.g. gadolinium) and their compounds. Ferromagnetism is very important to industry and technology, with applications in electric motors, generators, transformers, magnetic storages such as hard disks, and so on. Unit 8 - Magnetic Field and Electromagnetism (Part I) 10
Bar magnets are examples of permanent magnets that attract iron, nickel, cobalt or gadolinium. Each piece of bar magnet possesses the following properties: 1. It contains a north pole and a south pole. The north pole points north while the south pole points south. 2. Two nearby magnets exert force on each other. Like poles repel while unlike poles attract. 3. Cutting a magnet in half will create two magnets, each of which having a north pole and a south pole. Unit 8 - Magnetic Field and Electromagnetism (Part I) 11
2. Magnetic field Like gravitational or electrostatic fields, magnetic field can be represented by magnetic field lines which point from a north pole to a south pole. Unit 8 - Magnetic Field and Electromagnetism (Part I) 12
A magnetic field line indicates how a hypothetical, isolated north pole will move when it is placed at that position subject to the magnetic field due to a nearby magnet. The directions of the field lines can be determined using a compass needle. Its north end points in the direction of the field. Unit 8 - Magnetic Field and Electromagnetism (Part I) 13
Magnetic field lines are disturbed when two magnets are placed together. The way of constructing the resulting field lines is similar to that of constructing the electric field lines due to multiple point charges. Opposite poles Same poles Unit 8 - Magnetic Field and Electromagnetism (Part I) 14
The Earth possesses the liquid outer core made of iron and nickel; this layer is responsible for the Earth s magnetic field. Some interesting features: 1. The geographic poles do not coincide with the magnetic poles and the geomagnetic poles. 2. The polarity of Earth s magnetic field reverse for every 500,000 years. 3. The northern magnetic pole is actually the south pole of the Earth s magnetic field. Unit 8 - Magnetic Field and Electromagnetism (Part I) 15
3. Nature of magnetic field Source of magnetic field Since the discovery till early 1800 s, people treated magnetism as a set of phenomena independent of, and unrelated to, the phenomena due to electricity. In 1820, Hans Christian Ørsted (1777-1851) accidentally observed that a compass was deflected when it was placed near a current-carrying wire. Unit 8 - Magnetic Field and Electromagnetism (Part I) 16
Magnetic interaction with current can also be illustrated in the following experiments: 1. Magnetic interaction on wires Unit 8 - Magnetic Field and Electromagnetism (Part I) 17
2. Force between two parallel wires There exists interaction between current-carrying wires. For parallel current: attraction For anti-parallel current: repulsion Unit 8 - Magnetic Field and Electromagnetism (Part I) 18
These experiments demonstrate that magnetic interaction is induced between the following pairs: Magnet to magnet Current to magnet Magnet to current Current to current These reveal the nature of electricity and magnetism: They are indeed different manifestations of the same phenomenon called electromagnetic force, which is one of the four fundamental interactions (the other three being weak force, strong force and gravitational force) in nature. Magnetic field originates from moving electric charges. Unit 8 - Magnetic Field and Electromagnetism (Part I) 19
Strength of magnetic field Recall that electrostatic field can be defined as the force experienced per Coulomb of charge. E = F e q We can define magnetic field, called magnetic induction for some historical reasons, in the similar way. Experimentally, the force acting on a charge moving in a static magnetic field depends on two factors: 1. The velocity component of the charged particle perpendicular to the field 2. The magnitude of the charge Unit 8 - Magnetic Field and Electromagnetism (Part I) 20
Mathematically, B = F m qv where B is magnetic induction, q is the charge, v is the velocity component, and F m is the magnetic force. Note that F m and B are always perpendicular to each other. Their directions can be described using the Fleming s right-hand motor rule (RHR): Unit 8 - Magnetic Field and Electromagnetism (Part I) 21
The unit of magnetic field is Testa (T). By definition, 1 T = 1 N 1 C 1 m/s = 1 N 1 A m Permanent magnets usually possess a magnetic induction of about 0.001 T to 0.1 T. The magnetic field due to Earth has a magnitude of only 0.5 10-4 T near the surface. Another common unit of magnetic field used in engineering is called Gauss (G). Its conversion factor to Tesla is: 1 T = 10 4 G Unit 8 - Magnetic Field and Electromagnetism (Part I) 22
There are several common types of magnetic field generated by electric current depending on the shapes of the wires in which current is flowing through: Magnetic field due to loop current Magnetic field due to straight wire Unit 8 - Magnetic Field and Electromagnetism (Part I) 23
Magnetic field due to solenoid Magnetic field due to magnets The magnetic induction in these situations can be described using appropriate mathematical approaches. (We will discuss these cases individually in later sections.) Unit 8 - Magnetic Field and Electromagnetism (Part I) 24
The orientations of the magnetic field created by wires can be determined by a convenient method called Ampère s right-handed screw rule. There are two versions: For straight wire For solenoid Unit 8 - Magnetic Field and Electromagnetism (Part I) 25
4. Magnetic force on wires In previous sections, we have mentioned that the force experienced by a charged particle moving within a static magnetic field depends on its speed and its direction of motion. If the particle is moving along the same line as the magnetic field, it will feel no magnetic force. However, if it is moving at the direction perpendicular to the field, the force it feels will be strongest. What will happen if the charged particle is moving at an arbitrary angle θ with respect to the magnetic field? Unit 8 - Magnetic Field and Electromagnetism (Part I) 26
We can use vector resolution to solve this problem. Let θ be the angle between the velocity of the particle and the magnetic field. The component v = v cos θ is parallel to the field, and therefore it will not participate in the magnetic force. The component v = v sin θ is perpendicular to the field, and thus it is involved in the resulting magnetic force. Recall the definition of a magnetic field: B = F m qv Unit 8 - Magnetic Field and Electromagnetism (Part I) 27
Hence, F m = qv B = qvb sin θ This force is called the Lorentz force. It has the direction perpendicular to both the velocity vector and the magnetic field. Given the directions of v and B, we can determine the direction of F m using the Fleming s right hand rule. Mathematically, the Lorentz force can be defined in a more compact way using the cross product: F m = qv B Unit 8 - Magnetic Field and Electromagnetism (Part I) 28
When a conducting wire, with the length l and a conventional current I flowing within it, is placed in a magnetic field such that they are perpendicular, a Lorentz force due to the field acts on the wire. Recall that F m = qvb Since v = l/t, we have F m = q l t B Note, however, that q = It; hence F m = B q t l = BIl Unit 8 - Magnetic Field and Electromagnetism (Part I) 29
Example: The magnetic field strength inside a solenoid is 0.025 T. If a 3.2-cm long conducting strip positioned at right angles to the magnetic field inside the solenoid experiences a force of 5.9 10-4 N, what is the current in the conducting strip? Unit 8 - Magnetic Field and Electromagnetism (Part I) 30
In general, if the conducting wire and the magnetic field make an angle of θ, the Lorentz force exerted on the wire by the field will be given by F m = BIl sin θ Unit 8 - Magnetic Field and Electromagnetism (Part I) 31
Example: A wire carrying a steady 30 A current has a length of 12 cm between the pole faces of a magnet. The wire is at an angle of 60 to the field. The magnetic field is approximately uniform at 0.90 T. We ignore the field beyond the pole pieces. Determine the magnitude and direction of the force on the wire. Unit 8 - Magnetic Field and Electromagnetism (Part I) 32
If we want to represent a magnetic field which is pointing in a direction perpendicular to the page, we will use the following symbols: Pointing into the page Pointing out of the page Unit 8 - Magnetic Field and Electromagnetism (Part I) 33
The idea of Lorentz force allows us to explain the magnetic interaction between two parallel current carrying wires in close proximity. Suppose that there are two parallel wires with the current flowing in opposite direction. Unit 8 - Magnetic Field and Electromagnetism (Part I) 34
The effect of the magnetic force exerted on I 2 due to I 1 can be analyzed using the Ampère s right hand rule: Unit 8 - Magnetic Field and Electromagnetism (Part I) 35
And According to the RHR, the magnetic force on I 2 due to I 1 is pointing to the right. Similarly, we can show that the force on I 1 due to I 2 is pointing to the left according to the RHR. Consequently, these wires repel each other. Unit 8 - Magnetic Field and Electromagnetism (Part I) 36
Practice: Prove that two parallel wires carrying current in the same direction are attracted to each other. Unit 8 - Magnetic Field and Electromagnetism (Part I) 37
5. Magnetic force on charged particles Recall that a charged particle moving in a magnetic field experiences the Lorentz force in the direction perpendicular to its direction of motion. For instance, an applied magnetic field can bend the cathode ray beam which is made of electrons: But how much the beam will be bent by the magnetic field? Unit 8 - Magnetic Field and Electromagnetism (Part I) 38
Recall that F m = qvb, and F m is perpendicular to v. Hence, if an electron enters into a static magnetic field as follows: The electron will be deflected to the right of its original path. Unit 8 - Magnetic Field and Electromagnetism (Part I) 39
Since the Lorentz force is always acting on the electron in the direction perpendicular to its motion: 1. The magnitude of v is unchanged 2. The Lorentz force is centripetal 3. The electron travels in a circular path as long as it stays within the field Hence, F m = F c qvb = mv2 r qb = mv r = p r Unit 8 - Magnetic Field and Electromagnetism (Part I) 40
Example: An electron travelling at 2.5 10 7 m/s enters a magnetic field of strength 4.1 10-3 T as shown below. What is the radius of the circular path taken by the electron once it enters the field? Unit 8 - Magnetic Field and Electromagnetism (Part I) 41
Note that Rearranging this expression yields qb = mv r r = mv qb This relation is useful because it shows that, given q and B: If m is fixed (i.e., same type of charged particles), then faster particles (i.e., larger v) will pass through a larger circular path. r v If v is fixed (i.e., velocity selection), then heavier particles (i.e., larger m) will pass through a larger circular path. r m Unit 8 - Magnetic Field and Electromagnetism (Part I) 42
The following diagram shows how the circular path of a particle is related to its mass. All particles leaving this spot have the same velocity. Unit 8 - Magnetic Field and Electromagnetism (Part I) 43
The same velocity of the charged particles entering into the magnetic field is warranted by the modulator consisting of a combined electric and magnetic field in perpendicular directions. Electric field Magnetic field Overall Unit 8 - Magnetic Field and Electromagnetism (Part I) 44
For the charged particles to travel straight through, the two forces have to be balanced by one another. That means, F e = F m qe = qvb Hence, the velocity of the particles that leave this modulator is v = E B Technically, the velocity selection of the charged particles can be achieved by choosing appropriate E and B. Unit 8 - Magnetic Field and Electromagnetism (Part I) 45
A significant application of this phenomenon is the development of mass spectrometer, a machine used to determine the masses of particles of various sizes and compositions. v = E B 1 m = qb 2r v m q = B 1B 2 E r Mass-to-charge ratio of particle Unit 8 - Magnetic Field and Electromagnetism (Part I) 46
The diagram below shows the mass spectrum for different isotopes of zirconium: Isotope Relative abundance Zr-90 51.5% Zr-91 11.2% Zr-92 17.1% Zr-94 17.4% Zr-96 2.8% Unit 8 - Magnetic Field and Electromagnetism (Part I) 47
Example: Carbon atoms of atomic mass 12.0 a.m.u. are mixed with atoms of another unknown material. In a mass spectrometer, the C- 12 atoms follow a path of radius 22.4 cm, while the unknown atoms produces a 26.2-cm radius path. Assuming identical charges, what is the atomic mass of the unknown material? Unit 8 - Magnetic Field and Electromagnetism (Part I) 48
6. Biot-Savart Law In Chapter 6, we have seen that the electric field, and thus electric force, due to a point charge can be determined using the Coulomb s law. This gives rise to electrostatics. Similarly, one can determine the magnetic field, and thus magnetic force, due to a steady current (or moving charges with constant velocity). The mathematical formulation is called Biot-Savart law, and the associated physics is called magnetostatics. The actual form of Bito-Savart law is very complicated, but in simple words, it states that B μ 0I r Unit 8 - Magnetic Field and Electromagnetism (Part I) 49
In this expression μ 0 Magnetic permeability (4π 10 7 N/A 2 ) I r Steady current For a straight wire, the magnetic field at a distance a from it is given by B P = μ 0I 2πa Perpendicular distance from the wire Unit 8 - Magnetic Field and Electromagnetism (Part I) 50
Example: What is the force experienced by two 10-cm long straight wires separated by 5 cm, assuming that a 1.5-A current is flowing in them in parallel? Unit 8 - Magnetic Field and Electromagnetism (Part I) 51
The Biot-Savart law can also be used to derive the expression for the magnetic field within a solenoid. It can be proven that for a solenoid consisting of N loops, each carrying a current I, the magnetic field inside it is constant and is given by B = μ 0NI l Alternatively where B = μ 0 ni n = N l Unit 8 - Magnetic Field and Electromagnetism (Part I) 52
Example: A solenoid 15 cm long has 600 turns and carries a current of 5.0 A. What is the magnetic field strength inside this coil? Unit 8 - Magnetic Field and Electromagnetism (Part I) 53
Example: A 30.0 cm long solenoid 1.25 cm in diameter is to produce a field of 4.65 mt at its center. How much current should the solenoid carry if it has 935 turns of the wire? Unit 8 - Magnetic Field and Electromagnetism (Part I) 54
An application of solenoids: doorbell When the circuit is open When the circuit is closed Unit 8 - Magnetic Field and Electromagnetism (Part I) 55
7. Torque on a current loop Consider a closed loop of wire placed in a static magnetic field as depicted. When a current I is flowing through it, the magnetic field induces a Lorentz force that acts on the loop. According to the RHR: The force F 1 points down The force F 2 points up They produce a torque that makes the loop rotate. Unit 8 - Magnetic Field and Electromagnetism (Part I) 56
The magnitude of the Lorentz force is F 1 = F 2 = BIa Therefore, the total torque is τ = F 1 b 2 + F 2 b 2 = BIab = BIA If the coil consists of N loops, then τ = BNIA = BM The quantity M is called magnetic dipole moment. Unit 8 - Magnetic Field and Electromagnetism (Part I) 57
Example: A circular loop of wire has a diameter of 20.0 cm and contains 10 loops. The current in each loop is 3.00 A, and the coil is placed in a 2.00 T external magnetic field. Determine the maximum and minimum torque exerted on the coil by the field. Unit 8 - Magnetic Field and Electromagnetism (Part I) 58
There are many practical applications of magnetic force. We will consider the following examples. (1) Galvanometer The magnetic torque is τ m = NIAB sin θ The torque due to spring is At equilibrium τ s = kφ NIAB sin θ = kφ Hence the deflection of the pointer is: NIAB sin θ φ = I k Unit 8 - Magnetic Field and Electromagnetism (Part I) 59
(2) Electric motor It changes electric energy to mechanical energy. Coils are mounted on an armature which turns continuously in one direction within a magnetic field. A dc motor employs commutators and brushes to ensure the alternation of current that is necessary for maintaining the same direction of torque. An ac motor does not need commutators as the current itself alternates. Unit 8 - Magnetic Field and Electromagnetism (Part I) 60
(3) Loudspeakers A speaker cone is connected to a coil of wires which is placed within a magnetic field. The electrical output of a stereo directs an alternating current to pass through the internal coil. The field then generates a force that drags the coil and speaker cone back and forth, thus producing sound waves. The intensities and frequencies of sound waves are controlled by the magnitudes and the frequencies of the alternating current, respectively. Unit 8 - Magnetic Field and Electromagnetism (Part I) 61