unit 7 magnetic field and electromagnetism (part 1)

Physics 12Unit 7 – Magnetic Field and Electromagnetism

(Part 1)

Dr. John Lo

Royal Canadian College

2020-2021

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 7 - Magnetic Field and Electromagnetism (Part I) 2RCC @ 2020/2021

• 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


• 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).


• 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.


• 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.


• Example: Attraction of liquid oxygen by magnets


• 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).


• 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.


• 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.


• 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.


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.


• 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.


• 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.

Unit 7 - Magnetic Field and Electromagnetism (Part I) 14

Opposite poles Same poles

RCC @ 2020/2021

• 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 polesand 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.


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.


• Magnetic interaction with current can also be illustrated in the following experiments:

1. Magnetic interaction on wires


2. Force between two parallel wires

• There exists interaction between current-carrying wires.

• For parallel current: attraction

• For anti-parallel current: repulsion


• 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.


Strength of magnetic field

• Recall that electrostatic field can be defined as the force experienced per Coulomb of charge.

• 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 perpendicularto the field

2. The magnitude of the charge


𝐸 =𝐹𝑒𝑞

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• Mathematically,

where 𝐵 is magnetic induction, 𝑞 is the charge, 𝑣 is the velocity component, and 𝐹𝑚 is the magnetic force.

• Note that 𝐹𝑚 and 𝐵 are always perpendicular to each other. Their directions can be described using the Fleming’s right-hand motor rule(RHR):


𝐵 =𝐹𝑚𝑞𝑣

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• The unit of magnetic field is Testa (T). By definition,

• 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 =1 N

1 C 1 m/s=

1 N

1 A ∙ m

1 T = 104 G

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• 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

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• The magnetic induction in these situations can be described using appropriate mathematical approaches. (We will discuss these cases individually in later sections.)


Magnetic field due to solenoid Magnetic field due to magnets

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• 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

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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?


• 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 parallelto the field, and therefore it will notparticipate in the magnetic force.

• The component 𝑣┴ = 𝑣 sin 𝜃 is perpendicular to the field, and thus it isinvolved in the resulting magnetic force.

• Recall the definition of a magnetic field:


𝐵 =𝐹𝑚𝑞𝑣┴

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• Hence,

• 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 𝐹𝑚 using the Fleming’s right hand rule.

• Mathematically, the Lorentz force can be defined in a more compact way using the cross product:


𝐹𝑚 = 𝑞𝑣┴𝐵 = 𝑞𝑣𝐵 sin 𝜃

𝑭𝒎 = 𝑞𝒗 × 𝑩

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• 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

• Since 𝑣 = 𝑙/𝑡, we have

• Note, however, that 𝑞 = 𝐼𝑡; hence


𝐹𝑚 = 𝑞𝑣𝐵

𝐹𝑚 = 𝑞𝑙

𝑡𝐵

𝐹𝑚 = 𝐵𝑞

𝑡𝑙 = 𝐵𝐼𝑙

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• 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?


• 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


𝐹𝑚 = 𝐵𝐼𝑙 sin 𝜃

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• 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.


• 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

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• 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.


• The effect of the magnetic force exerted on I2 due to I1 can be analyzed using the Ampère’s right hand rule:


• And

• Similarly, we can show that the force on I1 due to I2 is pointing to the left according to the RHR.

• Consequently, these wires repel each other.


According to the RHR, the magnetic force on I2 due to I1

is pointing to the right.

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• Practice: Prove that two parallel wires carrying current in the same direction are attracted to each other.


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?


• Recall that 𝐹𝑚 = 𝑞𝑣𝐵, and 𝐹𝑚 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.


• 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,


𝐹𝑚 = 𝐹𝑐

𝑞𝑣𝐵 =𝑚𝑣2

𝑟

𝑞𝐵 =𝑚𝑣

𝑟=𝑝

𝑟

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• Example: An electron travelling at 2.5 × 107 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?


• Note that

• Rearranging this expression yields

• 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.

• If 𝑣 is fixed (i.e., velocity selection), then heavier particles (i.e.,

larger 𝑚) will pass through a larger circular path.


𝑞𝐵 =𝑚𝑣

𝑟

𝑟 =𝑚𝑣

𝑞𝐵

𝑟 ∝ 𝑣

𝑟 ∝ 𝑚RCC @ 2020/2021

• 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.

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• 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

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• For the charged particles to travel straight through, the two forces have to be balanced by one another. That means,

• Hence, the velocity of the particles that leave this modulator is

• Technically, the velocity selection of the charged particles can be achieved by choosing appropriate 𝐸 and 𝐵.


𝐹𝑒 = 𝐹𝑚

𝑞𝐸 = 𝑞𝑣𝐵

𝑣 =𝐸

𝐵

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• 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.


𝑚 =𝑞𝐵2𝑟

𝑣

𝑣 =𝐸

𝐵1

𝑚

𝑞=𝐵1𝐵2𝐸

𝑟

Mass-to-charge ratio of particle

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• 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%

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• 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?


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 ∝𝜇0𝐼

𝑟

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• In this expression

• For a straight wire, the magnetic field at a distance 𝑎 from it is given by


𝜇0 Magnetic permeability (4𝜋 × 10−7 N/A2)

𝐼 Steady current

𝑟 Perpendicular distance from the wire

𝐵𝑃 =𝜇0𝐼

2𝜋𝑎

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• 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?


• 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

• Alternatively

where


𝐵 =𝜇0𝑁𝐼

𝑙

𝐵 = 𝜇0𝑛𝐼

𝑛 =𝑁

𝑙RCC @ 2020/2021

• 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?


• 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?


• An application of solenoids: doorbell


When the circuit is open When the circuit is closed

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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 𝐹1 points down

• The force 𝐹2 points up

• They produce a torque that makes the loop rotate.


• The magnitude of the Lorentz force is

• Therefore, the total torque is

• If the coil consists of 𝑁 loops, then

• The quantity 𝑀 is called magnetic dipole moment.


𝐹1 = 𝐹2 = 𝐵𝐼𝑎

𝜏 = 𝐹1𝑏

2+ 𝐹2

𝑏

2= 𝐵𝐼𝑎𝑏 = 𝐵𝐼𝐴

𝜏 = 𝐵𝑁𝐼𝐴 = 𝐵𝑀

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• 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.


• There are many practical applications of magnetic force. We will consider the following examples.

(1) Galvanometer

• The magnetic torque is

• The torque due to spring is

• At equilibrium

• Hence the deflection of the pointer is:


𝜏𝑚 = 𝑁𝐼𝐴𝐵 sin 𝜃

𝜏𝑠 = 𝑘𝜑

𝑁𝐼𝐴𝐵 sin 𝜃 = 𝑘𝜑

𝜑 =𝑁𝐼𝐴𝐵 sin 𝜃

𝑘∝ 𝐼

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(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.


(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 7 magnetic field and electromagnetism (part 1)

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