College Physics 2e - Chapter 22 Summarized Notes

Chapter 22: Magnetism

22.1 Magnets

• Aurora Borealis:

  • Also known as northern lights.

  • Caused by the interaction of the Earth’s magnetic field with radiation from solar storms.

• Characteristics of Magnets:

  • Magnets exist in various shapes, sizes, and strengths.

  • All magnets possess both a north pole and a south pole.

  • Isolated poles (monopoles) do not exist.

• Behavior of Magnets:

  • Like poles repel each other, whereas unlike poles attract.

  • This behavior is analogous to electrostatics: unlike charges attract, and like charges repel.

• Pole Pairing:

  • North and south poles always appear in pairs.

  • It is impossible to separate a north pole from a south pole in the same way that positive and negative charges can be separated.

• Earth's Magnetic Field:

  • The Earth acts like a large bar magnet, with its south-seeking magnetic pole located near the geographic North Pole.

  • Thus, the north pole of a compass is attracted towards the geographic North Pole of the Earth.

  • The separation distance between the Earth's geographic north pole and its magnetic south pole is approximately 1300 km (about 800 miles).

  • The geographic south pole is located near the magnetic north pole.

22.2 Ferromagnets and Electromagnets

• Ferromagnetic Materials:

  • Show strong magnetic effects.

  • Can be magnetized to form permanent magnets.

  • Examples include iron, cobalt, nickel, and gadolinium.

• Unmagnetized Ferromagnets:

  • Comprise randomly oriented domains.

  • When exposed to an external magnetic field, these domains align, resulting in magnetization.

• Rare Earth Elements:

  • Certain rare earth elements, such as neodymium (atomic number 60), exhibit strong permanent magnet properties.

22.3 Magnetic Fields and Magnetic Field Lines

• Definition of Magnetic Field Lines:

  • Magnetic field lines indicate the direction a small compass would point when placed at a given location.

  • Connecting the direction arrows creates continuous magnetic field lines that form closed loops.

• Iron Filings Demonstration:

  • When iron filings are placed near a magnet, they align along the shape of the magnetic field lines, resembling tiny compass needles.

22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

• Lorentz Force:

  • A magnetic field exerts a force on moving electric charges.

  • The magnitude of the force is given by the formula:
    $F = qvB ext{ sin } heta$

  • Where:

  • $F$ = magnetic force,

  • $q$ = charge,

  • $v$ = velocity of the charge,

  • $B$ = magnetic field strength,

  • $heta$ = angle between velocity and magnetic field direction.

• Direction of the Lorentz Force:

  • Determined using Right Hand Rule 1 (RHR-1):

  • Point four fingers in the direction of the magnetic field.

  • Point the thumb in the direction of the charge's motion.

  • The direction of the Lorentz force is indicated by the palm of the hand.

• Negative Charge Behavior:

  • A negative charge moving in the same direction as the magnetic field experiences a perpendicular force directed upwards.

• Example of Positive Charge:

  • A positively charged particle moving due west in a magnetic field pointing due north will experience a downward Lorentz force.

• Units of Magnetic Field:

  • The Lorentz Force allows the definition of magnetic field strength as:
    $B = rac{F}{qv ext{ sin } heta}$

  • SI unit of magnetic field strength is the Tesla (T).

  • Equivalence:

  • $1 T = rac{N}{C imes m/s} = rac{N}{A imes m}$

  • The Gauss (G) is a smaller unit of magnetic field strength:

  • $1 G = 10^{-4} T$

  • Strength of different magnetic fields:

  • Strongest permanent magnets: ~ 2 T

  • Superconducting electromagnets: ~ 10 T

  • Earth's magnetic field: ~ 0.5 G or $5 imes 10^{-5} T$.

22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications

• Circular Motion:

  • The Lorentz force causes charged particles to travel along a circular path, governed by the formula for radius:
    $r = rac{mv}{qB}$

  • Students should know how to derive this from foundational principles.

• Spiraling Motion:

  • When a charged particle moves at an angle to the magnetic field, it spirals along the direction of the field.

• Cosmic Rays and Earth's Field:

  • Energetic electrons and protons (components of cosmic rays) from the Sun and outer space often follow the Earth’s magnetic field lines, leading to visible effects such as the Northern Lights.

22.7 Magnetic Force on a Current-Carrying Conductor

• Magnetic Force on Current-Carrying Wires:

  • The magnetic field exerts a force on a wire carrying an electric current.

  • This force direction follows Right Hand Rule 1 and can be substantial due to the large number of charges in typical currents.

• Lorentz Force in Context:

  • The equation is represented as:
    $F_{B} = I L B ext{ sin } heta$

  • Where:

  • $F_{B}$ = magnetic force,

  • $I$ = current,

  • $L$ = length of wire in the magnetic field,

  • $B$ = magnetic field strength,

  • $heta$ = angle between the wire direction and the magnetic field direction.

22.8 Torque on a Current Loop: Electric Motors

• Electric Motors:

  • Convert electrical energy into mechanical energy.

  • A current-carrying loop of wire connected to a rotating shaft experiences magnetic forces.

  • These forces generate torque, which turns the shaft clockwise when observed from above.

• Torque Equation:

  • Torque can be quantified using the formula:
    $au = r F ext{ sin } heta$

  • For a wire loop around an axis, the derived torque is:
    $au = B I A ext{ sin } heta$

  • Where:

  • $au$ = torque,

  • $B$ = magnetic field strength,

  • $I$ = current,

  • $A$ = area of loop (calculated as width multiplied by length).

• Maximum Torque Condition:

  • Maximum Torque is given by:
    $au_{max} = N B I A$

  • Where $N$ is the number of loops.

• Example of Torque Calculation:

  • Example 22.5:

  • For a 100-turn square loop with a 10.0 cm side carrying 15.0 A in a 2.00 T field, the maximum torque obtained is:
    $au_{max} = 30 ext{ N.m}$.

22.9 Magnetic Fields Produced by Currents: Ampere’s Law

• Induced Magnetic Fields:

  • Magnetic fields are produced by electric currents in wires.

• Field Strength from a Long Straight Wire:

  • The magnetic field strength can be experimentally derived as:
    $B = rac{ ext{μ}_0 I}{2 ext{π}R}$

  • Where:

  • $I$ is the current,

  • $R$ is the distance from the wire,

  • $ext{μ}_0 = 4 ext{π} imes 10^{-7} T imes m/A$ (the permeability of free space).

• Direction of the Magnetic Field:

  • Determined using Right Hand Rule 2 (RHR-2).

  • If the right-hand thumb points in the direction of the current, the fingers curl around the wire in the direction of the magnetic field lines.

• Example of Current Calculation:

  • Example 22.6:

  • Find current to generate a magnetic field twice the Earth's strength (0.5 G at 5.0 cm).

  • Solution provides:
    $I = 25 ext{ A}$.

• Strength of Field from Current-Coil:

  • For a current-carrying loop, the field strength is:
    $B = N rac{ ext{μ}_0 I}{2R}$

  • Increasing number of coils enhances the magnetic field.

22.10 Magnetic Force between Two Parallel Conductors

• Force Between Conductors:

  • According to RHR-1, two parallel conductors with currents in the same direction attract each other, while currents in opposite directions repel each other.

• Force Per Unit Length Calculation:

  • The force between two parallel wires carrying currents is given by:
    $F_{12} = k rac{I_1 I_2}{d}$

  • Where $k$ is a constant of proportionality, $I_1$ and $I_2$ are the currents, and $d$ is the separation distance between the wires.

22.11 Applications of Magnetism

• Magnetism in Technology:

  • Many applications arise from the properties of magnetism, such as mass spectrometers that employ both a velocity selector and a detection chamber with a uniform magnetic field.

  • The velocity selector determines the speed of charged particles, allowing for their analysis in a controlled magnetic field.

Practice Problems

• Problem 1:

  • Given: Mass $m = 1.00 ext{ kg}$ suspended by a circuit in a magnetic field of $B_{in} = 2.00 T$.

  • (a) Find current $I$, (b) Find resistance $R$.

• Problem 2:

  • Two long parallel wires with currents $I_1 = 3.00 A$ and $I_2 = 5.00 A$.

  • (a) Calculate magnetic field halfway between them, (b) at point $P$ above the 5 A wire.

• Problem 3:

  • An electron in a circular path with radius 2.0 cm.

  • Calculate the magnetic field strength and current in the solenoid with 25 turns per cm.

• Problem 4:

  • Magnitude and direction of magnetic field at a specific point due to two currents.

• Problem 5:

  • A rectangular loop with specific properties experiences a magnetic force. Calculate the magnetic field magnitude $B$.

• Problem 6 & 7:

  • A proton's path in a defined magnetic field is analyzed for the force experienced, engaging the velocity and direction components.