Chapter 2: Magnetism

2.1 Introduction to Magnetism

• Definition: Magnetism is the force of attraction or repulsion experienced by certain materials in the presence of a magnetic field.

• Not all materials are affected by magnetism.

  • Non-magnetic materials: wood, glass, paper, plastic

  • Magnetic materials (common metals): Iron, nickel, cobalt

2.1.1 Magnetic Field

• A Magnetic Field (B) is defined as a vector field that exerts a magnetic force on moving electric charges.

• Steady Magnetic Field: A magnetic field that remains constant over time, resulting from a steady electrical current.

• Unstable Magnetic Field: A magnetic field that varies over time.

2.1.2 Historical Perspective

• In 1820, Hans Christian Oersted conducted an experiment demonstrating the relationship between electricity and magnetism.

  • Oersted placed a compass needle near a wire carrying an electric current.

  • When the electric current was activated (switch closed), the compass needle deflected as though influenced by a magnet.

2.1.3 Lorentz Force

• The Lorentz Force describes the force acting on a moving charge in a magnetic field.

  • The force depends on three factors:

  1. Charge of the particle (electric quantity)

  2. Velocity of the particle

  3. Instantaneous position of the particle

  • This force is always perpendicular to both the magnetic field and the velocity of the charge.

Right-Hand Rule

• A convention (right-hand rule) to determine the direction of the Lorentz force.

  • Point the thumb of your right hand in the direction of the velocity of the charge.

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

  • The palm will face the direction of the force acting on a positive charge.

2.1.4 Properties of Magnetic Fields

• The magnetic field strength is characterized by the density of magnetic field lines: more lines indicate a stronger field.

• Direction of Magnetic Field Lines: Defined to be the direction in which the north end of a compass needle points, tangent to the field lines at any point.

• Properties:

  • Magnetic field lines cannot cross, indicating a unique magnetic field at any given point.

  • Magnetic field lines form continuous closed loops without beginning or end, distinguishing them from electric field lines that begin and end on charges.

2.1.5 Units of Magnetic Field

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

• Another smaller unit is Gauss (G):

  • 1 T = 10^4 G

2.1.6 Magnetic Field Strength Examples

• Pulsar magnetic field: approximately $10^8 T$

• Near certain atomic nuclei: $10^4 T$

• Superconducting magnet: can generate up to $25 T$

• Large laboratory magnets: greater than $2 T$

• Earth’s magnetic field: approximately $10^{-4} T$

• Indoor magnetic fields: range from $10^{-7} ext{ to } 10^{-6} T$

• Human heart: generates a magnetic field of approximately $3 imes 10^{-10} T$

2.1.7 Magnetic Induction

• The integral of magnetic induction over a curved surface within a magnetic field defines magnetic flux.

  • Unit of Magnetic Flux: Weber (Wb)

  • Mathematical representation: $ext{Flux} ext{ } (Φ) = B imes A$ where A is the area and B is the magnetic field strength.

2.2 Ampere’s Circulation Law

• Ampere’s Law states that the line integral of the magnetic field around any closed path equals $ext{μ}_0$ times the total steady current passing through any surface bounded by that path.

  • $B \bullet dl = ext{μ}<em>0 I</em>{total}$

• This law is essential for calculating magnetic fields in configurations exhibiting high degrees of symmetry.

Applications and Examples

1. Long Straight Wire:

   • For a straight wire carrying current $I$, Ampere's Law can be applied to a circular path to find the magnetic field strength at a distance $r$ from the wire.

   • Inside the wire (r<R):
     $B = rac{μI}{2 ext{π}R^2}$

   • Outside the wire ($r≥R$):
     $B = rac{μI}{2 ext{π}r}$

2. Infinite Current-Carrying Plate:

   • Considered as multiple infinite straight wires contributing to the overall magnetic field.

   • For such a plate, the magnetic induction can be calculated as being parallel to the plate and perpendicular to the direction of current.

3. Ideal Solenoid:

   • A long, straight solenoid with closely spaced turns creates a uniform magnetic field inside while being zero outside.

   • Applying Ampere’s Law to the rectangular path leads to uniform field expressions.

Conclusion

• Understanding the principles of magnetism, magnetic fields, Lorentz force, Ampere's Law, and their applications in various configurations is essential for the comprehensive study of electromagnetism.

• The unique properties of magnetic fields set them apart from electric fields, forming the basis for various technological applications in science and engineering.