Magnetism

Magnetism

• Magnetism is the study of magnets and their properties.

Magnet

• A magnet is a material that produces a magnetic field.
• Magnets have two poles: South (S) and North (N).
• Like poles repel each other; opposite poles attract.

How to Make a Magnet

1. Ferromagnet:

   • Start with a ferromagnetic material (e.g., iron).
   • Heat the material and tap it repeatedly while it cools.
   • This aligns the magnetic domains within the material, creating a permanent magnet.

2. Electromagnet:

   • Wrap a coil of wire around a ferromagnetic core (e.g., iron).
   • Pass an electric current (I) through the wire.
   • The current creates a magnetic field.
   • The strength of the magnetic field can be increased by increasing the current or the number of turns in the coil.
   • MRI (Magnetic Resonance Imaging) utilizes electromagnets.

Magnetic Field (B)

• A magnetic field is a region around a magnet or current-carrying wire where magnetic forces can be detected.
  • It has both direction and strength.
  • Symbol: $\vec{B}$
  • Unit: Tesla (T). Sometimes Gauss (G) is used (1 T = 10,000 G).

Magnetic Field Lines

• Magnetic field lines are used to visualize the direction and strength of a magnetic field.
  • They point from the north pole to the south pole outside the magnet.
  • The closer the lines, the stronger the field.

Notation Symbols for 3D Direction

• $\odot$: Out of the plane
• $\otimes$: Into the plane

Magnetic Moment (m)

• Magnetic moment is a measure of the strength and direction of a magnet.
  • It is a vector quantity.
  • Unit: $A \cdot m^2$

B-Field Strength

• Calculating B-field strength can be complex, but simplifications are possible in certain cases.
• Uniform Field: A magnetic field with constant magnitude and direction.
• Uniform Magnetic Field: A region where the magnetic field is uniform.

Magnetic Field Calculation

1. Straight Current Wire (Ampere’s Law):

   • Magnitude: $B = \frac{{\mu_0 I}}{{2 \pi r}}$
   • Direction: Determined by the right-hand rule (thumb in the direction of current, fingers curl in the direction of the magnetic field).
   • $\mu_0 = 4 \pi \times 10^{-7} T \cdot m/A$ (permeability of free space).

2. Center of a Current Loop:

   • Magnitude: $B = \frac{{\mu_0 I}}{{2r}}$, where r is the radius of the loop.
   • Direction: Perpendicular to the loop, determined by the right-hand rule.

3. Inside of a Solenoid:

   • Magnitude: $B = \mu_0 n I$, where n is the number of loops per unit length of the solenoid ($n = N/L$, with N being the number of loops and L the length).
   • Direction: Along the axis of the solenoid, determined by the right-hand rule.

Examples

• Direction of Magnetic Field Around a Wire:

  • The magnetic field lines wrap around the current in a counterclockwise fashion (based on the right-hand rule).

• Current in a Wire Pointing into the Page:

  • The magnetic field direction is Left.

• Current in a Wire Pointing to the Right:

  • The magnetic field above the wire is out of the page.

• Current Carrying Loop (Magnetic Dipole Moment):

  • The magnetic field at the center of the loop points along the direction of the magnetic dipole moment.

• Current Carrying Loop as a Magnetic Dipole:

  • Magnetic field lines leave a north pole and enter a south pole.
  • The direction of the current determines which pole faces towards you.

• Current Carrying Loop in an External Magnetic Field:

  • A current carrying loop placed in an external magnetic field will experience a torque.
  • The loop will rotate to align its magnetic dipole moment with the external magnetic field.

Example: Comparing Magnetic Field Magnitudes

• To determine which case has a larger magnetic field magnitude (at the center point), you need to compare the arrangements and distances.

Example: Find Magnetic Field at Specific Points

• Given a current $I$ in a wire, calculate the magnetic field at points A, B, and C using the formulas discussed earlier.

• Solution:

  • A: $B = \frac{\mu_0 I}{2 \pi (2)} \uparrow$
  • B: $B = \frac{\mu_0 I}{2 \pi (4)} \uparrow$
  • C:
    • The distance from the wire to point C is $\sqrt{2^2 + 4^2} = \sqrt{20} = 2\sqrt{5}$ meters.
    • $\tan(\theta) = \frac{4}{2}$, so $\theta = \tan^{-1}(\frac{1}{2}) = 63.43$ degrees.
    • The direction is such that the direction of the magnetic field at point C makes 63 degrees versus the x axis.
    • $B<em>x = \frac{\mu</em>0 I}{2 \pi (2*\sqrt{5})} \cos(90-63)$
    • $B<em>y = \frac{\mu</em>0 I}{2 \pi (2*\sqrt{5})} \sin(90-63)$