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.
      • Bx = \frac{\mu0 I}{2 \pi (2*\sqrt{5})} \cos(90-63)
      • By = \frac{\mu0 I}{2 \pi (2*\sqrt{5})} \sin(90-63)