Electromagnetism Notes

ELECTROMAGNETISM

Electric Current and Magnetic Fields

  • Electric current creates a magnetic field.
  • B∝IrB \propto \frac{I}{r}
  • B∝IB \propto I
  • Formula for the magnetic field around a long, straight wire: B=μ0I2πrB = \frac{\mu_0 I}{2 \pi r}
    • BB = magnetic field strength, in teslas [T]
    • II = current in wire, in [A]
    • rr = radial distance, in [m]
    • μ0\mu_0 = magnetic permeability
  • Magnetic field lines are concentric around the wire.
  • We use B to simplify the math, as B∝FB \propto F
  • B=μ×HB = \mu \times H

Biot-Savart Law

  • BB = "magnetic field"
  • Note: Magnetic field is related to force (F), really, but B∝FB \propto F

Visualizing Magnetic Fields (Cross-Sections)

  • Field out of the page is represented by a dot inside a circle.
  • Field into the page is represented by a cross inside a circle.

Coil of Wire (Solenoid)

  • A coil of wire creates a magnetic field. The more turns a coil has, the stronger the magnetic field it creates. The magnetic field inside the wire is uniform.
  • Approximation: The field outside the solenoid is practically zero.
  • How strong is the field inside?

Ampère's Law

  • ∮B⋅dl=μ0I′\oint B \cdot dl = \mu_0 I'
  • Applying Ampère's Law to a Solenoid:
    • ∮B⋅dl=μ0I′\oint B \cdot dl = \mu_0 I'
    • B⋅dl=0B \cdot dl = 0 (B is inside)
    • B⋅dl=0B \cdot dl = 0 (Outside)
    • B⋅dl=0B \cdot dl = 0 (B is perpendicular to l)
    • ∮B⋅dl=μ0I′\oint B \cdot dl = \mu_0 I'
  • I′I' = total current enclosed by the path
  • I′=I⋅NI' = I \cdot N where N is the number of turns.

Strength of Field Inside an Air-Cored Solenoid

  • Formula for the strength of the magnetic field inside an air-cored solenoid: B=μ0NIB = \mu_0 NI
    • BB = magnetic field strength, in teslas [T]
    • μ<em>0\mu<em>0 = permeability of free space (μ</em>0=4π×10−7T⋅mA\mu</em>0 = 4 \pi \times 10^{-7} \frac{T \cdot m}{A})
    • NN = number of turns
    • II = current in wire, in [A]
    • ll = length of solenoid, in [m]

Magnetic Force

  • Aligned fields result in an attractive force.
  • Opposing fields result in a repulsive force.

Force on a Current-Carrying Wire

  • A current-carrying wire in a magnetic field experiences a force.
  • F=B×IlF = B \times I l
  • F=BILF = BIL (BLIC)

Motor

  • Illustrates the force on a current-carrying wire in a magnetic field, which is the basis for how motors work.
  • Torque is generated by the force on the wire.
  • When the force is aligned there is no torque and when it is perpendicular, there is a torque.

Force on a Moving Charge

  • The force on a moving charge is represented by: F=qvBF = qvB
  • Relationship between current, charge, and velocity: I = q/t, so F=BIlF=BIl
  • l=vtl=vt thus F=BqvF = Bqv
  • Equating the two formulas: F=BIlF = BIl is the equivalent to F=BqvF=Bqv

Circular Motion in a Magnetic Field

  • A charged particle moving in a magnetic field experiences circular motion.
  • F=FcF = F_c
  • qvB=mv2rqvB = \frac{mv^2}{r}
  • qBr=mvqBr = mv
  • r=mvqBr = \frac{mv}{qB}

Speed Selector

  • Only particles with a specific velocity get through the gate.
  • If Fnet=0F_{net} = 0, then the particle is not deflected.

Crossed Fields

  • Crossed fields act as a "speed selector."
  • F<em>B=F</em>eF<em>B = F</em>e
  • qvB=EqqvB = Eq
  • v=EBv = \frac{E}{B}
  • Constants: e, d, μ0\mu_0, N.

Mass Spectrometer

  • r=mvqBr = \frac{mv}{qB}
  • v=EBv = \frac{E}{B}
  • mq=Brv\frac{m}{q} = \frac{Br}{v}
  • mq=B2rE\frac{m}{q} = \frac{B^2r}{E}