Chapter 4: Moving Charges and Magnetism

MAGNETIC FORCE AND FIELD

Magnetic Force

  • Definition: Force experienced by a moving charge in a magnetic field
  • Lorentz Force: F = q(v × B)
  • Direction: Determined by right-hand rule
  • Unit of Magnetic Field (B): Tesla (T) = N/(A·m) = Wb/m²

Magnetic Force on Current-Carrying Conductor

\[ \vec{F} = I(\vec{L} \times \vec{B}) \]
  • L = length vector in direction of current
  • Force is perpendicular to both current and magnetic field

Motion of Charged Particle in Magnetic Field

Circular Motion: When v ⊥ B

  • Radius: r = mv/(qB)
  • Angular frequency: ω = qB/m
  • Time period: T = 2πm/(qB)

Helical Motion: When v has components both parallel and perpendicular to B

  • Pitch of helix: p = 2πm(v·B̂)/(qB)

MAGNETIC FIELD DUE TO CURRENT

Biot-Savart Law

\[ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} \]
  • μ₀: Permeability of free space = 4π × 10⁻⁷ T·m/A
  • Analogous to Coulomb's law in electrostatics

Magnetic Field Due to Straight Current-Carrying Wire

\[ B = \frac{\mu_0 I}{2\pi r} \]

Direction determined by right-hand rule (thumb along current, fingers curl in direction of field)

Magnetic Field on the Axis of Circular Current Loop

\[ B = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}} \]

At center (x = 0): B = μ₀I/(2R)

Direction determined by right-hand rule

Magnetic Field Due to Solenoid

Inside (far from ends): B = μ₀nI

  • n = number of turns per unit length

Outside: B ≈ 0

Magnetic Field Due to Toroid

Inside: B = (μ₀NI)/(2πr)

  • N = total number of turns

Outside: B = 0

AMPERE'S CIRCUITAL LAW

Statement

Line integral of magnetic field around any closed loop equals μ₀ times the current enclosed

\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \]

Applications

  • Straight wire: B = (μ₀I)/(2πr)
  • Solenoid: B = μ₀nI
  • Toroid: B = (μ₀NI)/(2πr)

FORCE BETWEEN CURRENT-CARRYING CONDUCTORS

Parallel Conductors

\[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r} \]
  • Attractive if currents are in same direction
  • Repulsive if currents are in opposite directions

Definition of Ampere

1 Ampere: Current that, when flowing through each of two parallel conductors 1m apart, produces a force of 2 × 10⁻⁷ N/m

TORQUE ON CURRENT LOOP

Magnetic Dipole Moment

\[ \vec{\mu} = I\vec{A} \]
  • A = area vector of loop
  • Direction determined by right-hand rule

Torque on Current Loop in Uniform Magnetic Field

\[ \vec{\tau} = \vec{\mu} \times \vec{B} \]

Tends to align magnetic moment with field

Potential Energy

\[ U = -\vec{\mu} \cdot \vec{B} \]

Minimum when magnetic moment is aligned with field

MOVING COIL GALVANOMETER

Working Principle

  • Current-carrying coil experiences torque in magnetic field
  • Deflection proportional to current

Construction

  • Rectangular coil suspended between poles of magnet
  • Phosphor-bronze suspension provides restoring torque
  • Soft iron core enhances magnetic field

Conversion to Ammeter

  • Low resistance shunt in parallel with galvanometer
  • Ishunt = (Ig × Rg)/Rshunt

Conversion to Voltmeter

  • High resistance in series with galvanometer
  • Rseries = (V/Ig) - Rg

EARTH'S MAGNETISM

Components of Earth's Magnetic Field

  • Horizontal Component (BH)
  • Vertical Component (BV)
  • Total Field (B)
  • Angle of Dip (δ): tan δ = BV/BH
  • Angle of Declination: Angle between magnetic north and geographic north

KEY FORMULAS

  • Lorentz Force: F = q(v × B)
  • Force on current-carrying wire: F = I(L × B)
  • Radius of circular path: r = mv/(qB)
  • Biot-Savart Law: dB = (μ₀/4π) × (I dl × r̂)/r²
  • Field due to straight wire: B = (μ₀I)/(2πr)
  • Field at center of circular loop: B = μ₀I/(2R)
  • Field inside solenoid: B = μ₀nI
  • Ampere's Law: ∮B·dl = μ₀Ienc
  • Force between parallel conductors: F/L = (μ₀I₁I₂)/(2πr)
  • Torque on current loop: τ = μ × B
  • Magnetic dipole moment: μ = IA