Magnetic Dipole Moments and Torque in a Coil

Magnetic Dipole of a Coil

  • The magnetic dipole of a coil refers to its ability to create a magnetic field when electric current flows through it.
  • It can be understood as a vector quantity represented by the magnetic dipole moment (μ).
  • A magnetic dipole consists of a north pole and a south pole, analogous to a bar magnet.

Problem 1: Calculating Current for Given Dipole Moment

  • Given parameters:
    • Number of turns (N) = 160
    • Radius of the coil (r) = 1.90 cm = 0.019 m
    • Desired magnetic dipole moment (μ) = 2.30 A·m²
  • Formula for magnetic dipole moment: μ=NimesIimesAμ = N imes I imes A where:
    • I = Current (in Amperes)
    • A = Area of the coil (in m²)
Step 1: Calculate the Area of the Coil
  • The area A of a circular coil is calculated as: A=extπr2A = ext{π} r^2
    • $r = 0.019 m$
  • Substituting the radius into the area formula:
    A=extπimes(0.019)2=1.134imes10−3extm2A = ext{π} imes (0.019)^2 = 1.134 imes 10^{-3} ext{ m}^2
Step 2: Calculating the Current (I)
  • Rearranging the formula for magnetic dipole moment to solve for current:
    I=μNimesAI = \frac{μ}{N imes A}
  • Plugging in values:
    I=2.30160imes1.134imes10−3I = \frac{2.30}{160 imes 1.134 imes 10^{-3}}
  • Calculation yields:
    I=67extAI = 67 ext{ A}

Problem 2: Calculating Maximum Torque in a Magnetic Field

  • Given:
    • External magnetic field (B) = 35.0 mT = 0.035 T
    • Current (calculated in previous section) = 67 A
  • Maximum torque (τ) in a uniform magnetic field is given by:
    τmax=μBimesextsine(heta)τ_{max} = μB imes ext{sine}( heta)
  • Where θ is the angle between the magnetic dipole moment and the magnetic field.
  • The maximum torque occurs when sine(θ) = 1 (θ = 90 degrees):
    τmax=μBτ_{max} = μB
  • Substituting values:
    τmax=2.30imes0.035=0.0805N⋅mτ_{max} = 2.30 imes 0.035 = 0.0805 N·m

Problem 3: Potential Energy of an Electric Dipole in a Magnetic Field

  • Potential energy (U) of an electric dipole in an external magnetic field can be represented as:
    U = - oldsymbol{p} ullet oldsymbol{E}
  • Here,
    • p = electric dipole moment (in Coulomb·m)
    • E = external electric field (in N/C)
  • When considering work done (dw) in rotating the dipole under a conservative field: U=extPE[ext−Cos(θ)]U = ext{PE}[ ext{-Cos}(θ)]
    • where θ is the angle between dipole moment and field.

Problem 4: Relationship Between Torque, Magnetic Dipole, and Field

  • The torque (T) produced on a magnetic dipole (μ) in a magnetic field (B) is expressed as:
    T=μBimesextsine(θ)T = μB imes ext{sine}(θ)
  • This relationship highlights how the torque depends on the angle between the magnetic moment and the magnetic field.