#### Transcript General Case Magnetic Field - McMaster Physics and Astronomy

Magnetic Forces and Torques Review: Charged Particle in an external field: F qv B Straight wire of length L with current I in a uniform external field B: F I LB Note: L points in direction of positive current flow Force on a current-carrying wire (general case) If B not uniform, and/or wire not straight: the force dF on a short segment of vector length dL is dF = IdL x B The total force on the wire is: I dF B Segment of length dL dL I F I dL B along wire Example 1 Find the force on: y x x x x I R x x a) The straight wire b) The semicircular wire c) The whole circuit x x x x B (uniform) For (b): start with force dF due to an infinitesimal piece, and do the integral. Solution Theorem: For any closed current loop in a uniform magnetic field, Total magnetic force on the loop = 0 Proof: F I dL B I dL B (if B is a constant vector) But: dL 0 for a closed loop, so F = 0 Torque: Although there is no net force on a circuit in a uniform field, there may be a net torque. The torques due to equal and opposite forces applied at different locations do not necessarily cancel. We can calculate the torque directly for a rectangular loop. There is also a simple rule, which actually applies to a plane loop of any shape. First we need to define the “magnetic dipole moment” (a vector) for a current loop. Magnetic Dipole Moment (or “magnetic moment”) Define the magnetic moment of a small current loop by IA (A " vector area" of loop) Area of loop current Note the right-hand rule! – fingers follow I, thumb points in μ ( surface) Torque on a Current Loop (Uniform B) Example: a rectangular loop B A h B C D w Forces: FAD= I h B FAB= I w B sinθ x FBC= I h B FCD= I w B sinθ Top view: FBC= I h B I x B B I into the page A FAD = I h B w•sinθ Torque (about any pivot; e.g., at A) IhB w sin ( Ihw) B sin ( I area ) B sin B B The torque on a current loop with magnetic dipole moment in a uniform external field B is: B Where μ = IA Example 2 Find the torque on a flat, horizontal, circular coil due to the magnetic field of the Earth. B north B 50o 0.5 x 10-4 T Circular loop, 2 turns, R = 1 m, I = 20 A, CCW (from above) Find: Torque (magnitude and direction) Example 2 B B 50o I Which direction does the torque vector point in for a flat horizontal coin on the surface of earth ? Example 3 Find the torque on a flat, horizontal, circular coil due to the magnetic field of the Earth. B north B 50o 0.5 x 10-4 T Circular loop, 2 turns, R = 1 m, I = 20 A (ccw from above) Example 4 Consider an electron orbiting a proton and maintained in a fixed circular path of radius R=5.29x10-11 m by the Coulomb force. Treating the orbiting charge as a current loop, calculate the resulting torque when the system is in an external magnetic field of B=0.4 T directed perpendicular to the magnetic moment of the orbiting electron. Summary Charged Particle: F qv B Wire: dF I dL B or Straight wire, Uniform B: F IL B Torque: B F I dL B