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## Magnetic Forces and Torques

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**Review:**Charged Particle in an external field: Straight wire of length L with current I in a uniform external field B: 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: dF B I Segment of length dL dL I**Example 1**• Find the force on: • The straight wire • The semicircular wire • The whole circuit y x x x x x x x x x I R x B (uniform) For (b): start with force dF due to an infinitesimal piece, and do the integral.**Theorem: For any closed current loop in a uniform magnetic**field, Total magnetic force on the loop = 0 Proof: (if B is a constant vector) But: 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 current Area of loop Note the right-hand rule! – fingers follow I, thumb points in μ m**Torque on a Current Loop (Uniform B)**Example: a rectangular loop B A h C D w Forces: FAB= I w B sinθ FAD= I h B FBC= I h B x FCD= I w B sinθ**Top view:**FBC= I h B x B I into the page I w A w•sinθ FAD = I h B • Torque (about any pivot; e.g., at A)**The torque on a current loop with magnetic dipole**momentin a uniform external field is: Where μ = IA**Example 2**Find the torque on a flat, horizontal, circular coil due to the magnetic field of the Earth. 50o north 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**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. 50o north 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: Wire: Straight wire, Uniform B:Torque: