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Download Lecture 10: Electromagnetic Forces
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Lecture 10 : Electromagnetic Forces β’ Forces felt in πΈ- and π΅-fields β’ Motion of particle in a π΅-field β’ Applications Recap πΈ β’ Charges set up an electric field πΈ, in which other charges feel an electric force β’ Currents (or magnets) set up a magnetic field π΅, in which other currents (or magnets) feel a magnetic force π΅ β’ In this lecture we will examine these electromagnetic forces in more detail Forces felt in πΈ- and π΅-fields β’ A charge π in electric field πΈ feels force πΉπΈ = ππΈ β’ This causes acceleration parallel to the πΈ-field Forces felt in πΈ- and π΅-fields β’ Other examples of electric forces! Forces felt in πΈ- and π΅-fields β’ A current π° or a charge π moving with velocity π in a magnetic field π΅ feels a force perpendicular to both the direction of current/motion and π΅ β’ Force on a current element : πΉπ΅ = πΌ π π × π΅ β’ Force on a charge : πΉπ΅ = π π£ × π΅ β’ (Recall, the cross product of two vectors is perpendicular to both) Forces felt in πΈ- and π΅-fields β’ Direction of the force from βFlemingβs left-hand ruleβ πΉπ΅ = π π£ × π΅ π£ πΉπ΅ = πΌπ π × π΅ ππ π π΅ πΌ Forces felt in πΈ- and π΅-fields β’ Force between 2 current-carrying wires: πΏ Magnetic field set up by π0 πΌ1 wire 1 is π΅1 = 2ππ Force felt by wire 2 is π0 πΌ1 πΌ2 πΏ πΉ = π΅1 πΌ2 πΏ = 2ππ Forces felt in πΈ- and π΅-fields β’ Other examples of magnetic forces! Motion of particle in a π΅-field β’ Consider a charge +π injected into a magnetic field π΅ (into the screen) with velocity π£ (to the right) Magnetic field π΅ (into the screen) π£ +π The charge will feel a magnetic force πΉπ΅ = ππ£ × π΅ (i.e., upward) Motion of particle in a π΅-field β’ Because the magnetic force is always perpendicular to π£, it will drive the charge in a circular orbit β’ This is a centripetal force ππ£ 2 , π β’ Hence πΉπ΅ = ππ£π΅ = where π is the mass of the charge π, and π is the radius β’ Radius of orbit π = β’ Time period = 2ππ π£ ππ£ ππ΅ = 2ππ ππ΅ Motion of particle in a π΅-field β’ If the charge also has a velocity component π£β₯ parallel to π΅, the motion will be a helix β’ πβ₯ drives a circular orbit ππ£β₯ with radius and time period π‘ = ππ΅ 2ππ ππ΅ β’ πβ₯ drives a translational motion with helix 2πππ£β₯ separation π£β₯ π‘ = ππ΅ Motion of particle in a π΅-field β’ A charge moving in a π΅-field feels a force but never increases its energy β’ This is because the force is always perpendicular to the motion and so never does any work β’ Mathematically, the force does work at a rate πΉ. π£ = π π£ × π΅ . π£ = 0 because π£ × π΅ is always perpendicular to π£ Applications β’ How does a mass spectrometer work? How does the motion in the magnetic field allow us to determine the mass of the ions? How does the velocity selector produce ions of a known velocity? Applications β’ What produces the aurorae and why do they occur at the poles? Applications β’ How is magnetic confinement used to drive a plasma to high temperatures for nuclear fusion? Summary β’ Magnetic forces : A charge π moving with velocity π£ in a magnetic field π΅ feels force πΉπ΅ = ππ£ × π΅ ; a current element πΌ ππ feels force πΉπ΅ = πΌ ππ × π΅ β’ A charged particle moving in a π΅-field will trace out a helical path β’ For general motion in crossed πΈand π΅-fields, πΉ = ππΈ + ππ£ × π΅