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Lecture 3/27/2017 • Magnetic force on a moving charge. • Motion of charged particles in magnetic fields. • Torque on a current loop. Force on an Electric Charge Moving in a Magnetic Field The force on a moving charge is related to the force on a current: Once again, the direction is given by a right-hand rule. A proton beam enters a magnetic field region as shown below. What is the direction of the magnetic field B? 1) + y 2) – y 3) + x 4) + z (out of page) 5) – z (into page) y x A electron beam enters a magnetic field region as shown below. What is the direction of the magnetic field B? 1) + y 2) – y 3) + x 4) + z (out of page) 5) – z (into page) y x Force on an Electric Charge Moving in a 𝒃 Magnetic Field 𝑾 = න 𝑭 ∙ 𝒅𝒔 𝒂 If a charged particle is moving perpendicular to a uniform magnetic field, its path will be a circle. Does the magnetic force do work on the charge? a)Yes b)No A proton moves in a circular orbit of radius 14cm in a uniform 0.35T magnetic field perpendicular to the velocity of the proton. Find the speed of the proton in Mm/s. 𝒎𝒑𝒓𝒐𝒕𝒐𝒏 = 𝟏. 𝟔𝟕 × 𝟏𝟎−𝟐𝟕 𝒌𝒈 Force on an Electric Charge Moving in a 𝒃 Magnetic Field 𝑾 = න 𝑭 ∙ 𝒅𝒔 𝒂 If a charged particle is moving perpendicular to a uniform magnetic field, its path will be a circle. Does the magnetic force do work on the charge? a)Yes b)No Force on an Electric Charge Moving in a Magnetic Field Can a magnetic field be used to stop a single charged particle, as an electric field can? A) Yes B) No C) Depending on which direction the charge is moving with respect to the magnetic field. Force on an Electric Charge Moving in a Magnetic Field What is the path of a charged particle in a uniform magnetic field if its velocity is not perpendicular to the magnetic field? Force on an Electric Charge Moving in a Magnetic Field 𝒗 𝒘𝒉𝒆𝒏 𝑭𝑬 = −𝑭𝑩 𝑬 𝒗= 𝑩 The Hall Effect When a current-carrying wire is placed in a magnetic field, there is a sideways force on the electrons in the wire. This tends to push them to one side and results in a potential difference from one side of the wire to the other; this is called the Hall effect. The emf differs in sign depending on the sign of the charge carriers; this is how it was first determined that the charge carriers in ordinary conductors are negatively charged. The Hall Effect If d is the width of the conductor the Hall voltage is: 𝑉𝐻 = 𝐸𝐻 𝑑 = 𝑣𝑑 𝐵𝑑 Recall: 𝐼 𝑣𝑑 = 𝑛𝑞𝐴 Then we can write (in terms of the current): 𝐼𝐵𝑑 𝑉𝐻 = 𝑛𝑞𝐴 A rectangular copper strip 1.5 cm wide and 0.10 cm thick carries a current of 5.0 A. Find the Hall voltage (in µV) for a 1.2 T magnetic field applied in a direction perpendicular to the strip. 𝐼𝐵𝑑 𝑉𝐻 = 𝑉𝐻 = 𝐸𝐻 𝑑 = 𝑣𝑑 𝐵𝑑 𝑛𝑞𝐴 Torque on a Current Loop; Magnetic Dipole Moment The forces on opposite sides of a current loop will be equal and opposite (if the field is uniform and the loop is symmetric), but there may be a torque. The magnitude of the torque is given by Torque on a Current Loop; Magnetic Dipole Moment The quantity NIA is called the magnetic dipole moment, μ: We can rewrite the torque as: 𝝉=𝝁×𝑩 The potential energy of the loop depends on its orientation in the field: At the Large Hadron Collider in Geneva Switzerland, protons having a momentum of 3.75 x 10-15 kg m/s (7 TeV/c) are held in a circular with a circumference of 27 km by an upward magnetic field. 𝒎𝒑𝒓𝒐𝒕𝒐𝒏 = 𝟏. 𝟔𝟕 × 𝟏𝟎−𝟐𝟕 𝒌𝒈 What is the magnitude of this field?