Physics Tutorial MF1 Magnetic Forces

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Russell Skinner
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1 Physics Tutorial MF1 Magnetic Forces 1 Magnetic Forces The force F on a charge q moving with velocity v in a magnetic field is: F = qv The force F on a straight conductor of length L carrying a current i in a uniform magnetic field is: F = il The torque on a current loop in a uniform magnetic field: τ = i A 1.1 A charge q = 27 mc is travelling at a speed of v = 10 m/s, and enters a magnetic field of magnitude = 1.3 T. (a) What are the magnitudes of the maximum and minimum possible forces on the charge? (b) If the charge is moving in the x-y plane at 30 to the x-axis, and the magnetic field is in the direction of the y-axis, calculate the magnitude of the magnetic force. (c) Repeat the previous calculation, this time with the magnetic field in the direction of the z-axis. 1.2 Calculate the force acting on a proton (q proton = C) travelling with velocity v = 5i + 3j 4k m/s in a magnetic field = 4k T. 1.3 Calculate the force acting on an electron with velocity v = 2i 4j m/s in a magnetic field = 3i + 2j T. 1

2 1.4 A proton (m = kg) moving perpendicular to a magnetic field of 5.0 mt experiences a force of magnitude N. (a) What is the speed of the proton? (b) What potential difference is required to accelerate the proton to this speed from rest? (c) What is the radius of the proton s circular path? 1.5 A singly charged positive ion with mass m = kg is accelerated through a potential difference of 833 V before entering a magnetic field with magnitude = 0.92 T. Calculate the radius of the circular path of the ion in the magnetic field. 1.6 A wire carries a current of 1.8 A. A straight section of this wire of length 0.57 m is positioned so that the current flows in the +x direction within a magnetic field = 0.35j T. What is the force on this wire? 1.7 What would be the force on the wire in the previous question if it was moved to be in the y direction, in the same magnetic field? 1.8 A conductor of linear mass density µ = 0.50 g/cm is suspended by two thin conducting wires within a magnetic field as shown: L (a) When a current of 2.0 A is run through the conductor, the tensions in the supporting wires are measured to be zero. What must be the direction of this current? What is the magnitude of the magnetic field? (b) When the current is reduced to 1.0 A, then the tension in one wire is measured to be 0.25 N. What is the length L of the conductor? (c) If the breaking tension in each of the supporting wires is 5.0 N, what is the magnitude and direction of the smallest current that will cause the wires to break? 2

3 1.9 A rectangular wire loop rests on a table-top, with a uniform magnetic field directed parallel to the table-top as shown below, looking down from above: I h d (a) If the linear mass density of the wire is µ kg/m, determine an expression for the minimum current I required for the loop to move. (b) What current would be required if the magnetic field s direction was changed to be vertically out of the table-top? 1.10 A proton has a kinetic energy of 20 MeV, and is travelling in the x-y plane at 45 to the x-axis when it enters a region of uniform magnetic field = 1.5k T as shown: 45 θ d Calculate: (a) the angle θ of its exit (b) the distance d between the entry and exit points of the proton into the region of field. (c) The change in kinetic energy of the proton A current i = 2 A flows in a loop within a uniform magnetic field = 0.6 T in the x direction, as shown below: x y z i 1 m 2 m 2 m Calculate: (a) The torque vector τ acting on the loop. (b) The flux φ of magnetic field throught the loop. 3

4 2 Mass Spectrometer Velocity selector: v selected = E Analyzer: r = ( ) m v q 2.1 Three particles with equal charge and speed enter a uniform magnetic field, where they follow semi-circular paths as shown below: 5 mm 7 mm A (a) What sign is the charge of these particles? (b) Which of the particles has the greatest mass? 7 mm C (c) What is the ratio of masses: m C m? 2.2 An ion source produces doubly ionized gold ions (Au ++ ), each with mass kg. The ions pass through a velocity selector with E = V/m and = T. Then a T magnetic field causes the ions to follow a circular path. What is the radius of this path? 2.3 A velocity selector with an electric field of V/m and magnetic field T is used to select the speed of an ion of charge +e in a mass spectrometer. A T magnetic field in the analyzer bends the ion with a radius of m. (a) What is the mass of the ion? (b) What is the molar mass of the ion? (Avogadro s number = ) (c) What element is the ion? 4

5 3 Cyclotron Cyclotron frequency: Maximum kinetic energy: f = 1 q 2π m KE max = R2 q 2 2 2m 3.1 A small cyclotron has a maximum radius of R = 0.25 m and a magnetic field strength of = 1.7 T. If protons are being accelerated, find: (a) the frequency f needed for the applied voltage (b) the magnitude of the applied voltage (c) the kinetic energy of the protons emitted (in ev) (d) the velocity of these protons 3.2 Recalculate the previous questions if the protons are replaced by π + mesons. (the π + is an unstable particle which can be produced in the lab - it has the same charge as a proton, but a mass of kg) 3.3 A beam of protons moves in a circular path with radius 0.25 m. The beam is perpendicular to a 0.30 T magnetic field. Calculate: (a) the speed of each proton? (b) the cyclotron frequency of these protons. (c) the centripetal force acting on each proton. 5

6 Answers 1.1 (a) 0.35 N, zero (b) 0.30 N (c) 0.35 N 1.2 (1.9i 3.2j) N k N 1.4 (a) m/s (b) 1.1 kv (c) 96 cm cm k N 1.7 zero 1.8 (a) to right; 0.25 T (b) 2.0 m (c) 18 A to left 1.9 (a) I = µg ( 1 + d h) (b) no motion for any value of I since ΣF = 0 and τ = (a) 45 (b) 61 cm (c) zero (magnetic fields never do work) 1.11 (a) 4.8k Nm (b) 1.2 Tm (a) - (b) C (c) cm 2.3 (a) kg (b) 197 g (c) probably gold (see previous question) 3.1 (a) 26 MHz (b) any value will work! (c) 8.75 MeV (d) m/s 3.2 (a) 175 MHz (c) 58 MeV (d) m/s 0.7c (relativity required!) 3.3 (a) m/s (b) 4.6 MHz (c) N 6

Force Due to Magnetic Field You will use

Force Due to Magnetic Field You will use Units: 1 N = 1C(m/s) (T) A magnetic field of one tesla is very powerful magnetic field. Sometimes it may be convenient to use the gauss, which is equal to 1/10,000