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Transcript
Physics 227: Lecture 15 Magnetic Fields from wires • Lecture 14 review: • • • • • • Force on current/wire loops = 0, in uniform field Current loop magnetic moment μ = I A. Torque on current loops: τ = μxB. Application: DC motor. Induced magnetic moments pull ferromagnetic materials into high field regions. Hall effect: current carriers pushed to side of wire. Forces between parallel wires Thursday, October 27, 2011 Field from a Moving Charge We have already discussed, and seen in demos, how the magnetic field from a wire carrying a current is circular. A current is a set of charges, so it is not surprising that the field of a moving charge and a current are the same shape. Use the RH hand to determine the direction. Thursday, October 27, 2011 iClicker: What is the Magnetic Field Direction? P +Q v When the particle is in the position shown to the left, moving to the right, what is the direction of the magnetic field at point P? A. Down to the right. B. Up to the right. C. Out of the board. D. In to the board. E. Vertical, up or down. Thursday, October 27, 2011 See previous page, or use RH rule. What is the Magnetic Field Magnitude? µ0 q�v × r̂ � B(�r) = 4π r2 Using cylindrical coordinates, you can see the formula and picture agree: the velocity is in the z direction, the radial coordinate is radial, so the Bfield is in the azimuthal (θ) direction. μ0 = 4πx10-7 Tm/A or Ns2/C2 Thursday, October 27, 2011 Note that for a single moving particle, the field is most intense near it, and decreases for the same distance from the wire when further from the particle, as 1/r2 is smaller. What is the Force Between Moving Charges? The lower + charge, moving in the +x direction, generates a magnetic field in the +z direction at the position of the upper + charge. The magnetic force on the upper charge is FM = qvB in the +y direction. There is also an electric field from the lower charge at the position of the upper charge, leading to an electric force FE = qE in the +y direction. What is the relative magnetic of the electric and magnetic forces? Thursday, October 27, 2011 iClicker: What is the Relative Magnitude of the Electric and Magnetic Forces between two Equal Charges? A. Electric force is always greater. B. Magnetic force is always greater. C. The two forces are always equal. D. The electric force is never less. E. The magnetic force is never less. Answer follows in two slides. Thursday, October 27, 2011 iClicker: What is the Relative Magnitude of the Electric and Magnetic Forces between two Equal Charges? A. Electric force is always greater. B. Magnetic force is always greater. C. The two forces are always equal. D. The electric force is never less. E. The magnetic force is never less. A note on ``relativity’’: What happens if you are moving along to +x with the lower particle? Thursday, October 27, 2011 What is the ratio of Electric to Magnetic Force? µ q� v × r̂ 0 � r) = B(� 4π r2 2 2 µ qv µ q v 0 0 � = qv |F�M | = |q�v × B| = 4π r2 4π r2 q2 FE = 4π�0 r2 FM = µ0 �0 v 2 FE What is μ0ε0v2? From their numerical values, you can see that μ0ε0 = 1/(3x108)2. Later on we will learn that the speed of light is c2 = 1/μ0ε0, so μ0ε0 = v2/c2 < 1 always, and FM < FE for v < c. Thursday, October 27, 2011 What if the Velocities are Parallel, Rather than Antiparallel? All the equations are the same, but some directions reverse. Thursday, October 27, 2011 Change of Pace - Paramagnetism Demo Ferromagnets - large magnetization aligned to applied field. Paramagetics - small magnetization aligned to applied field, attracted into strong field region. Diamagnetics - small magnetization anti-aligned to applied field, repelled out of strong field region. Thursday, October 27, 2011 What is the Magnetic Field of a Current? µ0 q�v × r̂ � B(�r) = 4π r2 We need to go from a single charge to integrating over many charges, for now all the same magnitude and moving along the same straight section of wire. � q� v × r̂ µ Id l × r̂ µ 0 0 � r) = nAdl = dB(� 4π r2 4π r2 ``Biot-Savart’’ Law. To calculate the field from an infinitely long wire, we need to do an integral ... Thursday, October 27, 2011 The Magnetic Field of an Infinitely long Wire / Current? � q� v × r̂ µ Id l × r̂ µ 0 0 � r) = nAdl = dB(� 4π r2 4π r2 distance from wire = y θ x=-a B= � x=0 a −a µ0 Idx 4π x2 + y 2 � x=a � y x2 + y 2 Do not need to worry about components of B, B is always out of page for all x Thursday, October 27, 2011 � →a→∞ µ0 I 2πy Usually written B = μ0I/2πr sinθ of cross product The Magnetic Field of Two Infinitely long Wires / Currents? Magnetic fields add, just as electric fields add, so BTotal = B1 + B2. Usually written B = μ0I/2πr Thursday, October 27, 2011 Magnitude of Force Between Parallel Conductors Consider two wires carrying current in the same direction. We have already seen the force is attractive. The magnetic field at one wire from the other is B = μ0I/2πr. The force on a length L of the 2nd wire is F = ILxB = μ0I2L/2πr, or the force per unit length is F/L = μ0I2/2πr. Thursday, October 27, 2011 Magnetic Field of a Wire Loop - on Axis As we go around the loop, the y components of the field rotate and cancel. Only the x component survives. Note that nothing in the integral depends on θ - it is an easy integral to do! If you have a coil with N loops, multiply by N. Bx = � 0 cross product sinθ = 1: dl ⊥ r Thursday, October 27, 2011 2π � � µ0 Iadθ a µ0 Ia2 √ = 2 + a2 )3/2 2 2 4π x2 + a2 2(x x +a µ0 I Bx component Bx (x = 0) = 2a Magnetic Field of a Wire Loop - on Axis Thursday, October 27, 2011 iClicker: What is the Magnetic Field...? direction at point P? A. Into page. Straight wires lead to no field at P. Loop field is into the page. B. Out of page. C. Infinite magnitude and undefined direction, since it is along wire. D. Right, the same direction as the current in the wire. E. Up. Thursday, October 27, 2011 Thank you. See you Monday. Thursday, October 27, 2011