2 Magnetic force on a wire LNote: If wire is not straight,compute force on differential elements and integrate:
3 Example Wire with current i. Magnetic field out of page. What is net force on wire?By symmetry, F2 will onlyhave a vertical component,Notice that the force is the same as that for a straight wire,and this would be true nomatter what the shape ofthe central segment!.LR
4 Torque on a current loop: Principle behind electric motors.Rectangular coil: a x b, current = iNet force on current loop = 0But: Net torque is NOT zero!For a coil with N turns,= N I A B sinq,where A is the area of coil
5 Magnetic Dipole Moment We just showed: t = NiABsinqRight hand rule:curl fingers in direction of current;thumb points along mDefine: magnetic dipole moment mN=number of turns in coilA=area of coil.As in the case of electric dipoles, magnetic dipoles tend to align with the magnetic field (where potential energy is a minimum).
7 Magnetic forces: summary Magnetic force on a moving electric charge: F = q v x BMagnetic Force on a wire: F = i L x BMagnetic Torque on a magnetic dipole:t = m x B (potential energy: U= - m· B )
8 ExampleA magnetic dipole with a dipole moment of magnitude 0.02 J/T is released from rest in a uniform magnetic field of magnitude 50 mT. The rotation of the dipole due to the magnetic force on it is unimpeded. When the dipole rotates through the orientation where its dipole moment is aligned with the magnetic field, its kinetic energy is 0.70 mJ.(a) What is the initial angle between the dipole moment and the magnetic field? (b) What is the angle when the dipole is next (momentarily) at rest?Notice that torque is not constant, so we cannot use kinematics! (we also don’t know the rotational inertia). Energy is conserved, so: K0+U0=Kf+Uf 0-mu*Bcos(t0)=Kf-mu*Bcos(0) cos(t0)=1-Kf/mu*EB = mJ/(0.02J/T *50mT)=0.3, so t0=72.5 deg or deg.If the initial angle is 72.5deg, it will momentarily stop again at -72.5deg, and oscillate back and forth.
9 ExampleA wire of 62.0 cm length and 13 g mass is suspended by a pair of flexible leads in a magnetic field of T. What are the magnitude and direction of the current required to remove the tension in the supporting leads?Draw a free body diagram. If there is no tension and wire is in equilibrium, the the weight is balanced by an upwards magnetic force.iLB=mg, so i=mg/LB=0.013kg*9.8m/s^2/(0.62m*0.44T)=0.47 A
10 Magnetic fields Electric fields are created: microscopically, by electric charges (fields) of elementary particles (electrons, protons)macroscopically, by adding the field of many elementary charges of the same signMagnetic fields are created :microscopically, by magnetic “moments” of elementary particles (electrons, protons, neutrons)macroscopically, byadding many microscopic magnetic moments (magnetic materials); or byelectric charges that move (electric currents)
11 Currents producing magnetic fields: Biot-Savart Law When we computed the electric field due to charges we usedCoulomb’s law. If one had a large irregular object, one broke itinto infinitesimal pieces and computed,Which we canwrite as,Magnetic fields are produced by electrical currents. If you wish to compute the magnetic field due to a current in a wire, you use the law of Biot and Savart.
12 ?The Biot-Savart LawJean-BaptisteBiot ( )Felix Savart ( )iQuantitative rule for computing the magnetic field from any electric currentChoose a differential element of wire of length dL and carrying a current iThe field dB from this element at a point located by the vector r is given by the Biot-Savart Law:m0 =4px10-7 T.m/A(permeability constant)Compare with: