A Circular Current Ring and Magnetic Dipoles - Notes | PHY 481, Study notes of Physics

Download A Circular Current Ring and Magnetic Dipoles - Notes | PHY 481 and more Study notes Physics in PDF only on Docsity! PHY481 - Lecture 23 Chapter 8 of PS, Chapter 5 of Griffiths A. A circular current ring and magnetic dipoles Consider a circular loop of radius r centered at the origin and lying in the x-y plane. A steady current, i, flows in the positive φ̂ direction. Find the magnetic field on the z- axis. We use the Biot-Savart where d~l = rdφφ̂ and R̂ is along the vector from a position on the circle to a point on the z-axis. If we take the angle to the z axis to be α, from the geometry we find that d~l = rdφφ̂ is perpendicular to R̂. The vector d~l ∧ R̂ makes an angle of 90− α to the z-axis and its projection onto the x-y plane is at angle φ to the x-axis. On the z-axis, by symmetry, the magnetic field points in the z-direction, and we find, Bz(z) = µ0i 4π ∫ 2π 0 sin(α)rdφ r2 + z2 = µ0i 2 r2 (r2 + z2)3/2 (1) Expanding the expression above at large distances, z, gives, Bz(z) ≈ µ0ir 2 2z3 (1− 3 2 r2 z2 ) ≈ µ0iA 2πz3 (2) The leading order behavior is thus like that of a electrostatic dipole - there is no (monopole) term like 1/r2 for a localized current loop. However if we consider a point magnetic charge, which we call N , and an opposite point magnetic charge, that we call S. Further imagine that this magnetic dipole where centered at the origin, with N and S separated by distance d. What would the magnetic field look like? First we have to decide what the field due to a magnetic monopole looks like. Though the magnetic monopole has never been found, it is assumed that if it existed, its magnetic field would be just like that of an electric charge, as we would have ∫ ~B · d ~A = µ0qm, so that, ~B = µ0 4π qmr̂ r2 (3) In that case, the magnetic field of a dipole would be, ~B(~r) = µ0 4π 3(~m · r̂)r̂ − m̂ r3 (4) Now note that on the z − axis, this reduces to, ~B(~r) = µ0 4π 2m̂ r3 z− axis. (5) 1 Comparing this expression with that for the current loop, we see that they are the same, provide we take the magnetic moment of the dipole to be, ~m = iAk̂. This is a general result for current loops and provides a general concept that connects current loops to magnetic dipoles. Notice that the origin of the difference between magnetic and electric fields is purely due to the difference in the sources of the fields (charges for the ~E field, current for the ~B field). The fields themselves are perfectly analogous. Since the magnetic and electric fields are completely analogous, we dipole formulae also apply, so that U = −~m · ~B; ~τ = ~m ∧ ~B. (6) The Lorentz Force law Lorentz derived a general expression for the force on a charge, q, moving with velocity ~v in electric and magnetic fields ~E and ~B, ~F = q( ~E + ~v ∧ ~B) (7) A charge moving in a magnetic field A charge q moving in a magnetic field ~B with velocity ~v experiences a force, ~FB = q~v ∧ ~B (8) Or equivalently, FB = qvBsin(θ) (9) where θ is the angle between the velocity vector and the magnetic field vector. The direction of ~FB is given by the right hand rule. Note the following: (i) The force on the charged particle is always perpendicular to both the velocity vector and to the magnetic field vector. (ii) If the particle moves in the direction of the magnetic field, it experiences no magnetic force. (iii) If the particle moves perpendicular to the magnetic field it experiences the maximum force. (iv) Since the force is perpendicular to the magnetic field lines and to the velocity vector, the particle “spirals” around the magnetic field lines. The larger the magnetic field the larger the magnetic force and the tighter the spiral. 2