Class 35: Magnetic moments and intrinsic spin A small current loop produces a dipole magnetic field. The dipole moment, m, of the current loop is a vector with direction perpendicular to the plane of the loop and magnitude equal to the product of the area of the loop, A, and the current, I, i.e. m = AI . A magnetic dipole placed in a magnetic field, B, τ experiences a torque =m × B , which tends to align an initially stationary N dipole with the field, as shown in the figure on the right, where the dipole is represented as a bar magnetic with North and South poles. The work done by the torque in turning through a small angle dθ (> 0) is θ dW==−τθ dmBsin θθ d = mB d ( cos θ ) . (35.1) S Because the work done is equal to the change in kinetic energy, by applying conservation of mechanical energy, we see that the potential energy of a dipole in a magnetic field is V =−mBcosθ =−⋅ mB , (35.2) where the zero point is taken to be when the direction of the dipole is orthogonal to the magnetic field. The minimum potential occurs when the dipole is aligned with the magnetic field. An electron of mass me moving in a circle of radius r with speed v is equivalent to current loop where the current is the electron charge divided by the period of the motion. The magnetic moment has magnitude π r2 e1 1 e m = =rve = L , (35.3) 2π r v 2 2 m e where L is the magnitude of the angular momentum. Taking into account the sign of the electron charge, this can be written in vector form as e m= − L . (35.4) 2me In the hydrogen atom, states with l ≠ 0 have magnetic moments. The z-component of the magnetic moment is e e ℏ mz=− Lm z =− =− m µ B , (35.5) 2me 2 m e where the quantity eℏ µ B = , (35.6) 2me is called the Bohr magneton . It has the numerical value 9.27× 10−24 J T − 1 . Here T is the symbol for the SI unit of magnetic flux density, the tesla. Force on a magnetic dipole in a non-uniform magnetic field The potential energy of a magnetic dipole in a magnetic field is given by equation (35.2). We see that if the magnetic field is not uniform, the potential energy will vary with position and hence there will be a force F acting on the dipole where F= ∇( m ⋅ B ). (35.7) Since the force depends on the magnetic moment m, a non-uniform magnetic field will split a beam of atoms into components of different m. Since for a given l, there are 2 l + 1 allowed values of m, a beam of hydrogen atoms should be split into an odd number of components. The Stern-Gerlach experiment In a 1922 experiment, Stern & Gerlach found that a non-uniform magnetic field split a beam of atoms of silver into two components. In the ground state silver atoms have l = 0, and so no splitting is expected. The observed splitting was later explained by the electron having intrinsic spin. Since the beam was split into two components, the electron must have two spin states. To differentiate spin from orbital angular momentum, s is used to denote the spin angular momentum quantum number of a particle. The electron has spin s = 1/2. The magnetic quantum number for the electron has the allowed values, ms = ±1/2. The electron magnetic moment Since the electron is charged and has intrinsic spin, it has a magnetic dipole moment. From equation (35.5), we might expect that the electron magnetic moment has magnitude equal to µB 2. In fact, it has a magnitude which is about twice this value. This difference is accounted for by introducing a g factor , such that the electron magnetic momentum is S m = g µ , (35.8) s e B ℏ where S is the spin angular momentum of the electron and the electron g factor has the value ge = − 2.002319. .