Assume that an electron is describing a circular orbit of radius a about a positively charged nucleus.

(a) By selecting an appropriate current and area, show that the equivalent orbital dipole moment is ea²ω/2, where ω is the electron’s angular velocity.

(b) Show that the torque produced by a magnetic field parallel to the plane of the orbit is ea²ωB/2.

(c) By equating the Coulomb and centrifugal forces, show that ω is (4π₀m_ea³/e²)^−1/2, where me is the electron mass.

(d) Find values for the angular velocity, torque, and the orbital magnetic moment for a hydrogen atom, where a is about 6 × 10⁻¹¹ m; let B = 0.5 T.