In a simple model, the electron in a hydrogen atom orbits...

In a simple model, the electron in a hydrogen atom orbits the proton at a radius of $53 \mathrm{pm}$ and at a constant speed of $2.2 \times 10^{6} \mathrm{~m} / \mathrm{s}$. The orbital motion of the electron gives it an orbital magnetic dipole moment. (a) What is the current $I$ in this current loop? [Hint: How long does it take the electron to make one revolution?] (b) What is the orbital dipole moment $I A ?$ (c) Compare the orbital dipole moment to the intrinsic magnetic dipole moment of the electron $\left(9.3 \times 10^{-24} \mathrm{~A} \cdot \mathrm{m}^{2}\right)$.

Key Concepts

Current Due to Orbital Motion

When a charge executes periodic motion, such as moving in a circle, it generates an effective current which can be calculated by dividing the charge by the period of the motion. This concept is crucial for understanding how moving charges can create magnetic fields, as embodied in the current loop model.

Period of Revolution

The period is the time required for a complete orbit or cycle. In circular motion, this is calculated by dividing the circumference of the path by the speed of the moving particle. This period is used to determine the effective current created by the charged particle in motion.

Area of a Circular Loop

In calculating the magnetic dipole moment of a current loop, the area of the loop is required. For circular motion, the area is given by ? times the square of the radius. This geometric factor is a key component of determining the overall magnetic effect of the orbiting charge.

Orbital Magnetic Dipole Moment

The orbital magnetic dipole moment is derived from the product of the effective current and the area of the loop traversed by the charge. It represents the magnetic effect due to the orbital motion of the charge and is a fundamental concept in classical electromagnetism and atomic physics.

Intrinsic Magnetic Dipole Moment

Apart from the orbital contribution, particles like electrons also have an intrinsic magnetic dipole moment, which is independent of their orbital motion and arises from quantum mechanical properties such as spin. This intrinsic moment is an important concept when comparing different sources of magnetic moments in particles.

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