Animated Solution for Physics - Magnetic Effects of Current: A charged particle going around in a circle can be considered to be a current loop. A particle of mass $m$ carrying charge $q$ is moving in a plane with speed $v$ under the influence of magnetic field $\mathbf{B}$. The magnetic moment of this moving particle

Imagine a charged particle, let's say a proton, injected into a uniform magnetic field. We know it will start moving in a circular path. But let's shift our perspective for a moment. Instead of just seeing a moving particle, think about what an electric current actually is: it's simply the flow of charge over time.

Therefore, a single charge continuously going around in a circle is fundamentally no different from a steady current flowing in a circular wire loop! This beautiful equivalence allows us to assign a magnetic dipole moment to the orbiting particle.

To find the magnetic moment, we first need to determine the equivalent current $i$. Current is defined as the total charge passing a point per unit time. For our orbiting particle, the charge $q$ passes any given point on the circle once every time period $T$.

The time period $T$ is the total distance of one orbit (the circumference, $2\pi R$) divided by the particle's constant speed $v$:

Substituting this back into our current equation gives:

The magnetic moment $M$ of a current loop is the product of the current and the area it encloses ($A=\pi R^2$). Let's multiply them together:

Simplifying this, we get a neat expression for the magnitude:

However, the problem doesn't give us the radius $R$. We need to express $R$ in terms of the given variables ($m,v,q,B$). We know that the magnetic Lorentz force provides the necessary centripetal force to keep the particle in its circular orbit:

Solving for $R$, we find the cyclotron radius:

Now, substitute this expression for $R$ back into our magnetic moment equation:

We have the magnitude, but magnetic moment is a vector quantity. This is where many students make a silly mistake. We must determine the direction of $\mathbf{M}$ relative to the magnetic field $\mathbf{B}$.

Let's assume the magnetic field $\mathbf{B}$ points into the page ($\otimes$). For the magnetic force to point towards the center (providing centripetal force), a positive charge must move counter-clockwise.

A counter-clockwise flow of positive charge means the equivalent current is also counter-clockwise. Using the Right-Hand Grip Rule for current loops, if you curl your fingers counter-clockwise, your thumb points out of the page ($\odot$).

Notice the profound result: The magnetic moment $\mathbf{M}$ points outwards, while the external magnetic field $\mathbf{B}$ points inwards. They are strictly anti-parallel.

To express this mathematically, we write $\mathbf{M}$ in the direction of $-\mathbf{B}$. We use the unit vector $\hat{\mathbf{B}}=\frac{\mathbf{B}}{B}$:

Multiplying the terms yields our final, elegant vector expression: