Self-Force of a Dirac String: An Explicit Calculation

Motivated by a recent comment, this paper provides an explicit and elementary derivation showing that a Dirac string, modeled as a semi-infinite solenoid, experiences a divergent self-force proportional to the square of its magnetic flux divided by the square of its radius as the radius approaches zero.

The Gist

Imagine you have a very long, magical garden hose that never ends. Inside this hose, water (representing magnetic force) is swirling around in a tight circle, creating a powerful flow. In the world of physics, this is like a solenoid—a coil of wire carrying electricity.

Now, imagine you cut this hose in half and throw away the bottom half, leaving only the top part that stretches up to infinity. This is what physicists call a semi-infinite solenoid. In a famous theory by Paul Dirac, this setup is used to model a "magnetic monopole" (a magnet with only a North pole and no South pole). The invisible "hose" leading to the monopole is called the Dirac string.

The Big Question: Does the Hose Squeeze Itself?

Here is the puzzle the paper solves:
If you have a normal, finite garden hose (one with two ends), the water pressure pushing out on the left end is perfectly balanced by the pressure pushing out on the right end. The hose doesn't move; the forces cancel out.

But what happens if you have a hose with only one end (the top) and the other end is missing (cut off at the bottom)?

• The top end has no "opposing" force to balance it.
• The paper calculates that this missing end creates a weird, invisible "self-force." The hose is essentially trying to push itself upward, away from the cut-off point.

The "Edge" Analogy

Think of a crowd of people holding hands in a circle, all pushing outward.

• Finite Solenoid (Normal Hose): If the circle is complete, everyone pushes against their neighbor. The net force is zero. Everyone stays put.
• Semi-Infinite Solenoid (Dirac String): Imagine the circle is cut open at the bottom. The people at the very bottom of the cut have no one to push against. They feel a sudden, unbalanced push. Because the hose is infinitely long, this "unbalanced push" happens all along the length, creating a massive, cumulative force trying to blow the hose apart.

The "Pinch" Problem (The Divergence)

The most surprising part of the paper is what happens when you try to make this hose thinner and thinner.

In the Dirac model, physicists want to shrink the hose down to a single, infinitely thin line (a mathematical string).

• The Math: The paper shows that the force pushing the hose apart depends on how thin the hose is. Specifically, the force gets stronger as the square of the thickness decreases ($1/a^2$).
• The Result: As the hose gets thinner and thinner (approaching zero width), the force trying to blow it apart becomes infinite.

The Everyday Metaphor:
Imagine you have a balloon filled with air.

1. If the balloon is wide and fat, the air pressure pushes out gently.
2. If you squeeze the balloon into a tiny, needle-thin tube, the pressure inside becomes explosive.
3. If you try to squeeze it into a line with zero width, the pressure becomes infinite. It's physically impossible to hold that much force in a space that doesn't exist.

What Did the Author Actually Do?

Previous explanations of this problem were very complicated, using advanced math tricks to show that the force exists. The author, Alberto Rojo, decided to do it the "old-school" way:

1. He treated the hose as a stack of many tiny rings (like a stack of coins).
2. He calculated how each ring pushes on the others.
3. He showed that because the hose is cut off at the bottom, the very last ring (at the cut) creates a magnetic field that pushes on all the rings above it.
4. By adding up all these tiny pushes, he derived a simple formula showing that the force is huge and blows up to infinity as the hose gets thinner.

The Takeaway

The paper confirms a worry that the famous physicist Paul Dirac had decades ago: You cannot have a perfect, infinitely thin magnetic string without it experiencing an infinite, self-destructive force.

The "self-force" isn't a mystery; it's simply the result of removing one end of a balanced system. When you take away the "counter-weight" (the other end of the solenoid), the remaining structure is under immense, unbalanced stress. If you try to make that structure infinitely thin, the stress becomes infinite, proving that the "Dirac string" is a useful mathematical idea, but a physically impossible object to build in the real world.

Technical Summary

1. Problem Statement

The paper addresses a subtle limitation in the Dirac monopole model regarding the self-force experienced by the associated "Dirac string."

• Context: While the force between two distinct Dirac strings has been analyzed, the self-force of a single string remains a point of theoretical contention.
• The Paradox: A Dirac string is modeled as a semi-infinite solenoid. Intuitively, one might expect the self-force to vanish, similar to an ordinary finite solenoid where internal magnetic stresses cancel out. However, previous work (McDonald) and Dirac himself noted that the string experiences a non-vanishing, divergent self-force.
• Goal: The author aims to provide an elementary, explicit derivation of this self-force to clarify the physical origin of the divergence, moving beyond the sophisticated arguments used in previous literature.

2. Methodology

The author models the Dirac string as a semi-infinite solenoid with the following parameters:

• Geometry: Radius $a$, extending from $z=0$ to $z=\infty$.
• Current: Carries a current $I$ with $n$ turns per unit length.
• Magnetic Flux: $\Phi = \mu_0 n I \pi a^2$.
• Dirac Limit: The physical monopole is recovered by taking the limit $a \to 0$ and $I \to \infty$ while keeping the magnetic charge $g = \Phi/\mu_0$ constant.

Derivation Steps:

1. Force Mechanism: The axial self-force ($F_z$) arises from the interaction between the current in each loop of the solenoid and the radial magnetic field ($B_\rho$) generated by all other loops.
2. Vector Potential Approach: Instead of calculating the full field, the author utilizes the azimuthal vector potential $A_\phi$. The radial field at the solenoid surface is derived via the relation:
   $$B_\rho(a, z) = -\frac{\partial A_\phi}{\partial z}$$
3. Semi-Infinite Simplification: A key insight is that for a semi-infinite solenoid, the difference in vector potential between $z$ and $z+dz$ is solely due to the contribution of the current loop at the "bottom" end ($z=0$). This simplifies the calculation of $B_\rho$ to the vector potential generated by a single loop at the origin.
4. Integration:
   • The radial field $B_\rho(z)$ is expressed as an integral over the angle $\phi'$.
   • The total force is calculated by integrating the force on each differential loop ($dF = -2\pi a I B_\rho dz$) from $z=0$ to $\infty$.
   • This results in a double integral involving the geometry of the loops.

3. Key Results

The calculation yields a compact, exact expression for the total self-force along the axis:

$$F_z = \frac{\Phi^2}{2\pi \mu_0 a^2}$$

Analysis of the Result:

• Divergence: In the Dirac string limit where the radius $a \to 0$ (while keeping flux $\Phi$ fixed), the self-force diverges as $1/a^2$.
• Confirmation: This explicitly confirms the observation by McDonald and Dirac that the self-force is not only non-zero but infinite for an idealized string of zero thickness.
• Physical Interpretation: The divergence represents the "unavoidable price" of concentrating a finite magnetic flux into a vanishing cross-sectional area.

4. Significance and Physical Interpretation

The paper provides crucial physical intuition for why the self-force exists and diverges:

• Edge Effect vs. Cancellation: A finite solenoid experiences no net self-force because the magnetic stresses at the two ends cancel each other out. However, a semi-infinite solenoid (the Dirac string model) lacks one end. Consequently, the compensating stress is missing, leaving a net "edge force" at the open end ($z=0$).
• Idealization: The non-zero result highlights that the semi-infinite solenoid is a mathematical idealization rather than a physically realizable object. The divergence arises because the model removes the necessary boundary condition (the second end) required for momentum balance.
• Pedagogical Value: The derivation is presented as an "elementary" alternative to previous sophisticated methods, making the mechanism of the self-force accessible to students and researchers without requiring advanced tensor calculus (though the Maxwell stress tensor is noted as an alternative method in the supplementary material).

Conclusion

Rojo's paper successfully demystifies the self-force of a Dirac string. By modeling it as a semi-infinite solenoid, the author demonstrates that the divergent self-force is a direct consequence of the missing boundary at infinity. The result $F_z \propto 1/a^2$ quantifies the infinite magnetic pressure required to maintain a finite flux in a zero-radius tube, validating the theoretical concerns regarding the Dirac monopole model's singular nature.