Maxwell Stress Tensor: Magnetic Pressure and Hoop Stress

1. From Lorentz Force to the Maxwell Stress Tensor

The fundamental origin of the force on a current-carrying material in a magnetic field is the Lorentz force. The force density (force per unit volume) within a material is given by:

where ${\rho }_f$ and ${\vec{J}}_f$ are the free charge and current densities, respectively. In the quasi-static conditions of magnet operation, the magnetic term ${\vec{J}}_f\times \vec{B}$ is overwhelmingly dominant. Our goal is to reformulate this force density entirely in terms of field quantities, which are often easier to work with than the source currents themselves.

This reformulation leads to the Maxwell stress tensor. To understand its derivation and physical meaning, please study the following material from an MIT OpenCourseWare text on electromechanical dynamics.

Derivation Summary

As you've seen in the reading, the derivation hinges on using Maxwell's equations to replace the source term ${\vec{J}}_f$. In a region with no time-varying electric fields and constant permeability $\mu$, Ampere's Law is $\nabla \times \vec{H}={\vec{J}}_f$, and $\vec{B}=\mu \vec{H}$. The force density becomes:

Using a vector identity and the fact that $\nabla \cdot \vec{H}=0$ in this context, this expression can be rewritten as the divergence of a tensor. In index notation, the $m$-th component of the force density is:

where $T_{mn}$ is the Maxwell stress tensor. For a linear magnetic material, its components are:

Here, ${\delta }_{mn}$ is the Kronecker delta. This elegant result states that the force density at a point is determined by the spatial gradients of the stress tensor components at that point.

2. The Physical Meaning of Stress and Traction

What is this tensor? The Maxwell stress tensor describes the state of stress within the electromagnetic field itself. It allows us to calculate the force on an object without needing to know the detailed current distribution inside it.

By integrating the force density $\vec{F}=\nabla \cdot \mathbf{T}$ over a volume $V$ and applying the divergence theorem, we find the total force on the material within that volume:

where $S$ is the surface enclosing the volume $V$ and $\hat{n}$ is the outward normal vector. This is a powerful result. It means we can find the total force on an object by integrating a function of the fields on any convenient surface that encloses it.

The vector quantity $\vec{\tau }=\mathbf{T}\cdot \hat{n}$ is called the traction, which is the vector force per unit area acting on the surface element $da$.

The components of the tensor have direct physical interpretations:

• Diagonal components ($T_{mm}$): These represent normal stresses. For example, $T_{xx}$ is the force per unit area in the $x$-direction on a surface whose normal is also in the $x$-direction. A positive value corresponds to tension, while a negative value signifies pressure.
• Off-diagonal components ($T_{mn},m\ne n$): These represent shear stresses. For example, $T_{xy}$ is the force per unit area in the $x$-direction on a surface whose normal is in the $y$-direction.

To solidify this interpretation, let's return to the Electromechanical Dynamics text.

3. Magnetic Pressure: The Field Pushing Outward

Let's apply these ideas to the core of a high-field solenoid. Imagine a long solenoid generating a strong, uniform axial magnetic field $\vec{B}=B_0\hat{z}$ within its bore and a negligible field outside. What is the force exerted on the windings?

We could try to use the stress tensor directly, but an equivalent and highly intuitive method is to use a "virtual work" argument, which connects the force to the energy stored in the field.

The derivation shows that by considering the energy balance when a solenoid of radius $a$ expands by an infinitesimal amount $da$, the outward pressure exerted by the magnetic field on its container is exactly equal to the magnetic energy density in the bore:

This is one of the most fundamental results in high-field magnet design. The pressure scales with the square of the magnetic field. This non-linear scaling is why pushing from 20 T to 40 T doesn't just double the challenge; it quadruples the pressure the structure must contain.

Let's put some numbers to this. The constant $\frac{1}{2{\mu }_0}\approx 397,887{\text{J/m}}^3\cdot {\text{T}}^2$.

• For a 20 T field: $P_{mag}\approx 3.98\times 10^5\times (20{)}^2=1.59\times 10^8\text{ Pa}=159\mathbf{MPa}$
  (For comparison, atmospheric pressure is ~0.1 MPa. This is nearly 1600 atmospheres.)
• For a 40 T field: $P_{mag}\approx 3.98\times 10^5\times (40{)}^2=6.36\times 10^8\text{ Pa}=636\mathbf{MPa}$

These are enormous pressures, comparable to the yield strength of many engineering materials. This calculation single-handedly explains why constructing high-field magnets is primarily a problem of mechanical engineering and material science.

4. Hoop Stress: Containing the Pressure

Now that we know the outward magnetic pressure, we can calculate the resulting mechanical stress within the conductor winding itself. Imagine the cylindrical coil as a pressure vessel. The internal magnetic pressure tries to expand the cylinder, and this expansion is resisted by tension in the material along its circumference. This circumferential tension is called hoop stress.

For a thin-walled cylinder of inner radius $a$ and wall thickness $t$, we can find the hoop stress, ${\sigma }_{\theta }$, by considering a force balance on one half of the cylinder (per unit length).

The total outward force on the half-cylinder (per unit length) is the pressure multiplied by the projected area ($2a\times 1$):
$F_{\text{out}}=P_{mag}\cdot 2a$
This force must be balanced by the tensile stress in the two sections of the cylinder wall that we "cut" to make the half-cylinder. The area of each cut section is $t\times 1$.
$F_{\text{in}}={\sigma }_{\theta }\cdot (t\times 1)+{\sigma }_{\theta }\cdot (t\times 1)=2{\sigma }_{\theta }t$
Equating the forces, $P_{mag}\cdot 2a=2{\sigma }_{\theta }t$, gives the classic formula for hoop stress in a thin-walled cylinder:

${\sigma }_{\theta }=\frac{P_{mag}\cdot a}{t}=\frac{B_0^{2}a}{2{\mu }_{0}t}$

This formula is a first-order approximation but is incredibly useful for initial design estimates. It tells you that for a given field and bore radius, the stress is inversely proportional to the thickness of your winding. To reduce the stress, you need to make the magnet walls thicker.

Conclusion

Key Takeaways:

• The Maxwell stress tensor, $T_{mn}$, provides a way to describe electromagnetic forces in terms of field quantities alone, offering a powerful alternative to integrating the Lorentz force density.
• The total force on an object can be found by integrating the traction, $\vec{\tau }=\mathbf{T}\cdot \hat{n}$, over any surface enclosing the object. Diagonal tensor components represent normal stresses (tension/pressure), while off-diagonal components represent shear stresses.
• A magnetic field exerts an outward magnetic pressure on its confining structure equal to its energy density: $P_{mag}=B^2/(2{\mu }_0)$. This pressure scales quadratically with the field strength.
• This pressure creates a circumferential hoop stress within a cylindrical coil. For a thin-walled cylinder, this is approximated by ${\sigma }_{\theta }=(P_{mag}\cdot a)/t$.

Next Lesson Preview:
We have just calculated the stress generated by the magnetic field. The next crucial question is: can the material withstand it? In our next lesson, we will formally dive into the mechanical properties of materials. We will define stress, strain, and elastic modulus, and apply Hooke's law to understand how materials deform under the immense loads we've just calculated. This will lay the groundwork for selecting the high-strength materials needed to build robust pulsed magnets.