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Cham
Cham
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I'll take a simple case as an example. You have a constant and uniform magnetic field inside an ideal infinitely long solenoid, with currents circulating all around the thickless coils (so there's a surface current with a field discontinuity there). \begin{equation}\tag{1} \vec{B}_{\text{inside}} = \mu_0 \, n \, I \: \vec{z}, \end{equation} where $\vec{z}$ is the unit vector oriented along the solenoid's main axis. The field is 0 outside the solenoid (ideal case) : $\vec{B}_{\text{outside}} = 0$.

The magnetic field exerts a Lorentz force density $\vec{f} = \vec{J}_{\text{sol}} \times \vec{B}$ on the currents that are creating that same field, so there is magnetic pressure acting on the solenoid. That pressure should be proportional to the field's energy density.

Now, how do you evaluate the magnetic field, at the solenoid's surface, that should act on the current density $\vec{J}_{\text{sol}}$ ? $\vec{B}_{\text{inside}}$ given above ? $\vec{B}_{\text{outside}} = 0$ ? The average defined as this ?: \begin{equation}\tag{2} \vec{B}_{\text{average}} = \frac{\vec{B}_{\text{inside}} + \vec{B}_{\text{outside}}}{2} = \frac{1}{2} \, \mu_0 \, n \, I \: \vec{z}. \end{equation}\begin{equation}\tag{2} \vec{B}_{\text{average}} = \frac{\vec{B}_{\text{inside}} + \vec{B}_{\text{outside}}}{2} = \frac{1}{2} \, \mu_0 \, n \, I \: \vec{z} \quad ? \end{equation} or what else ? If it's the average (2), how can you justify it ?

I'll take a simple case. You have a constant and uniform magnetic field inside an ideal infinitely long solenoid, with currents circulating all around the thickless coils (so there's a surface current with a field discontinuity there). \begin{equation}\tag{1} \vec{B}_{\text{inside}} = \mu_0 \, n \, I \: \vec{z}, \end{equation} where $\vec{z}$ is the unit vector oriented along the solenoid's main axis. The field is 0 outside the solenoid (ideal case) : $\vec{B}_{\text{outside}} = 0$.

The magnetic field exerts a Lorentz force density $\vec{f} = \vec{J}_{\text{sol}} \times \vec{B}$ on the currents that are creating that same field, so there is magnetic pressure acting on the solenoid. That pressure should be proportional to the field's energy density.

Now, how do you evaluate the field, at the solenoid's surface, that should act on the current density $\vec{J}_{\text{sol}}$ ? $\vec{B}_{\text{inside}}$ given above ? $\vec{B}_{\text{outside}} = 0$ ? The average ? \begin{equation}\tag{2} \vec{B}_{\text{average}} = \frac{\vec{B}_{\text{inside}} + \vec{B}_{\text{outside}}}{2} = \frac{1}{2} \, \mu_0 \, n \, I \: \vec{z}. \end{equation} or what else ? If it's the average (2), how can you justify it ?

I'll take a simple case as an example. You have a constant and uniform magnetic field inside an ideal infinitely long solenoid, with currents circulating all around the thickless coils (so there's a surface current with a field discontinuity there). \begin{equation}\tag{1} \vec{B}_{\text{inside}} = \mu_0 \, n \, I \: \vec{z}, \end{equation} where $\vec{z}$ is the unit vector oriented along the solenoid's main axis. The field is 0 outside the solenoid (ideal case) : $\vec{B}_{\text{outside}} = 0$.

The magnetic field exerts a Lorentz force density $\vec{f} = \vec{J}_{\text{sol}} \times \vec{B}$ on the currents that are creating that same field, so there is magnetic pressure acting on the solenoid. That pressure should be proportional to the field's energy density.

Now, how do you evaluate the magnetic field, at the solenoid's surface, that should act on the current density $\vec{J}_{\text{sol}}$ ? $\vec{B}_{\text{inside}}$ given above ? $\vec{B}_{\text{outside}} = 0$ ? The average defined as this : \begin{equation}\tag{2} \vec{B}_{\text{average}} = \frac{\vec{B}_{\text{inside}} + \vec{B}_{\text{outside}}}{2} = \frac{1}{2} \, \mu_0 \, n \, I \: \vec{z} \quad ? \end{equation} or what else ? If it's the average (2), how can you justify it ?

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Cham
Cham
  • 7.7k
  • 4
  • 28
  • 69

How to evaluate the Lorentz force at a surface where the field is discontinuous?

I'll take a simple case. You have a constant and uniform magnetic field inside an ideal infinitely long solenoid, with currents circulating all around the thickless coils (so there's a surface current with a field discontinuity there). \begin{equation}\tag{1} \vec{B}_{\text{inside}} = \mu_0 \, n \, I \: \vec{z}, \end{equation} where $\vec{z}$ is the unit vector oriented along the solenoid's main axis. The field is 0 outside the solenoid (ideal case) : $\vec{B}_{\text{outside}} = 0$.

The magnetic field exerts a Lorentz force density $\vec{f} = \vec{J}_{\text{sol}} \times \vec{B}$ on the currents that are creating that same field, so there is magnetic pressure acting on the solenoid. That pressure should be proportional to the field's energy density.

Now, how do you evaluate the field, at the solenoid's surface, that should act on the current density $\vec{J}_{\text{sol}}$ ? $\vec{B}_{\text{inside}}$ given above ? $\vec{B}_{\text{outside}} = 0$ ? The average ? \begin{equation}\tag{2} \vec{B}_{\text{average}} = \frac{\vec{B}_{\text{inside}} + \vec{B}_{\text{outside}}}{2} = \frac{1}{2} \, \mu_0 \, n \, I \: \vec{z}. \end{equation} or what else ? If it's the average (2), how can you justify it ?