In Purcell and Morin’s Electricity and Magnetism, the magnetic force on a current-carrying sheet is found by using the fact that the force on a segment of wire with length $l$ would be $IB_{avg}l$, where $B_{avg}$ is the average magnetic field. I understand why this is true when $J$ is constant across the surface; the magnetic field would change linearly across the surface.
What if, instead of a constant current density, the current density changed across the thickness of the surface (for example, if the two halves of the surface were made of materials of different resistances)?
Would the magnetic force per area still be $(B_1^2-B_2^2)/(2\mu)$, derived from the facts that the force on a segment of wire is $IB_{avg}l$, and $B_{avg} = (B_1+B_2)/2$? Or would there be a different expression? I am thinking that there may be a difference, since the currents are different in each segment of the sheet, so you would have to find the forces on each half separately, then average them to get the average force per area.
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$\begingroup$ The force density of an external magnetic field, $\mathbf{B}$, acting on a volume of current density, $\mathbf{J}$, is just $\mathbf{J} \times \mathbf{B}$. $\endgroup$– honeste_vivereCommented Apr 8, 2022 at 14:34
2 Answers
$\vec{F_{B}} = \iint (\vec{K} × \vec{B}) da$
$\vec{K}$ = surface current density
Which has units $\frac{A}{m}$ ( yes current, per unit length, not area)
$\vec{K} = \sigma \vec{v}$
where $\sigma$ is surface charge density and has units $\frac{Q}{m^2}$
$\vec{v}$ = the velocity of the surface charge density
Would the magnetic force per area on the surface charge still be $(B_1^2-B_2^2)/(2\mu)$, derived from the facts that the force on a segment of wire is $IB_{avg}l$, and $B_{avg} = (B_1+B_2)/2$? Or would there be a different expression? I am thinking that there may be a difference, since the currents are different in each segment of the sheet, so you would have to find the forces on each half separately, then average them to get the average force per area.
Yes, the formula is the same.
Macroscopic magnetic force acting on the current-carrying slab per unit area is given by the formula
$$ f = \frac{B_1^2-B_2^2}{2\mu}\tag{*}, $$
irrespective of how exactly the current is distributed across the thin dimension of the slab.
This can be derived as follows.
Macroscopic magnetic force on the slab consists of two parts:
- force on the slab due to magnetic field $\mathbf {B}_{int}$ produced by the current in the slab;
- force on the slab due to magnetic field $\mathbf{B}_{ext}$ produced by other bodies (external magnetic field).
If the slab is static (we are interested in magnetic force on a slab that does not move), and current density in all layers of the slab has the same direction, then Newton's third law applies; every internal magnetic force due to one layer acting on another has a counterpart of opposite direction and equal strength acting back on the first layer. This makes sum of all internal magnetic forces zero.
So in this case, the whole magnetic force is just due to the external magnetic field $\mathbf B_{ext}$. So force per unit area due to external magnetic field is
$$ \mathbf f = \int_0^d \mathbf J(x) \times {\mathbf B_{ext}} ~dx $$ where the integration is over the thickness of the slab $d$ and $\mathbf J(x)$ is current density at position $x$ in the slab.
Since $\mathbf B_{ext}$ is almost constant throughout the slab, it can be taken out of the integration, and we end up with expression for magnetic force per unit area
$$ \mathbf f = \boldsymbol{\mathcal{J}} \times \mathbf{B}_{ext} $$
where $$ \boldsymbol{\mathcal{J}} = \int_0^d \mathbf J(x) ~dx $$
is the effective surface density of current for the slab.
Thus the force density has the same form as for 2D current sheet, the exact distribution across the thickness does not matter. This is because the external magnetic field varies negligibly across the slab thickness.
The rest of the derivation is the same as Purcell's and Morin's derivation for 2D current sheet. The external field $\mathbf B_{ext}$ can be expressed via the total field values at both faces of the slab $\mathbf {B}_1,\mathbf {B}_2$ as $(\mathbf {B}_1 + \mathbf {B}_2) /2$. This is because in this expression, the internal field component of $\mathbf {B}_1$ cancels out the internal field component of $\mathbf {B}_2$.
Surface current density can be expressed as $$ \boldsymbol{\mathcal{J}} = \frac{1}{\mu} (\mathbf {B}_1 - \mathbf {B}_2) \times \mathbf n. $$ where $\mathbf n$ is unit vector normal to the slab and oriented in direction of increasing magnetic field magnitude. This expression of $\boldsymbol{\mathcal{J}}$ can be derived from the definition above using Ampere's law (or Biot-Savart's law).
Combining these expressions, we obtain the formula (*).
