Understanding the Unusual Repulsion of Antiparallel Magnetic Fields

[superaznnerd]

If one wire has current running through it to the right, and another wire below it has current running it through the left...
Why would they repel each other??
Can someone explain mathematically/ using right hand rule??

Thanks

 

[Bob S]

There are two ways to calculate the repulsive force between two wires with opposing currents I in the z direction and separation a (The first way doesn't count)

1) Use Lorentz force, F = I x B = μ₀ I²/2πa

2) Calculate the total stored magnetic energy density over volume and take the partial derivative with respect to the wire separation.

The transverse magnetic field components for the two wires separated by distance a are given in Smythe, Static and Dynamic Electricity, third edition, Section 7.09 (3) (4). The stored magnetic energy per unit length is given by (See Smythe 8.02 (3))

W = 1/(2μ₀)∫_v B² dV where B²= B_x² + B_y²

and the force per unit length by (See Smythe 8.01 (4))

F_a = +∂W/∂a

It is obvious that that W is minimum when the two wires are close together, because the magnetic fields cancel, so the stored magnetic energy increases with increasing a, and F_a is therefore positive.

3) The inductance of the wire pair per unit length is (Smythe 8.12 (11)

L = (μ₀/4π)[1+ 4 Ln(a/c)]

where a is wire separation and c is wire radius.

The stored magnetic energy can then be written as (see Smythe 8.08 (1))

W = ½LI²

Then the force is

F_a = +∂W/∂a as before.

Bob S

 

[Bob S]

In post #2, the calculated force between the two conductors is

F_a = +∂/∂a [1/(2μ₀)∫_v B² dV] (with a plus sign)

This is very unusual, and unexpected, because the repulsive force is in the direction of increasing the stored magnetic energy.

In a mechanical system, like a compressed spring, W = ½ k x², so the force is

F_x = −∂W/∂x = −kx (with a minus sign).

So the force is in a direction to reduce the stored mechanical energy.

Smythe, in Static and Dynamic Electricity, third edition, sections 7.18 and 8.02, discusses the sign difference at some length. In the magnetic case, the external circuit provides energy to maintain a constant current in the conductors as the conductor is moved. Smythe states "The [magnetic case] is exactly the opposite of the electric case, where the force on equal and opposite charges tends to bring them together and destroy the electric field."

Bob S