Calculate pressure difference in current-carrying mercury

[Sam J]

Homework Statement

A vertical column of mercury, of cross-sectional area A, is contained in an insulating cylinder and carries a current I₀, with uniform current density.

By considering the column to be a series of concentric current carrying cylin-
ders, derive an expression for the difference in pressure at the centre of the column compared with the outer radius. Ignore end effects and assume that the mercury and
the cylinder are non-magnetic.

Homework Equations

Biot Savart
p=F/S (S=surface area)

The Attempt at a Solution

All I really need help with is a conceptual understanding.

In order to develop pressure difference we need some force to act. Assuming the mercury to be neutrally charged, it must be that this force is derived from a magnetostatic field.

Such a field at a radial distance r from the centre can be shown to be:

B(r)=μ₀I₀R/A

What I am struggling with is to understand what this field might be acting on. Is it the electrons flowing within the current?

If so, then I assume that I calculate the outward force acting on each cylindrical shell, integrating over them to find the total force on the outer-most shell and divide by its surface area to find the pressure. Is this correct?

In this case I get the result to be:

p=I₀²μ₀R²/A
=I₀²μ₀/π

ie. the pressure is independent of the radius of the mercury column!

 

[Charles Link]

With uniform current density per unit area, how much current flows inside a radius $r$ ? To begin the calculations for this problem, you need to compute the magnetic field $B$ at radius $r$ due to this current. Would suggest using Ampere's law. Additional comment is the degree of difficulty of the problem is more at the intermediate level. Once you get the magnetic field strength, computing the pressure from the force is somewhat routine, but non-trivial. I can give you a hint at this part as well: You need hydostatic forces to balance the electromagnetic forces. The equation that applies is $-\nabla P=F_v$ where $P$ is the pressure and $F_v$ is the force per unit volume. (Note: The $-\nabla P=F_v$ equation can also be used along with the gravitational forces to compute pressure as a function of height. That is actually a more common use for this equation.)