# Calculate pressure difference in current-carrying mercury

## 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

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