# Calculate pressure difference in current-carrying mercury

• Sam J
In summary, the problem at hand involves finding the pressure difference at the center of a mercury column compared to the outer radius. By considering the column as a series of concentric current carrying cylinders, the magnetic field at a radial distance can be shown to be B(r)=μ0I0R/A. This field acts on the electrons flowing within the current and can be used to calculate the outward force on each cylindrical shell and ultimately the pressure difference. To do this, Ampere's law and the equation -∇P=Fv are used, with the latter representing the balance between electromagnetic and hydrostatic forces. This problem is of intermediate level difficulty and requires a conceptual understanding of these principles.
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

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

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.)

Last edited:

## 1. How is pressure difference calculated in current-carrying mercury?

The pressure difference in current-carrying mercury can be calculated using the formula P = BIL, where P is the pressure difference, B is the magnetic field strength, I is the current, and L is the length of the mercury column.

## 2. What is the significance of calculating pressure difference in current-carrying mercury?

Calculating pressure difference in current-carrying mercury allows us to understand the relationship between electricity and magnetism, as well as the behavior of conductive materials in a magnetic field. It also has practical applications in devices such as electromagnets and electric motors.

## 3. How does the pressure difference in current-carrying mercury change with varying current or magnetic field?

The pressure difference in current-carrying mercury is directly proportional to both the current and the magnetic field strength. This means that as the current or magnetic field increases, the pressure difference will also increase.

## 4. Can pressure difference in current-carrying mercury be manipulated for practical use?

Yes, pressure difference in current-carrying mercury can be manipulated for practical use. By controlling the current and magnetic field, we can create devices such as valves and switches that utilize the pressure difference to control the flow of mercury.

## 5. Are there any safety concerns when working with current-carrying mercury and pressure difference?

Yes, there are safety concerns when working with current-carrying mercury and pressure difference. Mercury is a toxic substance and should be handled carefully. Additionally, high currents and magnetic fields can be dangerous, so proper precautions should be taken when conducting experiments involving these factors.

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