Problem 90 - University Physics with Modern Physics

Question

The Electromagnetic Pump. Magnetic forces acting on conducting fluids provide a convenient means of pumping these fluids. For example, this method can be used to pump blood without the damage to the cells that can be caused by a mechanical pump. A horizontal tube with rectangular cross section (height \(h ,\) width \(w )\) is placed at right angles to a uniform magnetic field with magnitude \(B\) so that a length \(l\) is in the field (Fig. P27.90). The tube is filled with a conducting liquid, and an electric current of density \(J\) is maintained in the third mutually perpendicular direction. (a) Show that the difference of pressure between a point in the liquid on a vertical plane through \(a b\) and a point in the liquid on another vertical plane through \(c d ,\) under conditions in which the liquid is prevented from flowing, is \(\Delta p = J / B\) . (b) What current density is needed to provide a pressure difference of 1.00 atm between these two points if \(B = 2.20 \mathrm { T }\) and \(l = 35.0 \mathrm { mm } ?\)

Step-by-Step Solution

Answer

(a) Pressure difference is \( \Delta p = JB \). (b) Current density required is approximately 46058 A/m².

Step 1: Identify the Forces

The problem involves a conducting liquid and an electromagnetic force. The magnetic force on the liquid is given by the Lorentz force equation: \( F = JLB \), where \( J \) is the current density, \( L \) is the length of the section of liquid in the magnetic field, and \( B \) is the magnetic field strength. This force acts as a pressure difference across the liquid.

Step 2: Calculate Pressure Difference

The magnetic force creates a pressure difference \( \Delta p \) across the vertical planes through points \( ab \) and \( cd \). Given that pressure \( \Delta p \) is force per unit area, the relationship can be written as \( \Delta p = \frac{F}{A} = \frac{JLB}{wL} \). Canceling \( L \) from the equation (because it appears in both the numerator as length and in the denominator due to area), we get \( \Delta p = JB \). This shows the required pressure difference as \( \Delta p = J/B \).

Step 3: Solve Part (a)

We have shown that the pressure difference \( \Delta p \) is given by \( \Delta p = JB \). This satisfies part (a) as the expression for the pressure difference.

Step 4: Solve Part (b)

For part (b), the required pressure difference is given as 1.00 atm, which is equivalent to 101325 Pa. Using the formula from part (a), \( J = \frac{\Delta p}{B} \). Substitute \( \Delta p = 101325 \) Pa and \( B = 2.20 \) T, give the current density \( J = \frac{101325}{2.20} \approx 46057.5 \) A/m².

Key Concepts

Magnetic Force

Magnetic force originates from the interaction between magnetic fields and electric charges. In an electromagnetic pump, this force plays a crucial role in moving conducting liquids, like blood, through a magnetic field. As a uniform magnetic field is applied perpendicular to the flow of the liquid, the magnetic force emerges from this cross-interaction.
This force is not surface dependent but is distributed throughout the volume of the liquid. Through proper control of magnetic fields and electric currents, electromagnetic pumps generate consistent forces.

• Magnetic forces are predictable based on field orientation and strength.
• These forces are essential for non-invasive liquid movements.

Understanding the behavior of magnetic forces helps in effectively designing systems for handling conductive liquids without the need for mechanical parts that may cause damage.

Current Density

Current density is crucial for characterizing how much electric current flows through a given area. It is defined as the amount of current per unit area through a surface that is perpendicular to the flow.
The formula for current density is:\[J = \frac{I}{A}\]where:

• \( J \) is the current density (in amperes per square meter).
• \( I \) is the total current (in amperes).
• \( A \) is the cross-sectional area through which the current flows (in square meters).

In electromagnetic pumps, the current density determines the efficiency of the magnetic force generation. Adjusting current density impacts the pressure difference across the liquid, enabling precise control of the pump's behavior.

Pressure Difference

Pressure difference is a critical concept in understanding how electromagnetic pumps function. This refers to the difference in pressure between two points along the conducting liquid in the setup.
The pressure difference is created by the magnetic force acting across the liquid's length:\[\Delta p = \frac{J}{B}\]where:

• \( \Delta p \) is the pressure difference.
• \( J \) is the current density.
• \( B \) is the magnetic field strength.

Manipulating this pressure difference is essential for controlling the flow of the liquid. It directly influences the force that moves the fluid without any mechanical intervention, thus preserving the integrity of sensitive materials like blood.

Lorentz Force

The Lorentz Force is a foundational principle that describes the force experienced by a charged particle moving through both electric and magnetic fields. This principle is vital in the operation of electromagnetic pumps.
For a conducting liquid in a magnetic field, the Lorentz force can be expressed as:\[F = JLB\]where:

• \( F \) is the force exerted on the liquid.
• \( J \) is the current density.
• \( L \) is the length of the liquid in the magnetic field.
• \( B \) is the magnetic field strength.

This force transforms the electrical energy into mechanical energy, pushing the liquid through the system. Understanding the Lorentz force allows engineers to design pumps capable of moving fluids efficiently and uniformly.

Conducting Liquid

Conducting liquids are substances that can transport electric charge. They play a central role in electromagnetic pumps, because it's their interaction with magnetic fields that enables the pump operation.
The nature of the liquid, along with its electrical conductivity, affects how efficiently the pump can function. Conducting liquids include:

• Electrolyte solutions.
• Liquid metals.
• Blood and other biofluids.

The selected conducting liquid must possess adequate conductivity to carry the current effectively. This behavior allows for the uniform application of the Lorentz force across the liquid, leading to optimal pumping performance without physical contact that may degrade its integrity.