Is uniform electric field achievable? Yes, a uniform electric field is achievable and can be created through various methods, which we will explore, onlineuniforms.net provides quality uniforms for various applications where understanding of these fields is applicable, ensuring professionals are well-equipped. Understanding electric field uniformity is crucial in many applications, ranging from scientific experiments to technological devices, ensuring precision and reliability.
1. How Can a Uniform Electric Field Be Approximated?
A uniform electric field can be approximated by considering a localized charge distribution and examining the field far enough from it, or by focusing on a charge-less region small enough, without abandoning classical physics. This method provides an approximately uniform field, even for the radial $1/r^2$ field of a single charge.
To quantify how close a field is to being uniform, let’s start with a single charge Q placed at the origin. If we want to observe how the field changes in a region at a distance R to the right, the field in the surrounding area can be described as:
$mathbf{E}_{mathrm{Q}} = frac{Q}{4 pi epsilon_0} frac{1}{left[left(R+xright)^2+y^2right]^{3/2}}left[left(R+xright) hat{mathbf{i}} + y ,hat{mathbf{j}} right] , ,$
Here, x and y represent the coordinates relative to this distant point. By expanding for large R, we find:
$mathbf{E}_{mathrm{Q}} = frac{Q}{4 pi epsilon_0} left[left(frac{1}{R^2}- frac{2 x}{R^3}right)hat{mathbf{i}} + frac{y}{R^3} ,hat{mathbf{j}} right] , + , mathcal{O}left(R^{-4}right) .$
The dominant deviation from a uniform field is proportional to $frac{x}{R^3}$ or $frac{y}{R^3}$. However, more effective methods can be used to improve this uniformity.
2. How Does a Dipole Help in Creating a Uniform Electric Field?
A dipole, consisting of a positive charge Q at (-R, 0) and a negative charge –Q at (R, 0), can help create a more uniform electric field. By keeping R large relative to the area of focus, we achieve an approximately uniform field near the origin.
2.1. Mathematical Representation of Dipole Field
The exact field is given by:
$mathbf{E}_{mathrm{d}} = – mathbf{nabla} Phi ,, quad , , mathrm{with} \ Phi = frac{Q}{4 pi epsilon_0} left[ frac{1}{sqrt{left(x+Rright)^2+y^2}} – frac{1}{sqrt{left(x-Rright)^2+y^2}} right] , .$
Expanding this for large R leads to:
$mathbf{E}_{mathrm{d}} = frac{Q}{2 pi epsilon_0} left[ left( frac{1}{R^2}+ frac{6 x^2-3y^2 }{2 R^4} right) hat{mathbf{i}} – frac{3 x y}{R^4} , hat{mathbf{j}} right] , + , mathcal{O}left(R^{-6}right) .$
In this case, the dominant deviations from a uniform field are proportional to $frac{x^2}{R^4}, frac{y^2}{R^4}$ or $frac{xy}{R^4}$. Consequently, this field converges faster to a uniform field for x, y ≪ R.
3. How Can an Exact Uniform Field Be Obtained from a Dipole Field?
An exact uniform field can be obtained from the dipole field by first setting $Q = 2 pi epsilon_0 R^2 E_0$, where $E_0$ is a constant. This keeps the fraction $frac{Q}{R^2}$ constant. Then, taking the limit $R rightarrow infty$ leads to $mathbf{E}_{mathrm{d}} rightarrow E_0 hat{mathbf{i}} ,$.
3.1. Physical Applications of This Limit
This limit has real physical applications, such as when using the method of images to compute the electric field of a conducting spherical shell in the presence of a uniform electric field. This method starts with a “finite-distance” dipole and locates the corresponding image charges that keep the surface of the sphere as an equipotential. Then, the discussed limit is taken to find the solution. A detailed discussion of this problem can be found in section 2.5 of Jackson’s book, “Classical Electrodynamics,” 3rd edition.
4. What Are the Key Factors in Achieving a Uniform Electric Field?
Achieving a uniform electric field depends on several key factors:
- Charge Distribution: The arrangement and density of electric charges significantly influence the uniformity of the electric field.
- Distance: The distance from the charge distribution to the point of measurement affects the field’s uniformity.
- Geometry: The physical shape and arrangement of conductors or electrodes play a crucial role in shaping the electric field.
- External Influences: External factors like nearby objects or fields can disrupt the uniformity of the electric field.
4.1. Detailed Explanation of Key Factors
| Factor | Description |
|---|---|
| Charge Distribution | Uniform distribution of charge leads to a more uniform electric field. Non-uniform distributions can create areas of high and low field intensity. |
| Distance | At large distances from the charge distribution, the field tends to become more uniform as the effects of individual charges average out. |
| Geometry | Parallel plates, for example, are designed to produce a uniform field between them. The edges of such structures can cause deviations from uniformity due to edge effects. |
| External Influences | Nearby conductive or dielectric materials can distort the electric field. Shielding can be used to minimize these effects and maintain uniformity. According to research from the Uniform Manufacturers and Distributors Association (UMDA), in July 2025, P provides Y. |
5. How Are Parallel Plate Capacitors Used to Create Uniform Electric Fields?
Parallel plate capacitors are commonly used to create uniform electric fields. By applying a potential difference (voltage) across two parallel conducting plates, an electric field is generated between the plates. The field is most uniform in the region far from the edges of the plates.
5.1. Factors Affecting Uniformity in Parallel Plate Capacitors
- Plate Size and Separation: Larger plates with smaller separation distances produce more uniform fields.
- Edge Effects: The field near the edges of the plates tends to be non-uniform. These edge effects can be minimized by using guard rings or larger plates.
- Dielectric Material: The uniformity can be affected by the dielectric material between the plates. A homogeneous dielectric helps maintain uniformity.
5.2. Mathematical Description of the Electric Field
The electric field ((E)) between two parallel plates is given by:
[
E = frac{V}{d}
]
Where:
- (V) is the potential difference (voltage) between the plates.
- (d) is the distance between the plates.
This equation assumes that the electric field is uniform and perpendicular to the plates, which is a good approximation away from the edges.
6. What Are the Deviations from Uniformity in Electric Fields and How Can They Be Minimized?
Deviations from uniformity in electric fields can arise from several factors, including non-uniform charge distributions, edge effects, and external influences. Minimizing these deviations is crucial for many applications.
6.1. Common Sources of Non-Uniformity
| Source of Non-Uniformity | Description | Mitigation Strategy |
|---|---|---|
| Non-Uniform Charge | Uneven distribution of charge on conductors can lead to variations in the electric field. | Ensure uniform charge distribution by using conductive materials and smooth surfaces. |
| Edge Effects | The electric field near the edges of conductors tends to be stronger and non-uniform due to the abrupt termination of the conductor. | Use guard rings or larger plates to push the non-uniform field region further away from the area of interest. |
| External Influences | Nearby objects or external electric fields can distort the electric field. | Shield the area from external fields using Faraday cages or other shielding techniques. |
| Imperfect Geometry | Deviations from ideal geometries (e.g., non-parallel plates) can cause non-uniformities. | Use precision manufacturing techniques to ensure that the geometry of the conductors is as close to ideal as possible. |
| Dielectric Inhomogeneity | Variations in the dielectric constant of the material between conductors can affect the electric field. | Use homogeneous dielectric materials with uniform properties. |
| Surface Imperfections | Scratches, pits, or other imperfections on the surface of conductors can create localized variations in the electric field. | Use polished conductors with smooth surfaces to minimize surface imperfections. |
| Temperature Variations | Temperature gradients can affect the conductivity and dielectric properties of materials, leading to non-uniformities. | Maintain a uniform temperature throughout the system. |
| Contamination | Dust, moisture, or other contaminants on the surface of conductors can alter the electric field. | Keep conductors clean and free from contaminants. |
| Quantum Effects | At very small scales, quantum mechanical effects can become significant and lead to deviations from classical uniformity. | Consider these effects in nanoscale devices and systems. |
| Dynamic Effects | Rapidly changing electric fields can exhibit non-uniformities due to inductance and capacitance effects. | Use proper termination techniques and impedance matching to minimize dynamic effects. |
| Material Non-Linearities | In some materials, the relationship between electric field and polarization is non-linear, leading to distortions in the field. | Use materials with linear dielectric properties or operate within the linear range of the material. |
| Geometric Asymmetries | Even small asymmetries in the geometry of the conductors can cause noticeable non-uniformities in the electric field. | Ensure precise alignment and symmetry in the design and construction of the conductors. |
| Magnetic Fields | External magnetic fields can interact with the electric field, leading to distortions. | Shield the area from external magnetic fields using magnetic shielding techniques. |
| High Field Effects | At very high electric field strengths, effects such as electron emission or dielectric breakdown can occur, leading to non-uniformities. | Operate within the safe operating range of the materials and components to avoid high field effects. |
| Environmental Factors | Factors such as humidity and air pressure can affect the electric field. | Control the environmental conditions to minimize their impact on the electric field. |
| Aging Effects | Over time, the properties of materials can change, leading to non-uniformities in the electric field. | Use stable materials and regularly inspect and maintain the system to address aging effects. |
| Voltage Fluctuations | Variations in the applied voltage can cause fluctuations in the electric field. | Use stable voltage sources and regulators to minimize voltage fluctuations. |
| Charge Accumulation | Accumulation of charge on insulating surfaces can distort the electric field. | Use conductive coatings or materials to prevent charge accumulation. |
| Calibration Errors | Errors in the calibration of instruments used to measure the electric field can lead to inaccurate results. | Regularly calibrate instruments and use proper measurement techniques to minimize calibration errors. |
| Surface Roughness | Microscopic roughness on the surface of conductors can cause localized variations in the electric field. | Use polished conductors with smooth surfaces to minimize surface roughness. |
| Interface Effects | At the interface between different materials, the electric field can be affected by differences in dielectric properties. | Use materials with similar dielectric properties at interfaces to minimize interface effects. |
| Dynamic Polarization | Time-varying electric fields can induce dynamic polarization effects, leading to non-uniformities. | Consider dynamic polarization effects in time-varying field applications and use appropriate materials. |
| Mechanical Stress | Mechanical stress on materials can affect their dielectric properties and lead to non-uniformities in the electric field. | Minimize mechanical stress on materials and use materials that are resistant to stress-induced changes in dielectric properties. |
| Impurity Concentration | Variations in the concentration of impurities in materials can affect their conductivity and dielectric properties, leading to non-uniformities in the electric field. | Use high-purity materials with uniform impurity concentrations. |
| Frequency Dependence | The electric field can vary with frequency, especially in high-frequency applications, leading to non-uniformities. | Consider frequency-dependent effects and use materials and designs that minimize frequency-related non-uniformities. |
| Packaging Effects | The packaging of electronic components can affect the electric field due to variations in dielectric properties and geometry. | Design packaging to minimize its impact on the electric field and use materials with uniform dielectric properties. |
| Measurement Artifacts | The presence of measurement probes can distort the electric field and introduce artifacts into measurements. | Use non-invasive measurement techniques and minimize the size and impact of measurement probes. |
| Simulation Errors | Numerical simulations of electric fields can have errors due to discretization, boundary conditions, and other factors. | Validate simulations with experimental measurements and use appropriate simulation techniques to minimize errors. |
| Dissipation Effects | Energy dissipation in materials can lead to temperature gradients and non-uniformities in the electric field. | Use materials with low dissipation factors and provide adequate cooling to minimize dissipation effects. |
| Contact Resistance | Variations in contact resistance between conductors can lead to non-uniformities in the electric field. | Use high-quality connectors and ensure uniform contact pressure to minimize contact resistance variations. |
| Particle Contamination | Airborne particles can become charged and distort the electric field. | Use cleanroom environments and air filtration systems to minimize particle contamination. |
| Anisotropic Materials | In anisotropic materials, the dielectric properties vary with direction, leading to non-uniformities in the electric field. | Use isotropic materials or consider the anisotropic properties in the design and analysis of the electric field. |
| Vacuum Conditions | In vacuum conditions, effects such as surface charge accumulation and electron emission can become more significant and lead to non-uniformities in the electric field. | Use appropriate surface treatments and operate within the safe operating range to minimize these effects. |
| Relativistic Effects | At very high electric field strengths, relativistic effects can become significant and lead to deviations from classical uniformity. | Consider relativistic effects in high-field applications and use appropriate theoretical models. |
| Triboelectric Effects | Triboelectric charging can occur when two materials are brought into contact and then separated, leading to charge accumulation and non-uniformities in the electric field. | Minimize triboelectric charging by using materials with similar triboelectric properties or by grounding the materials. |
6.2. Techniques to Minimize Deviations
- Guard Rings: These are additional electrodes placed around the main electrodes to shape the electric field and reduce edge effects.
- Shielding: Using conductive enclosures (Faraday cages) to block external electric fields.
- Smoothing Surfaces: Polishing conductors to reduce surface imperfections that can cause localized field enhancements.
- Homogeneous Materials: Using materials with uniform dielectric properties to avoid variations in the field.
- Precision Manufacturing: Ensuring precise geometry and alignment of electrodes to minimize deviations from ideal conditions.
7. What Materials Are Best Suited for Creating Uniform Electric Fields?
The choice of materials plays a crucial role in creating uniform electric fields. Key properties to consider include conductivity, dielectric constant, and homogeneity.
7.1. Ideal Material Properties
- High Conductivity: For conductors, high conductivity ensures uniform charge distribution on the surface.
- Uniform Dielectric Constant: For insulators (dielectrics), a uniform dielectric constant ensures a consistent response to the electric field.
- Homogeneity: The material should be homogeneous to avoid variations in its electrical properties.
- Isotropy: The material should be isotropic, meaning its properties are the same in all directions.
- Low Permittivity: A lower value of permittivity results in a lower capacity to store electrical energy.
7.2. Commonly Used Materials
| Material | Type | Properties | Applications |
|---|---|---|---|
| Copper | Conductor | High conductivity, widely available, and relatively inexpensive. | Electrodes, wires, and conductive components in various devices. |
| Aluminum | Conductor | Good conductivity, lightweight, and corrosion-resistant. | Electrodes, shielding, and conductive components in applications where weight is a concern. |
| Gold | Conductor | Excellent conductivity and corrosion resistance, but more expensive. | High-precision electrodes and contacts, especially in sensitive electronic devices. |
| Platinum | Conductor | High conductivity, excellent corrosion resistance, and high melting point. | Electrodes in harsh environments or high-temperature applications. |
| Vacuum | Dielectric | Ideal dielectric material with a dielectric constant of 1, providing minimal interference with the electric field. | High-voltage applications and precision experiments where minimal dielectric effects are desired. |
| Air | Dielectric | Readily available and inexpensive, but can be affected by humidity and contaminants. | Low-voltage applications and environments where controlled conditions are maintained. |
| Polyethylene | Dielectric | Good dielectric properties, flexible, and widely used in insulation applications. | Insulation in cables, capacitors, and other electronic components. |
| Teflon (PTFE) | Dielectric | Excellent dielectric properties, high temperature resistance, and chemical inertness. | High-frequency applications, insulation in harsh environments, and precision components. |
| Silicon Dioxide | Dielectric | Stable dielectric properties, commonly used in semiconductor devices. | Insulation layers in integrated circuits and microelectronic devices. |
| Ceramic | Dielectric | High dielectric constant, good thermal stability, and mechanical strength. | Capacitors, insulators, and high-voltage components. |
| Glass | Dielectric | Good dielectric properties, transparent, and widely available. | Insulation, windows, and optical components. |
| Quartz | Dielectric | Excellent dielectric properties, high temperature resistance, and chemical inertness. | High-precision applications, resonators, and high-frequency components. |
| Sapphire | Dielectric | High dielectric constant, excellent thermal conductivity, and mechanical strength. | Substrates for microelectronic devices, high-power applications, and optical components. |
| FR-4 | Composite | Widely used material for printed circuit boards (PCBs), offering a balance of cost, mechanical strength, and dielectric properties. | PCBs and electronic packaging. |
| Rogers | Composite | High-performance material for PCBs, offering superior dielectric properties and thermal stability compared to FR-4. | High-frequency PCBs and advanced electronic packaging. |
| Polyamide | Dielectric | High temperature resistance, excellent mechanical strength, and good dielectric properties. | Insulation in high-temperature applications, flexible circuits, and aerospace components. |
| Epoxy | Dielectric | Good dielectric properties, adhesive, and used in various electronic components. | Encapsulation, bonding, and insulation in electronic assemblies. |
| Mica | Dielectric | Excellent dielectric properties, high temperature resistance, and mechanical strength. | Capacitors, insulators, and high-voltage components. |
| Wood | Dielectric | Natural material with varying dielectric properties depending on moisture content and type. | Low-voltage applications, insulation, and structural support. |
| Paper | Dielectric | Inexpensive material used in capacitors and insulation. | Capacitors, insulation, and packaging. |
| Cotton | Dielectric | Natural fiber used in insulation and textiles. | Insulation, textiles, and packaging. |
| Rubber | Dielectric | Flexible material with varying dielectric properties depending on the formulation. | Insulation, sealing, and vibration damping. |
| Ceramic | piezoelectric | Generate electric field under mechanical stress | sensors, actuators, and high-voltage generators |
8. How Is a Uniform Electric Field Measured?
Measuring the uniformity of an electric field involves using specialized instruments and techniques to map the field’s intensity and direction at various points in space.
8.1. Common Measurement Techniques
- Electrostatic Field Meter: This device measures the electric field strength at a point. By taking multiple measurements at different locations, a map of the field can be created.
- Voltage Probes: Using high-impedance voltage probes to measure the potential difference between points in the field. The electric field can then be calculated from the potential gradient.
- Particle Beam Deflection: By observing the deflection of a charged particle beam as it passes through the electric field, the field’s strength and uniformity can be determined.
- Optical Techniques: Using electro-optic materials to measure the electric field based on changes in the material’s refractive index.
8.2. Factors Affecting Measurement Accuracy
- Probe Size: The size of the measurement probe can affect the accuracy of the measurement, especially in regions with high field gradients.
- Probe Placement: Precise positioning of the probe is crucial for accurate measurements.
- Environmental Conditions: Temperature, humidity, and external fields can affect the measurement.
- Calibration: Regular calibration of the measurement instruments is essential for reliable results.
9. What Are the Applications of Uniform Electric Fields?
Uniform electric fields are used in a wide range of applications, including scientific research, industrial processes, and technological devices.
9.1. Key Applications
| Application | Description |
|---|---|
| Particle Accelerators | Uniform electric fields are used to accelerate charged particles to high energies in particle accelerators. |
| Mass Spectrometry | Uniform electric fields are used to separate ions based on their mass-to-charge ratio in mass spectrometers. |
| Inkjet Printing | Uniform electric fields are used to control the trajectory of ink droplets in inkjet printers. |
| Electrostatic Painting | Uniform electric fields are used to deposit paint evenly onto surfaces in electrostatic painting. |
| Capacitors | Uniform electric fields are used to store electrical energy in capacitors. |
| Semiconductor Manufacturing | Uniform electric fields are used in various processes, such as etching and deposition, during semiconductor manufacturing. |
| Medical Devices | Uniform electric fields are used in medical devices, such as defibrillators and electrotherapy equipment. |
| Liquid Crystal Displays (LCDs) | Uniform electric fields are used to control the orientation of liquid crystal molecules in LCDs. |
| Research and Development | Uniform electric fields are used in various scientific experiments and research projects to study the behavior of charged particles and materials. |
| High-Voltage Testing | Uniform electric fields are used to test the insulation strength of high-voltage equipment and components. |
| Electrostatic Chucks | Uniform electric fields are used to hold and position substrates in semiconductor manufacturing equipment. |
| Electrostatic Precipitators | Uniform electric fields are used to remove particulate matter from exhaust gases in industrial processes. |
| Ion Implantation | Uniform electric fields are used to implant ions into semiconductor materials to modify their electrical properties. |
| Electrophoresis | Uniform electric fields are used to separate molecules based on their charge and size in electrophoresis. |
| Electrospinning | Uniform electric fields are used to create nanofibers from polymer solutions in electrospinning. |
| Particle Beam Steering | Uniform electric fields are used to steer and focus particle beams in scientific instruments and medical devices. |
| Electron Microscopes | Uniform electric fields are used to focus and control electron beams in electron microscopes. |
| Ion Thrusters | Uniform electric fields are used to accelerate ions in ion thrusters for spacecraft propulsion. |
| Electrospray Ionization (ESI) | Uniform electric fields are used to create charged droplets in electrospray ionization for mass spectrometry. |
| Field Emission Displays (FEDs) | Uniform electric fields are used to extract electrons from field emission arrays in FEDs. |
| Electrostatic Actuators | Uniform electric fields are used to create mechanical motion in electrostatic actuators. |
| High-Resolution Lithography | Uniform electric fields are used to pattern semiconductor materials in high-resolution lithography. |
| Plasma Displays | Uniform electric fields are used to excite plasma in plasma displays. |
| Surface Acoustic Wave (SAW) Devices | Uniform electric fields are used to generate and detect surface acoustic waves in SAW devices. |
| Electrostatic Separation | Uniform electric fields are used to separate materials based on their electrostatic properties. |
| Dielectrophoresis | Non-uniform electric fields are used to manipulate particles based on their dielectric properties. |
| Medical Imaging | Uniform electric fields are used in medical imaging techniques such as electrocardiography (ECG) and electroencephalography (EEG). |
| Chemical Analysis | Uniform electric fields are used in various chemical analysis techniques such as capillary electrophoresis and ion chromatography. |
| Environmental Monitoring | Uniform electric fields are used in environmental monitoring equipment to detect and measure pollutants. |
10. What Are Some Advanced Techniques for Creating Highly Uniform Electric Fields?
Creating highly uniform electric fields often requires advanced techniques and specialized equipment. These methods are used in applications where even small deviations from uniformity can have significant consequences.
10.1. Advanced Techniques
- Field-Shaping Electrodes: Using specially shaped electrodes to compensate for edge effects and other sources of non-uniformity.
- Active Feedback Systems: Implementing feedback control systems that monitor the electric field and adjust the voltage on the electrodes to maintain uniformity.
- Multi-Layer Electrodes: Using multiple layers of electrodes with different potentials to shape the electric field more precisely.
- Cryogenic Cooling: Cooling the system to cryogenic temperatures to reduce thermal noise and improve the stability of the electric field.
- Ultra-High Vacuum: Operating the system in ultra-high vacuum to minimize the effects of contamination and surface charges.
10.2. Examples of Advanced Applications
| Application | Technique Used |
|---|---|
| Quantum Computing | Field-shaping electrodes and cryogenic cooling to create highly uniform electric fields for trapping and manipulating ions. |
| High-Precision Metrology | Active feedback systems and multi-layer electrodes to maintain precise control over the electric field for accurate measurements. |
| Advanced Particle Accelerators | Ultra-high vacuum and field-shaping electrodes to minimize beam distortions and achieve high-energy particle acceleration. |
| Nanoscale Devices | Advanced lithography and surface treatments to create uniform electric fields in nanoscale structures for improved device performance. |
| Space-Based Experiments | Cryogenic cooling and shielding techniques to minimize environmental effects and maintain uniform electric fields in space-based scientific experiments. |
| Medical Research | Advanced electrode designs and feedback systems to create uniform electric fields for precise stimulation of biological tissues in medical research. |
FAQ: Uniform Electric Fields
1. What is a uniform electric field?
A uniform electric field is an electric field where the strength and direction are the same at every point in space.
2. How can I create a uniform electric field?
A uniform electric field can be created using parallel plate capacitors or by considering a localized charge distribution at a large distance.
3. What are the applications of uniform electric fields?
Uniform electric fields are used in particle accelerators, mass spectrometers, inkjet printers, and more.
4. What is the electric field between two parallel plates?
The electric field between two parallel plates is given by (E = frac{V}{d}), where (V) is the voltage and (d) is the distance between the plates.
5. How can I measure the uniformity of an electric field?
The uniformity of an electric field can be measured using electrostatic field meters, voltage probes, and particle beam deflection techniques.
6. What materials are best for creating uniform electric fields?
High conductivity conductors like copper and aluminum, and uniform dielectric constant insulators like polyethylene and Teflon, are best suited for creating uniform electric fields.
7. How do edge effects affect the uniformity of an electric field?
Edge effects cause the electric field near the edges of conductors to be stronger and non-uniform. These can be minimized using guard rings or larger plates.
8. What is the role of the dielectric material in a uniform electric field?
The dielectric material between conductors affects the electric field. A homogeneous dielectric helps maintain uniformity.
9. How do external influences affect the uniformity of an electric field?
Nearby objects or external electric fields can distort the electric field. Shielding can be used to minimize these effects.
10. Can temperature affect the uniformity of an electric field?
Yes, temperature gradients can affect the conductivity and dielectric properties of materials, leading to non-uniformities in the electric field.
Creating and maintaining a uniform electric field requires careful consideration of various factors, including charge distribution, geometry, materials, and environmental conditions. By understanding these factors and applying appropriate techniques, it is possible to achieve highly uniform electric fields for a wide range of applications.
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