# Magnetic field due to a long straight wire & force between two parallel wires

##### Intros
###### Lessons
1. Magnetic field due to a long straight wire
2. Magnetic field midway between two currents
3. Forces between two parallel wires
##### Examples
###### Lessons
1. In which diagram would an external magnetic field $\overline{B}$, cause two current-carrying wires to move towards one another?

1. Which of the four diagrams below correctly depicts the magnetic field found on either side of a current-carrying wire?

1. A long pair of wires conduct 25.0A of current to and from an instrument. If the insulated wires are of negligible diameter but are 2.88mm apart, what is the magnetic field 10.00cm from their midpoint?

## Introduction

The magnetic field due to a long straight wire and the force between two parallel wires are fundamental concepts in electromagnetism. Understanding these principles is crucial for grasping the behavior of electric currents and their interactions. The magnetic field around a long straight wire forms concentric circles, with its strength decreasing as distance from the wire increases. This field follows the right-hand rule, where the thumb points in the direction of current flow, and the fingers curl around the wire in the field's direction. When two parallel wires carry currents, they exert forces on each other due to their magnetic fields. These forces can be attractive or repulsive, depending on the current directions. The visual demonstration of magnetic fields provides a visual demonstration of these concepts, making it easier to comprehend the invisible magnetic fields and their effects. By exploring these electromagnetic phenomena, students can develop a solid foundation for more advanced topics in physics and electrical engineering.

## Magnetic Field Due to a Long Straight Wire

The concept of a magnetic field around a current-carrying wire is fundamental to understanding electromagnetism. When an electric current flows through a wire, it generates a magnetic field in the surrounding space. This field is characterized by invisible lines of force that form concentric circles around the wire, perpendicular to its length.

The circular nature of these field lines is a key feature of the magnetic field produced by a current-carrying wire. Unlike the field lines of a bar magnet, which run from north to south, these circular lines encircle the wire in a plane perpendicular to the current's direction. This circular pattern extends outward from the wire, with the field strength decreasing as the distance from the wire increases.

To determine the direction of these magnetic field lines, we use the right-hand rule. This simple yet powerful tool involves wrapping your right hand around the wire with your thumb pointing in the direction of the current flow. Your fingers will then naturally curl in the direction of the magnetic field lines. This rule provides an easy way to visualize and predict the orientation of the magnetic field around a current-carrying wire.

Several factors affect the strength of the magnetic field produced by a current-carrying wire. The most significant of these are the magnitude of the current and the distance from the wire. As the current increases, the magnetic field strength increases proportionally. Conversely, as you move farther away from the wire, the field strength decreases. This relationship is inverse, meaning that doubling the distance from the wire reduces the field strength by half.

The equation that describes the magnetic field strength (B) at a distance (r) from a long, straight current-carrying wire is:

B = μI / (2πr)

In this equation:

• B is the magnetic field strength, measured in teslas (T)
• μ (mu-zero) is the permeability of free space, a constant equal to 4π × 10 T·m/A
• I is the current flowing through the wire, measured in amperes (A)
• r is the perpendicular distance from the wire to the point where the field is measured, in meters (m)

This equation encapsulates the key relationships we've discussed. The direct proportionality between B and I shows that increasing the current strengthens the magnetic field. The inverse relationship between B and r demonstrates how the field weakens with distance. The presence of μ, the permeability of free space, reflects the fundamental nature of this magnetic phenomenon, independent of the wire's material.

To illustrate these concepts, let's consider some examples. Imagine a straight wire carrying a current of 5 amperes. Using the right-hand rule, we can determine that the magnetic field lines circle the wire counterclockwise when viewed from above if the current is flowing upward. If we measure the magnetic field strength at a distance of 10 cm from the wire, we can use the equation to calculate its value:

B = (4π × 10 T·m/A)(5 A) / (2π(0.1 m)) = 1 × 10 T

Now, if we double the current to 10 amperes while keeping the same distance, the magnetic field strength would also double to 2 × 10 T. Alternatively, if we keep the original current but measure at twice the distance (20 cm), the field strength would halve to 5 × 10 T.

Understanding the magnetic field around a current-carrying wire is crucial in many practical applications of magnetic fields. For instance, this principle is used in the design of electromagnets, electric motors, and transformers. In an electric motor, the interaction between the magnetic field of a current-carrying wire and a permanent magnet creates the rotational force that drives the motor. Similarly, in a transformer, the changing magnetic field around a current-carrying wire induces a voltage in a nearby coil, allowing for the transfer of electrical energy between circuits.

These practical applications of magnetic fields demonstrate the importance of understanding the principles of electromagnetism. By mastering these concepts, we can harness the power of magnetic fields to create innovative technologies and improve existing ones.

## Magnetic Field Midway Between Two Parallel Wires

The magnetic field created by two parallel wires carrying currents in opposite directions is a fascinating phenomenon in electromagnetism. This configuration is commonly encountered in various electrical and electronic applications, and understanding its behavior is crucial for engineers and physicists alike. In this section, we'll explore the characteristics of this magnetic field, learn how to determine its direction, and calculate its strength at the midpoint between the wires.

When two parallel wires carry currents in opposite directions, they each generate their own magnetic field. These individual fields interact to create a complex overall magnetic field pattern in the surrounding space. The strength and direction of this field vary depending on the location relative to the wires.

To determine the direction of the magnetic field for each wire, we can use the right-hand rule. This simple yet powerful technique allows us to visualize the field's orientation based on the current direction. Here's how to apply the right-hand rule for a straight wire:

1. Point your right thumb in the direction of the current flow.
2. Curl your fingers around the wire.
3. The direction your fingers point indicates the magnetic field direction around the wire.

For two parallel wires with opposite currents, applying this rule reveals that the magnetic fields between the wires are in the same direction. This results in a stronger field in the space between the wires compared to the outer regions.

Calculating the total magnetic field at the midpoint between the wires involves vector addition of magnetic fields from each wire. The process can be broken down into the following steps:

1. Calculate the magnetic field strength due to each wire at the midpoint.
2. Determine the direction of each field at the midpoint.
3. Add the field vectors to find the resultant field.

The magnetic field strength (B) at a distance (r) from a long, straight wire carrying a current (I) is given by the equation:

B = (μ * I) / (2π * r)

Where μ is the permeability of free space (4π × 10 T·m/A).

Let's work through a step-by-step example calculation to illustrate this process:

Example: Two parallel wires are separated by a distance of 10 cm. Wire A carries a current of 5 A upward, while Wire B carries a current of 3 A downward. Calculate the magnetic field at the midpoint between the wires.

1. Calculate the distance to the midpoint: r = 10 cm / 2 = 5 cm = 0.05 m
2. Calculate the field due to Wire A: B_A = (4π × 10 * 5) / (2π * 0.05) = 2 × 10 T
3. Calculate the field due to Wire B: B_B = (4π × 10 * 3) / (2π * 0.05) = 1.2 × 10 T
4. Determine field directions: Both fields point in the same direction at the midpoint (perpendicular to the line connecting the wires)
5. Add the fields: B_total = B_A + B_B = 2 × 10 T + 1.2 × 10 T = 3.2 × 10 T

The total magnetic field at the midpoint is 3.2 × 10 T, pointing perpendicular to the line connecting the wires.

Understanding the magnetic field created by parallel wires with opposite currents is essential in many practical applications. This knowledge involves the vector addition of magnetic fields and is crucial for designing and analyzing electrical systems.

## Force Between Two Parallel Current-Carrying Wires

The magnetic force between two parallel current-carrying wires is a fascinating phenomenon in electromagnetism that demonstrates the interaction between electric currents and magnetic fields. This concept is crucial in understanding various applications in electrical engineering and physics. When two wires carry electric currents, they generate magnetic fields around them, and these fields interact with each other, resulting in a force between the wires.

The direction of the currents in the wires plays a pivotal role in determining whether the magnetic force is attractive or repulsive. When the currents in both wires flow in the same direction, the magnetic force between them is attractive, causing the wires to be pulled towards each other. Conversely, when the currents flow in opposite directions, the force becomes repulsive, pushing the wires apart. This principle is fundamental in the design of electromagnetic devices and has significant implications in various technological applications.

The magnitude of this magnetic force can be quantified using the equation for force per unit length between the wires: F/L = (μII) / (2πd). In this equation, F represents the force, L is the length of the wires, μ is the permeability of free space (a constant), I and I are the currents in the first and second wires respectively, and d is the distance between the wires. This equation provides a mathematical framework to analyze and predict the behavior of current-carrying wires in proximity to each other.

Several factors influence the magnitude of the magnetic force between parallel current-carrying wires. The currents in both wires (I and I) have a direct proportional relationship with the force increasing either current will result in a stronger force. The distance between the wires (d) has an inverse relationship with the force; as the wires are brought closer together, the force between them increases. The permeability of the medium (μ) also affects the force, though this is typically constant in most practical scenarios involving air or vacuum.

To illustrate the concept of attractive force, consider two parallel wires carrying currents in the same direction. For example, if both wires carry a current of 10 amperes in the upward direction and are separated by a distance of 1 centimeter, they will experience an attractive force pulling them towards each other. This scenario is commonly observed in the design of electromagnetic coils and transformers, where the attractive force helps maintain the structural integrity of the device.

On the other hand, a repulsive force scenario can be observed when the currents in the parallel wires flow in opposite directions. For instance, if one wire carries a current of 5 amperes upward while the adjacent wire carries 5 amperes downward, and they are separated by 2 centimeters, they will experience a repulsive force pushing them apart. This principle is utilized in various applications, such as in the design of certain types of electric motors and in the construction of high-current carrying bus bars in power distribution systems, where repulsion helps prevent short circuits.

Understanding the magnetic force between parallel current-carrying wires is essential in numerous practical applications. In the field of electrical engineering, this concept is crucial for the design of transformers, electric motors, and power transmission lines. It also plays a significant role in the development of precision measurement instruments, such as the ampere balance, which was historically used to define the ampere as a unit of electric current. Moreover, this principle is fundamental in the study of plasma physics and in the design of particle accelerators, where the control of charged particle beams relies heavily on the manipulation of magnetic forces.

In conclusion, the magnetic force between two parallel current-carrying wires is a complex yet fascinating aspect of electromagnetism. The interplay between current directions, resulting in either attractive or repulsive forces, and the factors influencing the magnitude of these forces, provide a rich field of study with wide-ranging applications. As technology continues to advance, the principles governing this magnetic interaction remain crucial in shaping the future of electrical and electronic systems, from the smallest microchips to the largest power distribution networks.

## Applications and Real-World Examples

The magnetic fields and forces between current-carrying wires have numerous practical applications in various fields of engineering and technology. Understanding these concepts is crucial for developing and improving a wide range of devices and systems. Let's explore some of the most significant applications and real-world examples.

One of the fundamental applications of magnetic fields between current-carrying wires is the definition of the ampere, the SI unit of electric current. The ampere is defined based on the force between two parallel, infinitely long conductors carrying electric current. This definition provides a standardized way to measure and quantify electric current, which is essential for electrical engineering and metrology.

Electromagnetic rail guns represent an exciting application of magnetic forces between current-carrying conductors. These advanced weapons use powerful electromagnetic fields to accelerate projectiles to extremely high velocities. The rail gun consists of two parallel conducting rails connected to a power supply. When a conductive projectile is placed between the rails, it completes the circuit, and the resulting magnetic field accelerates the projectile along the rails. This technology has potential applications in both military and space exploration fields.

Electrical transformers are another crucial application that relies on the principles of magnetic fields between current-carrying wires. Transformers are used to step up or step down voltage levels in power distribution systems. They consist of two or more coils of wire wound around a common core. When an alternating current flows through the primary coil, it creates a changing magnetic field that induces a voltage in the secondary coil. The ratio of turns in the primary and secondary coils determines the voltage transformation. This principle is fundamental to the efficient transmission and distribution of electrical power over long distances.

In the field of electrical motors and generators, the interaction between current-carrying wires and magnetic fields is the basis for their operation. Electric motors convert electrical energy into mechanical energy by utilizing the force exerted on a current-carrying conductor in a magnetic field. Conversely, generators use the principle of electromagnetic induction to convert mechanical energy into electrical energy. These devices are essential in various industries, from manufacturing to renewable energy production.

Magnetic levitation (maglev) technology is another fascinating application of magnetic forces between current-carrying conductors. Maglev trains use powerful electromagnets to levitate and propel the train, eliminating friction with the tracks. This technology allows for high-speed transportation with reduced energy consumption and maintenance requirements. The principles of magnetic levitation are also applied in other fields, such as frictionless bearings and energy storage systems.

In the realm of scientific research, particle accelerators rely on the precise control of magnetic fields to guide and accelerate charged particles. These massive machines use a combination of electromagnets and radio-frequency cavities to accelerate particles to near-light speeds for studying fundamental physics and developing new technologies. The Large Hadron Collider at CERN is a prime example of how understanding magnetic fields and forces between current-carrying wires contributes to groundbreaking scientific discoveries.

The design of electromagnetic actuators and solenoids also heavily relies on the principles of magnetic fields and forces. These devices convert electrical energy into mechanical motion and are used in various applications, from automotive systems to industrial automation. Understanding the relationship between current, magnetic fields, and forces is crucial for optimizing the performance and efficiency of these components.

In conclusion, the practical applications of magnetic fields and forces between current-carrying wires are vast and diverse. From defining fundamental units of measurement to powering advanced transportation systems and enabling cutting-edge scientific research, these principles play a crucial role in shaping our modern technological landscape. As engineers and scientists continue to push the boundaries of what's possible, a deep understanding of these concepts will remain essential for driving innovation and solving complex challenges across various fields of engineering and technology.

## Problem-Solving Techniques for Magnetic Fields and Forces Between Wires

Approaching problems involving magnetic fields and forces between wires requires a systematic method and a solid understanding of fundamental principles. This guide will walk you through the essential steps for solving typical problems in this area of physics, focusing on field strength calculations, force determinations, and analysis of complex wire configurations.

### Step 1: Identify the Problem Type

Before diving into calculations, determine whether you're dealing with field strength at a point, forces between wires, or a more complex wire configuration. This initial assessment will guide your problem-solving approach.

### Step 2: Draw a Diagram

Visualize the problem by sketching a clear diagram. Include all relevant wires, their orientations, current directions, and any points of interest. This visual representation will help you apply the right-hand rule and identify the directions of magnetic fields and forces.

### Step 3: List Given Information and Constants

Write down all provided data, including currents, distances, and any relevant constants like the permeability of free space (μ). Ensure all units are consistent, converting if necessary.

### Step 4: Choose the Appropriate Equation

For field strength calculations, use the Biot-Savart law for single wires or the formula B = (μI)/(2πr) for long, straight wires. When dealing with forces between parallel wires, apply F/L = (μII)/(2πd), where L is the length of the wire segment.

### Step 5: Apply the Right-Hand Rule

Use the right-hand rule to determine the direction of magnetic fields and forces. For current-carrying wires, point your thumb in the direction of the current, and your fingers will curl in the direction of the magnetic field.

### Step 6: Perform Calculations

Substitute the values into your chosen equation and solve for the unknown variable. Pay close attention to units and significant figures.

### Step 7: Analyze Complex Configurations

For problems involving multiple wires or complex geometries, apply the principle of superposition. Calculate the field or force contribution from each wire separately, then combine them vectorially to find the total effect.

Verify that your result makes physical sense and has the correct units. Compare the magnitude of your answer to expected values for similar problems.

### Tips for Avoiding Common Pitfalls

• Always use SI units in your calculations to avoid errors.
• Be mindful of the direction of currents and their impact on field and force directions.
• For problems involving perpendicular wires, remember that the force on one wire due to the field of another is always perpendicular to both the current and the field.
• When dealing with loops or coils, consider the field at the center using the formula B = (μNI)/(2R) for a circular loop, where N is the number of turns and R is the radius.
• For solenoids, use B = μnI, where n is the number of turns per unit length.

### Practice Problems

To master these techniques, work through a variety of problems, including:

1. Calculating the magnetic field strength at various points around a long, straight wire.
2. Determining the force per unit length between two parallel current-carrying wires.
3. Analyzing the magnetic field at the center of a current-carrying loop or coil.
4. Solving for the force on a wire segment in a non-uniform magnetic field.
5. Calculating the magnetic field inside a long solenoid.

## Conclusion

In this article, we've explored the fundamental concepts of magnetic fields and forces in current-carrying wires, which are crucial for understanding electromagnetic principles. The introduction video provided a visual foundation for grasping these complex ideas. We discussed how moving charges create magnetic fields, the right-hand rule for determining field direction, and the interactions between current-carrying wires. Understanding these principles is essential for various technology applications, from simple electric motors to advanced particle accelerators. We encourage readers to delve deeper into electromagnetic concepts, as they form the backbone of many modern technologies. Further study can lead to exciting discoveries and innovations in fields such as renewable energy, medical imaging, and telecommunications. By mastering these principles, you'll gain valuable insights into the invisible forces shaping our technological world. Continue exploring resources on electromagnetism to expand your knowledge and unlock new possibilities in science and engineering.

### Magnetic Field Due to a Long Straight Wire

Understanding the magnetic field generated by a long straight wire carrying current is fundamental in electromagnetism. This guide will walk you through the step-by-step process of determining the magnetic field around such a wire.

#### Step 1: Introduction to Magnetic Fields Around a Wire

When a wire carries an electric current, it generates a magnetic field around it. This phenomenon is a direct consequence of Ampère's circuital law, which relates the magnetic field in a loop to the electric current passing through the loop. The magnetic field lines form concentric circles around the wire, with the wire at the center.

#### Step 2: Visualizing the Magnetic Field

To visualize the magnetic field, imagine placing a plotting compass around the wire. The compass needle will align tangentially to the circular magnetic field lines. The direction of these field lines can be determined using the right-hand rule: if you point the thumb of your right hand in the direction of the current, your fingers will curl in the direction of the magnetic field lines.

#### Step 3: Determining the Magnetic Field Strength

The strength of the magnetic field (B) at a given point around the wire depends on two main factors: the current (I) flowing through the wire and the distance (r) from the wire. The relationship is given by the formula:
B = (μ / 2π) * (I / r)
Here, μ (mu-zero) is the permeability of free space, a constant value of 4π × 10 T·m/A (tesla meter per ampere).

#### Step 4: Proportionality of Magnetic Field and Current

The magnetic field strength is directly proportional to the current flowing through the wire. This means that if you increase the current, the magnetic field strength will also increase. For example, if the current is doubled from 2 amps to 4 amps, the magnetic field strength will also double.

#### Step 5: Inverse Proportionality of Magnetic Field and Distance

The magnetic field strength is inversely proportional to the distance from the wire. As you move further away from the wire, the magnetic field strength decreases. For instance, if you measure the magnetic field at two points, A and B, where point B is farther from the wire than point A, the magnetic field at point B will be weaker than at point A.

#### Step 6: Combining the Relationships

By combining the direct proportionality to current and the inverse proportionality to distance, we get the complete formula for the magnetic field around a long straight wire:
B = (μ / 2π) * (I / r)
This formula allows you to calculate the magnetic field at any point around the wire, given the current and the distance from the wire.

#### Step 7: Understanding the Permeability of Free Space (μ)

The constant μ, known as the permeability of free space, is crucial in calculating the magnetic field. Its value is 4π × 10 T·m/A. This constant represents the extent to which a magnetic field can penetrate free space and is a fundamental constant in electromagnetism.

#### Step 8: Practical Application

Understanding the magnetic field due to a long straight wire has practical applications in various fields, including electrical engineering and physics. It helps in designing electromagnets, understanding the behavior of current-carrying conductors, and analyzing the forces between parallel wires carrying current.

### FAQs

Here are some frequently asked questions about magnetic fields due to a long straight wire and force between two parallel wires:

1. What is the right-hand rule for determining the direction of a magnetic field around a wire?

The right-hand rule states that if you wrap your right hand around the wire with your thumb pointing in the direction of the current, your fingers will curl in the direction of the magnetic field lines. This rule helps visualize the circular nature of the magnetic field around a current-carrying wire.

2. How does the strength of a magnetic field change with distance from a long straight wire?

The strength of the magnetic field decreases inversely with the distance from the wire. This relationship is described by the equation B = μI / (2πr), where B is the magnetic field strength, μ is the permeability of free space, I is the current, and r is the distance from the wire.

3. What determines whether parallel current-carrying wires attract or repel each other?

The direction of the currents in the wires determines whether they attract or repel. If the currents flow in the same direction, the wires attract each other. If the currents flow in opposite directions, the wires repel each other. This is due to the interaction of their magnetic fields.

4. How is the force between two parallel current-carrying wires calculated?

The force per unit length between two parallel current-carrying wires is given by the equation F/L = (μII) / (2πd), where F is the force, L is the length of the wires, μ is the permeability of free space, I and I are the currents in each wire, and d is the distance between the wires.

5. What are some practical applications of the magnetic fields and forces between current-carrying wires?

Practical applications include electric motors, generators, transformers, electromagnetic rail guns, maglev trains, and particle accelerators. These concepts are also fundamental in defining the ampere (the SI unit of electric current) and in various precision measurement instruments used in scientific research and industry.

### Prerequisite Topics

Understanding the fundamental concepts that lay the groundwork for more advanced topics in physics is crucial for students aiming to master complex subjects like "Magnetic field due to a long straight wire & force between two parallel wires." One of the most essential prerequisite topics for this subject is electric currents produce magnetic fields. This foundational concept is pivotal in comprehending the intricate relationship between electricity and magnetism, which forms the basis of electromagnetism.

The principle that electric currents generate magnetic fields is a cornerstone in electromagnetic theory. When studying the magnetic field due to a long straight wire, students must first grasp how moving charges (electric current) create a magnetic field around the conductor. This understanding is crucial because it explains the origin of the magnetic field that surrounds a current-carrying wire, which is the primary focus of the main topic.

Moreover, the concept of vector addition of magnetic fields is particularly relevant when analyzing the force between two parallel wires. As each wire produces its own magnetic field, the interaction between these fields results in a force that can be attractive or repulsive, depending on the direction of the currents. Students who have a solid understanding of how to add magnetic field vectors will find it much easier to visualize and calculate these interactions.

The prerequisite topic also introduces students to the right-hand rule, a fundamental tool used to determine the direction of magnetic fields around current-carrying conductors. This rule is indispensable when working with long straight wires and parallel conductors, as it helps predict the direction of the magnetic field and the resulting forces between wires.

Furthermore, understanding how electric currents produce magnetic fields lays the groundwork for more advanced concepts such as Ampère's law and Biot-Savart law, which are often used to calculate the strength and direction of magnetic fields around various current-carrying configurations, including long straight wires.

By mastering this prerequisite topic, students will be better equipped to tackle the complexities of magnetic fields due to long straight wires and the forces between parallel conductors. They will have the necessary tools to visualize the magnetic field lines, understand their behavior, and calculate their effects on neighboring current-carrying wires. This foundational knowledge not only facilitates a deeper understanding of the main topic but also prepares students for more advanced studies in electromagnetism and its numerous applications in modern technology and engineering.

In this lesson, we will learn:

• Magnetic field due to a long straight wire
• Magnetic field midway between two currents
• Forces between two parallel wires

Notes:

An electric current produces a magnetic field
• The magnetic field surrounding the electric current in a long straight wire is such that the field lines are circles with the wire at the center.
• The field strength at a given point would be greater if the current flowing in the wire were greater; $B \propto I$
• The filed strength would be less at points farther from the wire $B \propto \frac{1}{r}$;

• $\quad B \propto I$
$\quad B \propto \frac{1}{r} \qquad B \propto \frac{1}{r} \quad \Rightarrow \quad B =$ $\large \frac{\mu_{0}}{2 \pi} \frac{I}{r}$,

The value of the constant $\mu_{0}$, which is called the permeability of free space, is $\mu_{0}$ = 4$\pi$ × 10-7 $T.m/A$

Magnetic Field Midway Between Two Currents

Two parallel wires 10.0cm apart carry currents in opposite directions. Current $I_{1}$ = 5.0A is out of the page, $I_{2}$=7.0 A is into the page. Determine the magnitude and direction of the magnetic field halfway between the two wires.

$B_{1} =$ $\large \frac{\mu_{0} I_{1}} {2\pi r} = \frac{(4\pi \, \times \, 10^{-7} \, T\, . \, m/A) (5. 0A) } {2 \pi (0.050m) }$ = 2.0 × 10-5 $T$

$B_{2} =$ $\large \frac{\mu_{0} I_{2}} {2\pi r} = \frac{(4\pi \, \times \, 10^{-7} \, T\, . \, m/A) (7. 0A) } {2 \pi (0.050m) }$ = 2.8 × 10-5 $T$

The total filed is up with the magnitude of

$B = B_{1} +B_{2} = 4.8 \times 10^{-5} T$

Forces Between Two Parallel Wires
• Consider two long parallel wires separated by a distance $d$. They carry currents $I_{1}$ and $I_{2}$, respectively. Each current produces a magnetic field that is felt by the other, so each must exert a force on the other.

• Magnetic field $B_{1}$ produced by $I_{1}, \, B_{1} =$ $\large \frac{\mu_{0} I_{1}} {2 \pi d}$
• The force $F_{2}$ is exerted by $B_{1}$ on a length of $I_{2}$ of wire 2, carrying current $I_{2}$, $F_{2}$ = $I_{2}$ $B_{1}$ $I_{2}$
• Substitute $B_{1}$ into $F_{2}$ formula to get the final equation;

• $F_{2} =$ $\large \frac{\mu_{0}} {2 \pi} \frac{I_{1} I_{2}} {d} I_{2}$

• Parallel currents in the same direction exert an attractive force on each other
• Antiparallel currents (in opposite directions) exert a repulsive force on each other.