Consider Two Parallel Wires Where The Magnitude Of The Left Current Is 2 I O 2I_o 2 I O ​ And That Of The Right Current Is I O I_o I O ​ . Point A Is Midway Between The Wires, And Point B Is An Equal Distance On The Other Side Of The Wires.What Is The

Introduction

When two parallel wires carry electric currents, they interact with each other through the magnetic field generated by the currents. In this scenario, we have two parallel wires where the magnitude of the left current is $2I_o$ and that of the right current is $I_o$. We need to determine the magnetic field at point A, which is midway between the wires, and point B, which is an equal distance on the other side of the wires.

The Magnetic Field

The magnetic field generated by a current-carrying wire is given by the Biot-Savart law. The law states that the magnetic field at a point due to a current element is proportional to the current, the length of the element, and the sine of the angle between the element and the point. The direction of the magnetic field is given by the right-hand rule.

Magnetic Field at Point A

Point A is midway between the wires, so the distance from each wire to point A is equal. Let's denote this distance as $r$. The magnetic field at point A due to the left wire is given by:

$$B_{A,left} = \frac{\mu_0}{4\pi} \cdot \frac{2I_o}{r}$$

The magnetic field at point A due to the right wire is given by:

$$B_{A,right} = \frac{\mu_0}{4\pi} \cdot \frac{I_o}{r}$$

Since the wires are parallel, the magnetic fields at point A due to each wire are in the same direction. Therefore, the total magnetic field at point A is the sum of the magnetic fields due to each wire:

$$B_A = B_{A,left} + B_{A,right} = \frac{\mu_0}{4\pi} \cdot \frac{3I_o}{r}$$

Magnetic Field at Point B

Point B is an equal distance on the other side of the wires, so the distance from each wire to point B is also equal. Let's denote this distance as $r$. The magnetic field at point B due to the left wire is given by:

$$B_{B,left} = \frac{\mu_0}{4\pi} \cdot \frac{2I_o}{r}$$

The magnetic field at point B due to the right wire is given by:

$$B_{B,right} = \frac{\mu_0}{4\pi} \cdot \frac{I_o}{r}$$

Since the wires are parallel, the magnetic fields at point B due to each wire are in the same direction. Therefore, the total magnetic field at point B is the sum of the magnetic fields due to each wire:

$$B_B = B_{B,left} + B_{B,right} = \frac{\mu_0}{4\pi} \cdot \frac{3I_o}{r}$$

Comparison of Magnetic Fields

We can see that the magnetic fields at point A and point B are equal. This is because the distance from each wire to point A and point B is the same, and the magnetic fields due to each wire are in the same direction.

Conclusion

In conclusion, the magnetic field at point A and point B due to two parallel wires carrying currents of $2I_o$ and $I_o$ is given by:

$$B_A = B_B = \frac{\mu_0}{4\pi} \cdot \frac{3I_o}{r}$$

The magnetic fields at point A and point B are equal due to the symmetry of the situation.

References

• Biot-Savart law
• Magnetic field due to a current-carrying wire
• Right-hand rule

Further Reading

• Magnetic field due to multiple current-carrying wires
• Interaction between parallel wires
• Applications of magnetic fields in physics and engineering
  Frequently Asked Questions (FAQs) About Parallel Wires and Magnetic Fields ====================================================================

Q: What is the Biot-Savart law?

A: The Biot-Savart law is a mathematical formula that describes the magnetic field generated by a current-carrying wire. It states that the magnetic field at a point due to a current element is proportional to the current, the length of the element, and the sine of the angle between the element and the point.

Q: What is the right-hand rule?

A: The right-hand rule is a method used to determine the direction of the magnetic field generated by a current-carrying wire. To use the right-hand rule, point your thumb in the direction of the current and your fingers will curl in the direction of the magnetic field.

Q: How do parallel wires interact with each other?

A: Parallel wires interact with each other through the magnetic field generated by the currents. The magnetic field due to one wire can affect the current in the other wire, causing it to change direction or magnitude.

Q: What is the magnetic field at point A and point B due to two parallel wires carrying currents of $2I_o$ and $I_o$?

A: The magnetic field at point A and point B due to two parallel wires carrying currents of $2I_o$ and $I_o$ is given by:

$$B_A = B_B = \frac{\mu_0}{4\pi} \cdot \frac{3I_o}{r}$$

Q: Why are the magnetic fields at point A and point B equal?

A: The magnetic fields at point A and point B are equal due to the symmetry of the situation. The distance from each wire to point A and point B is the same, and the magnetic fields due to each wire are in the same direction.

Q: What are some applications of magnetic fields in physics and engineering?

A: Magnetic fields have many applications in physics and engineering, including:

• Magnetic resonance imaging (MRI)
• Magnetic storage devices (hard drives)
• Electric motors and generators
• Magnetic levitation (maglev) trains
• Particle accelerators

Q: What are some common mistakes to avoid when working with parallel wires and magnetic fields?

A: Some common mistakes to avoid when working with parallel wires and magnetic fields include:

• Not considering the direction of the magnetic field
• Not accounting for the distance between the wires and the point of interest
• Not using the correct units for the magnetic field
• Not considering the effects of other magnetic fields on the system

Q: How can I learn more about parallel wires and magnetic fields?

A: There are many resources available to learn more about parallel wires and magnetic fields, including:

• Textbooks on electromagnetism and physics
• Online courses and tutorials
• Research papers and articles
• Professional conferences and workshops

Conclusion

In conclusion, parallel wires and magnetic fields are an important topic in physics and engineering. By understanding the Biot-Savart law, the right-hand rule, and the interaction between parallel wires, you can gain a deeper appreciation for the complex relationships between electric currents and magnetic fields.