Two parallel wires carrying current in the same direction attract each other while two beams of electrons travelling in the same direction repel each other. Why?

Explanation

To understand why two parallel wires carrying current in the same direction attract each other while two beams of electrons traveling in the same direction repel each other, we can break down the explanation into a series of steps.

### Step-by-Step Solution:

1. **Understanding Current-Carrying Wires:**
- When an electric current flows through a wire, it generates a magnetic field around it. The direction of this magnetic field can be determined using the right-hand rule.

2. **Magnetic Field Due to a Current-Carrying Wire:**
- For two parallel wires carrying current in the same direction, let’s denote the currents as \( I_1 \) and \( I_2 \). The magnetic field \( B_1 \) created by wire 1 at the location of wire 2 is directed into the plane (using the right-hand rule).
- The magnetic field \( B_2 \) created by wire 2 at the location of wire 1 is also directed into the plane.

3. **Force on the Wires:**
- The force \( F \) on a current-carrying wire in a magnetic field is given by the equation:
\[ F = I \cdot L \cdot B \cdot \sin(\theta) \]
- In this case, since the angle \( \theta \) between the current direction and the magnetic field is 90 degrees, \( \sin(90^\circ) = 1 \). Thus, the force on wire 1 due to the magnetic field created by wire 2 is attractive, and vice versa.

4. **Conclusion for Wires:**
- Therefore, two parallel wires carrying current in the same direction attract each other due to the magnetic fields they create and the forces acting on them.

5. **Understanding Electron Beams:**
- Now, consider two beams of electrons traveling in the same direction. Electrons carry a negative charge and also create a magnetic field due to their motion.

6. **Magnetic Field Due to Electron Beams:**
- Similar to the wires, each beam of electrons generates its own magnetic field. However, since the electrons are negatively charged, the direction of the magnetic field they create is opposite to that of the current in the wires.

7. **Force on the Electron Beams:**
- The force acting on a moving charge in a magnetic field is given by:
\[ F = q \cdot v \cdot B \cdot \sin(\phi) \]
- Here, \( q \) is the charge of the electron, \( v \) is the velocity, and \( B \) is the magnetic field. For two beams of electrons moving in the same direction, the magnetic field created by one beam exerts a repulsive force on the other beam.

8. **Conclusion for Electron Beams:**
- Thus, two beams of electrons traveling in the same direction repel each other due to the magnetic forces acting between them.

### Summary:
- **Attraction of Wires:** Two parallel wires carrying current in the same direction attract each other due to the magnetic fields they create and the forces acting on them.
- **Repulsion of Electron Beams:** Two beams of electrons traveling in the same direction repel each other due to the magnetic fields generated by their motion.