The interaction between two moving charges generates a magnetic force, which can be understood through the principles of electromagnetism. When two currents flow in parallel wires, they exert a mutual force on each other, described by the equation:
\[ F = \frac{\mu_0 I_1 I_2 L}{2 \pi r} \]
Here, \( \mu_0 \) is the permeability of free space, \( I_1 \) and \( I_2 \) are the currents, \( L \) is the length of the wire, and \( r \) is the distance between the wires. If the currents flow in the same direction, they attract each other; if they flow in opposite directions, they repel each other. This principle extends to individual moving charges, which also experience a magnetic force based on their velocities and charges.
The magnetic force between two moving charges can be expressed with the formula:
\[ F = \frac{\mu_0 q_1 q_2 v_1 v_2}{4 \pi r^2} \]
In this equation, \( q_1 \) and \( q_2 \) represent the charges, \( v_1 \) and \( v_2 \) are their respective velocities, and \( r \) is the distance between them. The direction of the force is influenced by the signs of the charges and their velocities. There are 16 possible combinations of charge signs and directions, but key takeaways include that like charges moving in the same direction attract, while opposite charges moving in opposite directions also attract. All other combinations result in a repulsive force.
For example, consider an electron moving to the right with a velocity of \( 1 \times 10^8 \, \text{m/s} \) and a proton moving to the left with a velocity of \( 2 \times 10^8 \, \text{m/s} \), separated by a distance of \( 3 \, \mu m \). The magnetic force can be calculated using the aforementioned formula, substituting the appropriate values:
\[ F_B = \frac{\mu_0 (1.6 \times 10^{-19})^2 (1 \times 10^8)(2 \times 10^8)}{4 \pi (3 \times 10^{-6})^2} \]
After performing the calculations, the resulting magnetic force is approximately \( 5.7 \times 10^{-18} \, \text{N} \), indicating an attractive force due to the opposite charges and their directions.
In addition to magnetic forces, the electric force between the same two charges can be calculated using Coulomb's law:
\[ F_E = \frac{k q_1 q_2}{r^2} \]
Where \( k \) is Coulomb's constant (\( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)). Substituting the values yields an electric force of approximately \( 2.56 \times 10^{-17} \, \text{N} \). Comparing the two forces reveals that the electric force is stronger, but only by a factor of about 4.5, illustrating that both forces, while weak, play significant roles in the interactions of charged particles.
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