Write Coulomb’s law in vector form and show that it obeys Newton’s third law of motion.

Coulomb's law states the electrostatic force between two charged particles, and it can be expressed in vector form as follows:

Let's consider two point charges, q1 and q2, located at positions r1 and r2, respectively. The force (F) experienced by q2 due to q1 can be given by Coulomb's law:

F = k * (q1 * q2) * (r12 / |r12|^3),

where:

F is the electrostatic force vector experienced by q2,
k is the electrostatic constant (k = 1 / (4πε₀), where ε₀ is the vacuum permittivity),
q1 and q2 are the magnitudes of the charges,
r12 = r2 - r1 is the vector pointing from the position of q1 to q2,
|r12| represents the magnitude of the vector r12.
Now let's examine how Coulomb's law obeys Newton's third law of motion:

According to Newton's third law, for every action, there is an equal and opposite reaction. In the context of Coulomb's law, this means that if q1 exerts a force on q2, then q2 must also exert an equal and opposite force on q1.

To demonstrate this, let's calculate the force (F21) experienced by q1 due to q2:

F21 = k * (q2 * q1) * (r21 / |r21|^3),

where:

F21 is the electrostatic force vector experienced by q1,
r21 = r1 - r2 is the vector pointing from the position of q2 to q1.
Now, let's compare F with F21:

F = k * (q1 * q2) * (r12 / |r12|^3),
F21 = k * (q2 * q1) * (r21 / |r21|^3).

We can observe that the magnitudes of the charges q1 and q2 are the same in both expressions, and the vectors r12 and r21 have the same magnitude but opposite directions (r12 = -r21). Thus, we can rewrite F21 as:

F21 = -k * (q1 * q2) * (r12 / |r12|^3).

Comparing F and F21, we see that they have the same magnitudes but opposite directions, which confirms that Coulomb's law obeys Newton's third law of motion. The force exerted by q1 on q2 is equal in magnitude and opposite in direction to the force exerted by q2 on q1.