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When considering the concept of electric force, we're diving into the fundamental interaction of charges. Electric force, known as Coulomb's force, is what attracts or repels any two charged objects. It's governed by Coulomb's Law, which states that the force (\(F\)) between two charges (\(Q_1\) and \(Q_2\)) is directly proportional to the product of the charges and inversely proportional to the square of the distance (\(r\)) between them.

The formula for electric force is given by:\[ F = \frac{k \cdot Q_1 \cdot Q_2}{r^2} \]Here \(k\) is Coulomb's constant, related to the permittivity of free space (\(\varepsilon_0\)) as \(k = \frac{1}{4\pi \varepsilon_0}\). This highlights how permittivity affects the magnitude of the electric force. The greater the permittivity, the weaker the force for a given charge and separation.

In dimensional analysis, electric force carries the dimensions of:

• Mass - \(M\)
• Length - \(L\)
• Time - \(T\)

The dimensional formula for force is therefore \([M L T^{-2}]\). Understanding these dimensions enables us to assess the compatibility of equations and verify their correctness based on fundamental physical quantities.

Magnetic force is a force exerted by magnets when they attract or repel each other. This force is a result of the motion of electric charges, and is perceptible in the case of current-carrying conductors. The formula associated with magnetic force between two long, parallel current-carrying wires is \(F = \frac{\mu_0 I_1 I_2 L}{2\pi r}\), where \(I_1\) and \(I_2\) are the currents in the wires, \(L\) is the length of the wire segments, \(r\) is the distance between the wires, and \(\mu_0\) is the permeability of free space.

**Key aspects of magnetic force:**

• Directly proportional to the product of the currents.
• Inversely proportional to the distance between the wires.
• Affected by the magnetic permeability of the medium.

Dimensional analysis of magnetic force leads us to realize that it shares the same fundamental dimensions as electric force, \([M L T^{-2}]\). Moreover, understanding these dimensions helps confirm the validity of the expressions used in magnetic force calculations in terms of mass, length, time, and current.

Permittivity \(\varepsilon_0\) and permeability \(\mu_0\) are essential constants in physics that describe how electric and magnetic fields interact within a medium. Permittivity measures how much electric field "permitts" itself through a space, affecting how charges interact. It is crucial in determining the strength of the electric force between charges. Significantly, in a vacuum or free space, the permittivity is denoted as \(\varepsilon_0\)\ and influences calculations along with Coulomb's constant.

Permeability, on the other hand, measures how easily a magnetic field can penetrate a material. In the context of vacuum, it's represented by \(\mu_0\), which is termed "magnetic permeability of free space." It plays a critical role in defining the magnetic force and in the workings of electromagnetic devices like transformers and solenoids.

**Dimensional Roles:**

• Permittivity (\(\varepsilon_0\)): \([M^{-1} L^{-3} T^{4} I^{2}]\)\
• Permeability (\(\mu_0\)): \([M^1 L^1 T^{-2} I^{-2}]\)

The expression \(\frac{1}{\sqrt{\mu_0 \varepsilon_0}}\) shows how these constants balance to provide dimensions of a wave speed (specifically, the speed of light in vacuum), which is \([I^0 M^0 L^1 T^{-1}]\). This links back to their pivotal roles in electromagnetism and dimensional analysis, showing their effects in the formulas governing electric and magnetic interactions.