91Ó°ÊÓ

The speed of light, represented as \( c \), is an essential constant in physics. It is approximately \( 299,792,458 \) meters per second in a vacuum. This speed is a fundamental limit, meaning that nothing can travel faster than light in a vacuum. The expression \( c = \frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}} \) gives us insight into how electromagnetic forces govern the speed of light. Here, \( \mu_{0} \) is the permeability of free space, and \( \varepsilon_{0} \) is the permittivity of free space. Together, they define how electromagnetic waves, such as light, propagate through empty space. The dimensions of speed are \([M^{0}L^{1}T^{-1}]\), where:

• \( M^{0} \) refers to no dependency on mass,
• \( L^{1} \) implies a direct dependency on length (in meters, since light travels a certain distance),
• \( T^{-1} \) indicates an inverse dependency on time (as speed is measured as distance per time unit).

Understanding these dimensions helps clarify why options relating speed dimensions to \( \frac{E}{B} \) make sense in physics problems.

Electric fields, denoted as \( E \), are vector fields representing the electric force exerted per unit charge at every point in space. An electric field encompasses how charged particles interact at a distance. The dimensional formula for an electric field is \([M^{1}L^{1}T^{-3}A^{-1}]\). This illustrates the fundamental physical quantities that define electric fields:

• \( M^{1} \): Dependence on mass, emphasizing that electric fields relate to forces (which depend on mass),
• \( L^{1} \): Dependency on length, given that the field acts over distances,
• \( T^{-3} \): Indicates the relation to temporal changes in electric fields,
• \( A^{-1} \): Shows the dependency inversely on current (amperes), linking electric fields to the flow of charge.

These dimensions are integral when comparing to other fields, like magnetic fields, helping determine ratios such as \( \frac{E}{B} \). This understanding is useful to understand the exercise we've tackled, where it confirms the dimensional analysis.

Magnetic fields, denoted as \( B \), describe the magnetic force experienced by moving charged particles or magnetic poles. Like electric fields, magnetic fields are a cornerstone of electromagnetism. The dimensional formula of a magnetic field is \([M^{1}L^{0}T^{-2}A^{-1}]\). Let's break this down:

• \( M^{1} \): Implies the dependency on mass, as magnetic forces are a type of mechanical force,
• \( L^{0} \): Indicates no dependency on physical distance in its definition,
• \( T^{-2} \): Signifies a strong relation with time, showing how quickly magnetic fields change influences their effects,
• \( A^{-1} \): Reveals the inverse relationship with electrical current, essential in electromagnetism.

Knowing these dimensions provides a foundation to comprehend how magnetic fields interact with other entities, like electric fields, in physics problems. This understanding aids in determining how \( \frac{E}{B} \) matches the speed of light's dimensional analysis, enhancing our grasp of electromagnetic waves.