A dimensional equation is a mathematical expression that shows how a physical quantity relates to the seven base units of physics.

What are Dimensions?

In physics, every quantity is built from seven base units. These are length (L), mass (M), time (T), electric current (A), temperature (K), luminous intensity (cd), and amount of substance (mol). A dimension represents the power to which these base units are raised. For example, area is length multiplied by length, so its dimension is L². We write this in square brackets as [L²].

Writing Dimensional Equations

A dimensional equation is formed by equating a physical quantity with its dimensional formula. For instance, velocity is defined as distance divided by time. Distance has the dimension [L] and time has the dimension [T]. Therefore, the dimensional equation for velocity is v = [M⁰L¹T⁻¹]. The exponents (0, 1, and -1) show exactly how much of each base unit is inside the physical quantity.

Uses of Dimensional Equations

• Checking Correctness: You can check if a formula is correct by seeing if the dimensions on the left side match the right side. This is called the Principle of Homogeneity. If both sides match, the formula is likely correct.
• Unit Conversion: These equations help you change a value from one unit system to another, such as converting meters to centimeters.
• Deriving Formulas: If you know which quantities affect a result, you can use these equations to find the relationship between them.

A Real-World Example

Consider the formula for Force, which is mass multiplied by acceleration. Mass has the dimension [M]. Acceleration is velocity change over time, which gives it dimensions of [LT⁻²]. When you multiply them, you get the dimensional equation for Force: [F] = [M¹L¹T⁻²]. This tells us that Force depends equally on mass and length, but is affected inversely by the square of time.