What is the dimension of physical quantities?
Let us first understand what are physical quantities? A physical quantity is the attribute of a measuring substance or system. A value is the algebraic multiplication of a numerical value and a unit to express a physical quantity. The physical quantity mass, for example, may be expressed as n kg, where n denotes the numerical value and kg denotes the unit. At least two properties are shared by all physical quantities. The first is numerical magnitude, while the second is the unit of measurement.ISO/IEC 80000, the IUPAP red book, and the IUPAC green book set out international recommendations on the usage of symbols representing quantities. The preferred physical quantity mass sign, for example, is m while the suggested electric charge symbol is Q.
Physical quantities may have several “sizes,” such as the scalar, vector or tensor.
Scaler:In physical science, a scalar or scalar quantity can be defined by an element in a numerical field like a real number, typically accompanied with measuring units, as in the case of ’10 cm.’ This contrasts with the numbers of vectors, tensors, etc, describing their magnitude, direction and so on. In simple words, a scalar is a physical quantity of magnitude without direction. Symbols denoting physical amounts are generally used in the Latin or Greek alphabet to consist of a single letter and are displayed in italics.
The idea of a physics scalar is fundamentally the same as a mathematical scalar. The co-ordinate system transforms formally unchanged a scalar. This means that rotations or reflections maintain scalar in classical theories, such in Newtonian mechanics, whereas relativistic theories preserve scalar transformations or space-time translations.
Vector: A vector is an element of a vector space in mathematics and physics.The vectors have certain names, which are mentioned below, for many certain vector areas.
Historically, before the formalization of the idea of vector space, vectors have been introduced in geometry and physics (usually mechanics). Therefore, one speaks commonly about vectors without mentioning their vector space. In a specific case, spatial vectors are considered in a Euclidean space, also called Euclidean vectors, which may be added, subtracted and scaled (i.e., multiplied by a real number), as a quantity representing both a magnitude and a direction.
Tensors: In mathematics, the tensor is an algebraic object describing a (multilinear) connection in a vector space between the sets of algebraic objects. Objects between vectors and scalars, and even other tensors, are mapped by tensors. Many forms of tensors are available, including scalars and vectors (the most elementary tensors), dual vectors, multi-linear mappings of vectors, and even dot operations. Tensors are defined independently of any basis, even if their components are typically referred to on the basis of a specific coordination system.Tensors have become essential in physics because in fields like mechanics, general relativity and others they give a clear mathematical foundation for defining and resolving physics issues. In applications, circumstances where a distinct tensor may arise at each place of an item are frequent to be studied; for example, stress in an object might change from one position to another. The idea of a tensor field leads to this. Tensor fields are so prevalent in some sectors that they are often referred to simply as “tensors.”
Dimensions of physical quantities:
In 1822, Joseph Fourier established the concept of the dimension of a physical quantity. By convention, the physical quantities are arranged into a dimensional system, each of which consists of its own dimension, based on basic or base quantities.
The base quantitiesare those which are distinct in nature and have not been specified in other numbers in some circumstances historically. The base amounts are the quantities that can be represented by other quantities. The following table lists the seven basic quantities and associated SI units and dimensions of the International System of Quantities (ISQ). There may be various baseline units of other conventions (e.g., the CGS and MKS systems of units).
1)length
2)mass
3)time
4)electric current
5)temperature
6)amount of substance
7)luminous intensity.
The term “dimensional formulation” is used to indicate the power to which the basic units have to be increased in order to get a unit of a derived quantity.
If P is the unit of a derived quantity of P = XaYbZc, XaYbZc is termed the dimensional formula and the exponents a, b and c are called the dimensions.
It is necessary that we utilize [] to write a physical quantity dimension. Everything is inscribed in actual life in terms of mass, duration and temporal dimensions. See some of the following examples:
- Distance
For mass, “m” its dimension is [M], for times,”t” it is [T].
Now, we know that,
Distance = (Velocity) x (Time)
Here, we can write distance as length = [L]
And, Velocity = [L/T]
Time = [T]
(Velocity)(Time) = [T][L/T] = [L] = Distance
- Volume = Length × Breadth × Height
Volume = [L] × [L] × [L]
Volume = [L]3