## What is Dimension in Physics

**Definition of Dimension in Physics:** Dimension in physics refers to measurable attributes that define the extent and properties of objects and events in our universe. They include familiar spatial dimensions (length, width, height) and dimensions in other physical quantities like time, mass, and charge.

The dimension of a physical quantity shows the application of the method of fundamental quantities like **Mass (M), Length (L), and Time (T)** to relate the fundamental units of a physical quantity.

In dimension, we derive quantities by applying physical quantities. Understanding dimension, demands understanding the method of measurements, fundamental quantities, derived quantities, and even their corresponding units in physics.

By reading this article, readers can expect to gain a deeper understanding of what dimension in physics truly represent, how to calculate it, how they extend beyond the three spatial dimensions, and their role in shaping the fabric of reality.

Moreover, the article will explore the implications of higher dimensions, compactification, and how these concepts contribute to the unification of forces in the universe.

By the end of this journey, readers will not only comprehend the multidimensional nature of our existence but also appreciate how this understanding has profound implications for our comprehension of the cosmos and its many enigmas.

The dimension of a physical quantity allows us to find the relationship between quantities. As we mentioned earlier, we apply Mass in form of M, Length in form of L, and Time in form of T.

Additionally, units like temperature will be in form of K or θ, electric current in form of (A) which is an expression for ampere, and finally luminous intensity as (II).

## Table of Dimensions in Physics

QUANTITY | DIMENSION |

Speed | LT^{-1} |

Velocity | LT^{-1} |

Acceleration | LT^{-2} |

Force | MLT^{-2} |

Tensile stress | ML^{-1}T^{-2} |

Elastic Force | MLT^{-2} |

Young Modulus | ML^{-1}T^{-2} |

Work done in elastic spring/string | ML^{2}T^{-2} |

Work | ML^{2}T^{-2} |

Work done horizontal | ML^{2}T^{-2} |

Work done vertical | ML^{2}T^{-2} |

Kinetic energy | ML^{2}T^{-2} |

Potential energy | ML^{2}T^{-2} |

Power | ML^{2}T^{-3} |

Projectile time of flight | T |

Projectile maximum height | L |

Projectile range | L |

Moment of force | ML^{2}T^{-2} |

Density | ML^{-3} |

Upthrust | MLT^{-2} |

Centripetal acceleration | LT^{-2} |

Centripetal force | MLT^{-2} |

Linear velocity | LT^{-1} |

Linear acceleration | LT^{-2} |

Period | T |

Frequency | T^{-1} |

Period of simple pendulum | T |

Momentum | MLT^{-1} |

Impulse | |

Pressure | ML^{-1}T^{-2} |

Work done by expanding gas | ML^{2}T^{-2} |

Wavelength | M |

Wave velocity | LT^{-1} |

Gravitational Force | MLT^{-2} |

Gravitational potential | L^{2}T^{-2} |

Escape velocity | LT^{-1} |

Electric force | MLT^{-2} |

Magnetic force | MLT^{-2} |

## Understanding Dimension of Physical Quantities

Dimensions in physics refer to the basic parameters that define the extent and properties of objects and events in our three-dimensional reality.

These dimensions include the familiar spatial dimensions of length, width, and height, as well as dimensions in other physical quantities, such as time, mass, charge, and more.

When combined, they constitute what is known as the spacetime continuum, a four-dimensional structure that underlies all physical phenomena.

The knowledge of dimensions is very important in various physical theories. These theories, which include General Relativity, Quantum Mechanics, and many others, form the foundation of our understanding of the universe.

Therefore, dimensions play a central role in these theories, influencing the behavior of particles, the curvature of spacetime, and the interactions between forces.

Without a pure understanding of dimension, it would be impossible to understand the workings of the cosmos and the profound laws governing it.

The purpose of this article is to shed light on the concept of dimensions in physics and delve into its significance across various theoretical frameworks.

As we embark on this exploration of dimensions in physics, prepare to venture beyond the boundaries of everyday experience and into the extraordinary vistas of theoretical possibilities.

Through this knowledge, we can unlock the mysteries of the universe and glimpse the harmonious interplay between dimensions that governs the majestic dance of creation. Let us now embark on this intellectual odyssey and unravel the secrets that dimensions hold in the mesmerizing tapestry of our universe.

*You may also like to read:*

SS1 Lesson Note: Introduction to Physics For First Term

## How to Calculate Dimensions of Physical Quantity

To calculate the dimensions of a physical quantity, we need to understand that the measurement of quantities in units of Length, Mass, and Time involves the dimension Length (L), Mass (M), and Time (T) respectively.

To find the dimension of any quantity, we substitute the units of the quantity into L, M, and T. You insert the symbols into a bracket in this way: [L], [M], and [T]. For example, to find the dimension of speed, we first write the formula as

**Speed = Distance / Time**

and we know that the unit for

Distance = meter (m) and meter is a length, which can be

Dimension of Distance = [L]

Note: We only have m which is [L]. If we have m^{2}, we can now write [L^{2}]

and Time is in seconds which can also be T

Thus, the dimension of Time = T

To derive the dimension of speed, we say:

Speed = Distance / Time

This implies that

Speed = [L] / [T]

Therefore, the dimension of Speed = [LT^{-1}] or [L][T^{-1}]

## Worked Examples on Dimension in Physics

Here are examples to make you understand dimension in physics: how to calculate dimensions of physical quantities

### Example 1

- Find the dimension of an Area and Volume.

**Solution:**

For Area = Length x Breadth

Which shows that the dimension of Area = [L] x [L] = [L^{2}]

Now, since volume = Length x Base x Height

The dimension of volume = [L] x [L] x [L] = [L^{3}]

### Example 2

2. Deduce the dimension of (a) force and (b) Pressure

**Solution:**

(a) force = mass x acceleration

which shows that

Dimension of force = dimension of mass x dimension of acceleration

and the Dimension of mass = [M]

**while** the Dimension of acceleration = Dimension of Velocity / Dimension of Time

and Dimension of Velocity = Dimension of Displacement [L] / Dimension of Time [T]

Thus, Dimension of Velocity = [L] / [T] = [L] [T^{-1}] or [LT^{-1}]

Therefore, by substituting the dimensions of velocity and time into acceleration,

we have

dimension of acceleration = [LT^{-1}] / [T] = [LT^{-2}]

Now, to finally derive the acceleration of force,

we say

Dimension of force = [M] x [LT^{-2}] = [M] [LT^{-2}] or [MLT^{-2}]

(b) To find the dimension of pressure

We apply the formula pressure = force / area

which means Dimension of pressure = Dimension of force / Dimension of area

and Dimension of force = Dimension mass x Dimension of acceleration

you can remember that we earlier calculated the dimension of force as [MLT^{-2}]

Also, Dimension of area = [L] x [L] = [L^{2}]

Thus, Dimension of pressure = [MLT^{-2}] / [L^{2}]

This implies

Dimension of pressure = [ML^{-1}T^{-2}]

### Example 3

Find the dimension of the following:

a. work

b. Velocity

c. Linear acceleration

d. Power

e. Pressure

f. Momentum

g. Impulse

**Solution:**

**a.** Work = Force x Distance

Dimension of Work = Dimension of Force x Dimension of Distance

Therefore, Dimension of Work = MLT^{-2} x L

This implies that, Dimension of work = ML^{2} T^{-2}

**b.** Velocity = Displacement / Time = L / T = LT^{-1}

**c.** Acceleration = Velocity / Time = LT^{-1} / T = LT^{-2}

**d.** Power = work done / time = ML^{2} T^{-2} / T = ML^{2} T^{-3}

**e.** Pressure = Force / Area = MLT^{-2} / L^{3} = ML^{-2}T^{-2}

**f.** Momentum = mass x velocity = M x LT^{-1} = MLT^{-1}

**g.** Impulse = Force x Time = MLT^{-2} x T = MLT^{-1}

### Example 4

Find the dimension in the electrical system of the following:

**a.** Electric charge

**b. **Electrical Potential (voltage)

**c.** Capacitance (c)

**Solution:**

**a.** To find electric charge, we use the formula

electric charge = current (A) x time (T)

Thus, electric charge = [A] x [T] = [AT]

**b.** Electric potential (voltage) = power / current

Dimension voltage = ML^{2} T^{-3} / A = [MA^{-1}L^{2} T^{-3}]

**c.** Capacitance = charge / potential = [AT] / [MA^{-1}L^{2} T^{-3}]

Thus, capacitance = [M^{-1}A^{2}L^{-2} T^{4}]

## Dimensional Constants

A dimensional constant, also known as a fundamental physical constant, is a fixed numerical value that characterizes certain fundamental aspects of the physical universe. These constants play a crucial role in scientific equations and theories, providing a consistent framework for understanding the laws of physics. They remain invariant across different systems of measurement and are essential for making accurate scientific predictions and comparisons.

One well-known dimensional constant is the speed of light, denoted as “c,” which is approximately 299,792,458 meters per second in a vacuum. This constant is fundamental in Einstein’s theory of relativity and underpins our understanding of space and time.

Another example is Planck’s constant, denoted as “h,” which plays a central role in quantum mechanics, describing the relationship between the energy and frequency of photons and other quantum particles.

These constants are extensively documented and utilized in scientific research and engineering applications. For a comprehensive list of dimensional constants and their significance, you can refer to the National Institute of Standards and Technology (NIST) website: NIST Fundamental Constants. This resource provides up-to-date values and detailed information on these constants, serving as a reliable reference for scientists and researchers worldwide.

## Dimensionless Quantities

Dimensionless quantities, also known as dimensionless numbers or pure numbers, are mathematical ratios or constants that express a relationship between two physical quantities without any units of measurement. They play a crucial role in various branches of science and engineering by providing insights into the fundamental nature of physical phenomena, independent of the specific units used for measurement.

One well-known dimensionless quantity is the Reynolds number (Re), which characterizes the flow of fluids. It is the ratio of inertial forces to viscous forces and helps classify fluid flow regimes, such as laminar or turbulent.

Another example is the Mach number (Ma), which describes the speed of an object relative to the speed of sound in a medium. It is vital in aerodynamics and aerospace engineering.

To explore a comprehensive list of dimensionless quantities and their applications, you can refer to the National Aeronautics and Space Administration (NASA) website: NASA Glenn Research Center Dimensionless Numbers. This resource provides valuable information and examples of dimensionless numbers, illustrating their significance in scientific and engineering contexts.

### Understanding Dimension

Imagine you have a special pair of glasses that let you see and understand everything around you in a magical way. Well, dimensions are like those special glasses that physicists wear to understand how our universe works!

A. **What are Dimensions?** In the context of physics, dimensions are like special measuring sticks that help scientists describe and make sense of everything they see and study in the world. Just like you use a ruler to measure the length of a toy or a book, dimensions are used to measure different aspects of everything around us.

B. **Different Types of Dimensions** Now, here’s the exciting part: there are different types of dimensions! The ones you probably know well are the **spatial dimensions**. These are like the directions you can move in—up and down, left and right, and forward and backwards. We use these three spatial dimensions to figure out where things are in our world.

But guess what? There are other types of dimensions too! For example, there’s the **time dimension**. Time helps us understand when things happen, like when your favourite TV show starts or how long it takes for your ice cream to melt.

Then, there are even more mysterious dimensions, like those related to **mass** and **charge**. These special dimensions tell us how heavy something is (mass) or how it attracts or repels other things (charge).

C. **Let’s Talk About “n-Dimensional” Spaces** Now, close your eyes and imagine a world where you could move not just in three directions but many more—like up, down, left, right, forward, backward, and even more ways! That’s what we call an “n-dimensional” space. The “n” stands for the number of directions you can move.

**How Are These Dimensions Represented Mathematically?**

Now, we use some special math to show these dimensions. Imagine you have a treasure map, and you need to give the exact location of the hidden treasure. You would use numbers and symbols, right? Well, that is what we do with spatial dimensions!

For **length**, we use a number line that goes from left to right. The left side is like a big “0,” and the right side is like a big number, let’s say “10.” So, if your toy car is at “3” on the number line, it means it’s three units away from the “0” point.

For **width**, we have another number line that goes from front to back. The front side is “0,” and the back side might be “6.” So, if your favorite teddy bear is at “4” on this line, it’s four units away from the front.

For **height**, it’s like a number ladder going up and down. The floor is “0,” and maybe the ceiling is “5.” So, if your kite is floating at “2,” it’s two units above the floor.

C. **Real-World Examples of Each Spatial Dimension** Let us see how these dimensions help us in real life! Imagine you are building a cool Lego house. You need to know the length, width, and height of each brick to make the house just right!

For length, you use a ruler to measure how long each brick is. For width, you measure how wide the brick is. And for height, you stack the bricks up to see how tall your Lego tower is getting!

**Higher Dimensions**

Just like we have length, width, and height, there are more dimensions that exist beyond our normal senses. We call them higher dimensions, and they’re like secret passages that lead us to new and amazing places in the universe!

Picture this: If you had special glasses that let you see in four, five, or even more directions, you’d discover a whole new way to explore the world! These extra dimensions may be a bit tricky to imagine, but that’s where the magic of math and imagination comes in!

**Using Math and Imagination to Understand Higher Dimensions**

Physicists, who are like the greatest adventurers of all, use a special kind of math to understand these higher dimensions. It’s like having a treasure map that guides them on this exciting quest! With math, they can dream up wild and wondrous worlds beyond what we can see with our eyes.

But wait, how can you explore something you can’t see? That’s where imagination comes to the rescue! You know how you imagine flying on a dragon or exploring a mystical castle? It’s just like that! Physicists use their big imaginations to create vivid pictures of these hidden dimensions in their minds.

**Theories of Higher Dimensions**

We will now have a look at some extraordinary theories that talk about these higher dimensions:

**Kaluza-Klein Theory:**This theory suggests that there might be a fifth dimension, invisible to us, but it’s wrapped up so tiny that we can’t notice it. It’s like a sneaky mouse hiding in a tiny hole!**String Theory:**Hold on tight, because this theory is mind-blowing! It says that everything in the universe is made up of tiny, vibrating strings, like musical notes playing together. And guess what? To make the math work, string theory needs not three, not four, but a whopping ten dimensions!**Multidimensional Universes:**Imagine a bunch of soap bubbles floating in the air, each one a separate universe. Some scientists think that there might be many universes, each with its own set of dimensions, existing side by side.

These theories help scientists understand the deepest secrets of the universe and imagine what incredible things might be out there.

### Frequently Asked Questions

Certainly, here are some frequently asked questions (FAQs) related to the concept of “dimension” in physics:

**What is the meaning of “dimension” in physics?**

- In physics, dimension refers to the fundamental physical quantities or properties that describe the nature of space and time.

**How many dimensions are there in our universe?**

- Our universe is commonly described as having three spatial dimensions (length, width, height) and one temporal dimension (time), making it four-dimensional.

**What is meant by “dimensional analysis” in physics?**

- Dimensional analysis is a mathematical technique used in physics to check the consistency of equations and to derive relationships between physical quantities based on their dimensions.

**Can there be more than four dimensions in physics?**

- Some theoretical physics models, such as string theory and certain cosmological theories, suggest the possibility of additional spatial dimensions beyond the familiar three, but these are not directly observable in our everyday experience.

**How do dimensions play a role in units of measurement?**

- Dimensions are crucial in forming units of measurement. For example, distance is measured in meters (m), which is a unit derived from the dimension of length (L).

**What is a dimensionless quantity in physics?**

- A dimensionless quantity is a numerical value that expresses a physical relationship without any units. It represents a ratio of two quantities with the same dimensions and is often used to characterize various physical phenomena.

**Can you explain the concept of “spacetime dimensions” in physics?**

- Spacetime dimensions combine the three spatial dimensions and the temporal dimension into a four-dimensional continuum, where events in the universe are described in terms of their positions in this spacetime framework.

**How do physicists use dimensions to formulate and solve problems?**

- Physicists use dimensional analysis to check the correctness of equations and derive new equations or relationships between physical quantities. It helps simplify complex problems and ensures that units are consistent in equations.

Reference