JEE Main 2025 Physics Dimensions of Magnetic Field Induction Formula and Derivation

Understanding the dimensions of the magnetic field is essential for mastering Physics concepts in JEE Main. The magnetic field is a derived physical quantity that represents the influence of a magnet or moving electric charges in the surrounding region. This page provides a detailed explanation of the dimensional formula of the magnetic field, its derivation, and its significance.

Physical Quantities and Dimensional Analysis

A physical quantity is any quantity that can be measured with a device in terms of units. The entire list of physical quantities can be broadly classified into two types namely fundamental physical quantities and derived physical quantities.

The subsequent sections of this page give a brief description of the basic idea of physical quantities and the dimensional formula for the magnetic field. The page also provides a glimpse of different units of measurement. 

Physical Quantities 

All physical quantities can be broadly classified into two types namely fundamental or base physical quantities and derived physical quantities. There are seven fundamental physical quantities in Physics. They are Mass, time, length, electric current, temperature, luminous intensity, and amount of substance. These seven quantities are called the dimensions of the physical world. Instead of using the names, their first letters are used to represent the base quantities. The three mechanical quantities: mass, length and time are represented as M, L and T. In order to represent the dimensions, these symbols are used within a pair of square brackets.

It is important to note that the dimensions of a physical quantity and the dimensions of their unit are the same. It is only the nature of the unit represented as [M], [L] and [T], and not its magnitude. Every derived physical quantity is represented in terms of its fundamental physical quantities with appropriate power. The powers to which the fundamental physical quantity should be raised to obtain the derived quantity is called the dimensions of the derived quantity.

More about Dimensional Formula and Dimensional Equation

The dimensional formula is a complex expression that narrates which fundamental quantities are involved and how are they involved in modelling the derived physical quantity. The physical quantity and its dimensions are equated in an equation called the dimensional equation. In other words, dimensional equations are the equations that represent the dimensions of any physical quantity in terms of its fundamental quantities.

What is a Magnetic Field?

The region around the magnet, where there is an influence of the force by that magnet, is called the Magnetic field. Magnetic fields can be created by moving electric charges. The magnetic field penetrates through space and acts as a driving force to move electric charges and magnetic dipoles. 

In case of a bar magnet, the magnetic field lines are stronger at the poles and weaker at the middle of the bar magnet. The strength of the magnetic field at both poles is the same. A magnetic field is a derived physical quantity. The dimensional formula of a magnetic field is equal to [M]1 [T]-2 [I]-1. 

What is the Dimensions of Magnetic Field

The dimensions of the magnetic field can be derived from its formula and units. The magnetic field 𝐵 is defined in terms of the force 𝐹 on a charged particle moving with velocity v in the field.

Units and Dimensions of Magnetic Field

Units of Magnetic Field

The magnetic field is measured in tesla (T) in the SI system and in gauss (G) in the CGS system.

Dimensions of Magnetic Field

The magnetic field (B) is related to the force (F) on a moving charge (q):

$F = qvB \sin\theta$

Rearranging for B:

$B = \dfrac{F}{qv \sin\theta}$

Where:

• F (Force): Dimensions = $[M L T^{-2}]$

• q (Charge): Dimensions = $[A T]$,

• v (Velocity): Dimensions = $[L T^{-1}]$.

Substituting these, the dimensions of BBB are:

$[B] = \dfrac{[M L T^{-2}]}{[A T] \cdot [L T^{-1}]} = [M T^{-2} A^{-1}]$

Magnetic Field Dimensional Formula

The dimensional formula of a magnetic field can be defined as the representation of units of a magnetic field in terms of fundamental physical quantities with appropriate power. The dimensional formula of Magnetic field is given as [M]1 [T]-2 [I]-1. In this formula, ‘M’ represents the mass, ‘T’ represents the time, and ‘I’ represents the current.

Derivation of the Dimensional Formula of the Magnetic Field

The Lorentz force is given by the equation,

Force = Charge x Magnetic Field (B) x Velocity

The above equation can be rewritten to find the magnetic field as:

Magnetic field = Force x (Charge x Velocity)-1 → (1)

We know that the velocity can be written in terms of distance and time as:

Velocity = Distance x [Time]-1

The dimensional formula of velocity is [L T-1] → (2)

The basic formula for electric charge is 

Charge = Electric Current x time

The dimensional formula for electric charge is [I1 T1] → (3)

The Lorentz force is given by Newton’s second law of motion as:

Lorentz force = Mass x Acceleration → (4)

But acceleration is the rate of change of velocity and is given as:

Acceleration = Velocity x [Time]-1 → (5)

Substituting (5) in (4), we get

Lorentz force = Mass x Velocity x [Time]-1

The dimensional formula for Lorentz force is thus written as [M L T-2] → (6)

Substituting the dimensions of velocity, electric charge, and Lorentz force from (2), (3) and (6) in the formula for Magnetic field represented as (1), we get,

Therefore B = [M L T-2] {[I1 T1] [L T-1]}-1

B = [M1 L0 T-2 I-1]

Therefore, the magnetic field can be dimensionally represented as [M1 T2 I-1].

Importance of Dimensions Of Magnetic Field Induction Formula in JEE Main Preparation

1. Dimensional Consistency: Helps ensure accuracy in equations and solve complex problems.

2. Formula Validation: Verifies whether derived equations are dimensionally correct.

3. Unit Analysis: Assists in converting one system of units to another, such as CGS to SI units.

4. Quick Problem-Solving: Simplifies the derivation of formulas and helps in solving numerical questions efficiently.

Applications of Magnetic Field Dimensions

• Electromagnetism: Understanding magnetic forces in electric circuits.

• Motors and Generators: Designing systems where magnetic fields play a crucial role.

• Astrophysics: Studying magnetic fields in celestial phenomena.

• Material Science: Analysing the magnetic properties of materials.

Conclusion

The dimensional formula of the magnetic field, $[M^1 T^{-2} I^{-1}]$, is crucial for understanding its relationship with fundamental physical quantities. It simplifies complex derivations and ensures a deeper grasp of electromagnetic concepts, which are essential for JEE Main 2025. For more detailed explanations and study resources, explore Vedantu’s expert-curated content to strengthen your understanding of key Physics topics.

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