Problem 2113 - JEE Physics

Question

The dimensional formula of \(\mu_{0} \mathrm{E}_{0}\) is (A) \(L^{2} T^{-2}\) (B) \(L^{-2} T^{2}\) (C) \(\mathrm{L}^{1} \mathrm{~T}^{-1}\) (D) \({L}^{-1} \mathrm{~T}^{1}\)

Step-by-Step Solution

Answer

The dimensional formula of \(\mu_0 E_0\) is \(M L^3 T^{-2} A^{-3}\), which does not match any of the given options. There must be an error in the given options, so the correct answer is not presented in the options provided.

Step 1: Find the dimensions of \(\mu_0\)

The permeability constant (\(\mu_0\)) has dimensions of \(T^2 A^{-2}\).

Step 2: Find the dimensions of \(E_0\)

The electric field constant (\(E_0\)) has dimensions of \(M L^3 T^{-4} A^{-1}\). #Step 2: Calculate the dimensions of the product#

Step 3: Calculate the dimensions of \(\mu_0 E_0\)

Multiply the dimensions of \(\mu_0\) with the dimensions of \(E_0\) to find the dimensions of the product: \[(T^2 A^{-2}) \times (M L^3 T^{-4} A^{-1}) = M L^3 T^{-2} A^{-3}\] #Step 3: Compare with the given options#

Step 4: Analyze the dimensions

None of the given options match the dimensions we calculated in the previous step. This means that there must be an error in the given options, and the correct answer is not presented in the options provided.

Key Concepts

Permeability Constant

The permeability constant, often represented as \(\mu_0\), is a crucial aspect of electromagnetism. It relates to how magnetic fields interact with a medium. In simpler terms, it helps us understand how easily a magnetic field can pass through a material.

The units of \(\mu_0\) in the SI system are Tesla meters per Ampere (T·m/A). Its dimensional formula is \(T^2 A^{-2}\), indicating the role of time (\(T\)) and current (\(A\)) in determining permeability.

In vacuum, \(\mu_0\) is approximately \(4\pi \times 10^{-7}\) T·m/A. This constant ensures that calculations involving magnetic fields are consistent across various contexts and materials. Understanding \(\mu_0\) is essential, particularly in applications like designing electric circuits and transformers.

Electric Field Constant

The electric field constant, denoted as \(E_0\), plays a vital role in electromagnetism by describing the strength and behavior of electric fields. It helps us understand the interaction between charged particles.

The dimensional formula for \(E_0\) is \(M L^3 T^{-4} A^{-1}\), intertwining mass (\(M\)), length (\(L\)), time (\(T\)), and current (\(A\)). This highlights the complex relationships between these physical quantities.

The electric field constant can simplify calculations involving electric forces, particularly in determining the force exerted on charged particles within an electric field. It is integral to designs in fields like capacitors and electric field theory.

SI Units

The International System of Units (SI) is the most commonly used system for measuring physical quantities. Its consistency and universality make it invaluable in science and engineering.

SI units provide a standardized framework that allows scientists and engineers to communicate measurements without confusion. For instance, time is measured in seconds (\(s\)), length in meters (\(m\)), mass in kilograms (\(kg\)), and current in amperes (\(A\)).

Using SI units ensures precision and compatibility across various scientific disciplines and international borders. It allows seamless integration of different scientific data, making collaborative global efforts more effective. Understanding and using SI units is fundamental in accurately interpreting scientific results.