Problem 30 - Master Resource Book in JEE Main Physics

Question

If \(L\) denotes the inductance of an inductor through which a current \(I\) is flowing, then the dimensional formula of \(L I^{2}\) is (a) \(\left[\mathrm{MLT}^{-2}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (c) \(\left[\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right]\) (d) not expressible in terms of \(M, L, T\)

Step-by-Step Solution

Answer

The correct answer is (b) \([\text{ML}^2 \text{T}^{-2}] \).

Step 1: Recall the Dimensional Formula for Inductance

The dimensional formula for inductance, denoted by \( L \), is \( \left[ \text{ML}^2\text{T}^{-2}\text{I}^{-2} \right] \). This formula is derived from the energy stored in an inductor being equal to \( \frac{1}{2} L I^2 \), and relating this energy to other dimensions.

Step 2: Determine Dimensional Formula for Current

The current, denoted by \( I \), has a dimensional formula \( [\text{I}] \). This means that when we consider \( I^2 \), its dimensional formula will be \( [\text{I}^2] \).

Step 3: Combine Inductance and Current Squared

To find the dimensional formula of \( L I^2 \), multiply the dimensional formula of \( L \) by that of \( I^2 \). Thus, it becomes: \[ \text{Dimensional formula of } (L I^2) = [\text{ML}^2\text{T}^{-2}\text{I}^{-2}] \times [\text{I}^2] = [\text{ML}^2\text{T}^{-2}] \].

Step 4: Match with Given Options

Compare the result \( [\text{ML}^2\text{T}^{-2}] \) with the provided options. The option matching this dimensional formula is (b) \( [\text{ML}^2 \text{T}^{-2}] \).

Key Concepts

Dimensional Formula

The term "dimensional formula" is a key concept in physics, as it helps display the relationship between basic physical quantities like mass, length, and time. They provide insight into the way certain quantities are measured and observed. Dimensional formulas are typically expressed using symbols. For instance:

• \( [ ext{M}] \) represents mass
• \( [ ext{L}] \) for length
• \( [ ext{T}] \) for time
• \( [ ext{I}] \) for electric current

Dimensional analysis uses these formulas to ensure that mathematical equations are dimensionally consistent. For example, when considering the dimensional formula of quantities like inductance \( L \), expressed as \([\text{ML}^2\text{T}^{-2}\text{I}^{-2}]\), it becomes easier to manipulate them in equations. This formula indicates the relationship of inductance with energy, as it relates to the squared current passing through a device.

Inductance

Inductance is a fundamental concept in the study of electromagnetism. It refers to the property of an electrical conductor by which a change in current through it induces an electromotive force, either in the conductor itself or in a nearby conductor by mutual inductance. The unit of inductance is expressed in henrys (H).

The dimensional formula for inductance is \([\text{ML}^2\text{T}^{-2}\text{I}^{-2}]\). This reflects how inductance correlates with energy. For instance, the energy stored in an inductor is given by \( \frac{1}{2} L I^2 \), illustrating how inductance is connected to the magnetic energy when current flows. This formula helps in deducing how inductance, when multiplied by the square of current, yields a dimension of \([\text{ML}^2\text{T}^{-2}]\), excluding the current's dimension.

Understanding inductance is essential as it applies to various real-world applications, such as transformers, inductive sensors, and radio frequency transmission, making it core to the design of circuits.

Current

Current is one of the foundational elements in electricity. It refers to the flow of electric charge across a circuit and is measured in amperes \((A)\), showcasing how much charge is passing through the circuit over a specific period. The concept of current can be simplified by imagining water flowing through a pipe—a perfect analogy for how electrical current flows through wires.

Dimensional formula for current is simply \([\text{I}]\), representing how it is a basic unit in electric measurements. When you consider the square of the current \( I^2 \), the dimensional analysis becomes \([\text{I}^2]\). This aspect is important when analyzing energy relationships like in inductors, where current squared turns into energy terms.

Understanding the physics of current is integral for everyday applications like designing electrical appliances or analyzing circuits since it deeply impacts how devices operate and are modeled.