Problem 55 - Master Resource Book in JEE Main Physics

Question

In the formula, \(a=3 b c^{2}, a\) and \(c\) have dimensions of electric capacitance and magnetic induction respectively. What are dimensions of \(b\) in MKS system? (a) \(\left[\mathrm{M}^{-3} \mathrm{~L}^{-2} \mathrm{~T}^{4} \mathrm{Q}^{4}\right]\) (b) \(\left[\mathrm{M}^{-3} \mathrm{~T}^{4} \mathrm{Q}^{4}\right]\) (c) \(\left[\mathrm{M}^{-3} \mathrm{~T}^{3} \mathrm{Q}\right]\) (d) \(\left[\mathrm{M}^{-3} \mathrm{~L}^{2} \mathrm{~T}^{4} \mathrm{Q}^{-4}\right]\)

Step-by-Step Solution

Answer

None of the provided options match the calculated dimensions.

Step 1: Identify Dimensions of Known Quantities

First, let's identify the dimensions of the given quantities. The dimension of electric capacitance (\( a \)) is \([\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^{4} \mathrm{Q}^2]\) and the dimension of magnetic induction (\( c \)) is \([\mathrm{M}^{1} \mathrm{T}^{-2} \mathrm{Q}^{-1}]\).

Step 2: Write the Formula with Dimensions

The given formula is \( a = 3bc^2 \). Using the dimensions, we have:\[ [a] = [b][c]^2 \] or \[ [\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^{4} \mathrm{Q}^2] = [b]\cdot [\mathrm{M}^{1} \mathrm{T}^{-2} \mathrm{Q}^{-1}]^2 \]

Step 3: Calculate Dimensions of \( c^2 \)

Since \([c] = [\mathrm{M}^{1} \mathrm{T}^{-2} \mathrm{Q}^{-1}]\), we have:\[ [c]^2 = ([\mathrm{M}^{1} \mathrm{T}^{-2} \mathrm{Q}^{-1}])^2 = [\mathrm{M}^{2} \mathrm{T}^{-4} \mathrm{Q}^{-2}] \]

Step 4: Solve for Dimensions of \( b \)

Substitute \([c]^2\) into the dimension equation:\[ [\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^{4} \mathrm{Q}^2] = [b]\cdot [\mathrm{M}^{2} \mathrm{T}^{-4} \mathrm{Q}^{-2}] \]Rearrange to solve for \([b]\):\[ [b] = \frac{[\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^{4} \mathrm{Q}^2]}{[\mathrm{M}^{2} \mathrm{T}^{-4} \mathrm{Q}^{-2}]} \]\[ = [\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^{4} \mathrm{Q}^2] \cdot [\mathrm{M}^{-2} \mathrm{T}^{4} \mathrm{Q}^{2}] \]Simplifying, we get:\[ [b] = [\mathrm{M}^{-3} \mathrm{L}^{-2} \mathrm{T}^{8} \mathrm{Q}^4] \]

Step 5: Match the Answer to Options Given

Compare with provided options:(a) \([\mathrm{M}^{-3} \mathrm{L}^{-2} \mathrm{T}^{4} \mathrm{Q}^{4}]\)(b) \([\mathrm{M}^{-3} \mathrm{T}^{4} \mathrm{Q}^{4}]\)(c) \([\mathrm{M}^{-3} \mathrm{T}^{3} \mathrm{Q}]\)(d) \([\mathrm{M}^{-3} \mathrm{L}^{2} \mathrm{T}^{4} \mathrm{Q}^{-4}]\)The calculated dimension \([\mathrm{M}^{-3} \mathrm{L}^{-2} \mathrm{T}^{8} \mathrm{Q}^4]\) does not match exactly any of the provided options.

Key Concepts

Electric Capacitance

Electric capacitance is a concept that relates to the ability of a system to store electric charge. It essentially measures how much charge can be stored per unit voltage. A common example to think about is a capacitor, which is a device specifically designed to store electrical energy. In some more practical terms, you can think of electric capacitance as the potential to "hold" electric charge in an electrical field.

When dealing with dimensions in physics, each quantity can be expressed in terms of fundamental dimensions like mass (M), length (L), time (T), and charge (Q). For electric capacitance, the dimensions are \([\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^{4} \mathrm{Q}^{2}]\). This might seem daunting at first, but it essentially breaks down into components that fundamentally describe how electric capacitance behaves in physical terms.

• Mass (M): Indicates that there is an inverse mass dependency. More mass typically means lower capacitance, inversely speaking.
• Length (L): An inverse length squared component is usually due to the geometry of the capacitor.
• Time (T): The longer the time element, the higher the energy storage capacity.
• Charge (Q): Directly proportional to how much charge can be held.

Understanding these dimension components helps us understand the behavior of capacitors in practical electronic circuits.

Magnetic Induction

Magnetic induction refers to the process by which a material becomes magnetized in the presence of an external magnetic field. It's a key principle in electromagnetism and explains many natural phenomena, including how Earth’s magnetic field works.

The dimensional representation for magnetic induction is \([\mathrm{M}^{1} \mathrm{T}^{-2} \mathrm{Q}^{-1}]\). This notation can be dissected into broader components of mass, time, and charge, which only enhance our understanding.

• Mass (M): Indicates a direct relationship, meaning the effect increases with mass.
• Time (T): The inverse square of time indicates that as time increases, the force experienced due to magnetic fields decreases.
• Charge (Q): The inverse charge dimension means that the effect weakens with increasing charge quantity.

Magnetic induction is crucial for understanding how magnetic fields are created and manipulated, such as in transformers and electric motors. Recognizing the underlying dimensional attributes guides us in applications such as magnetic resonance imaging (MRI) and electromagnetic field generation.

Dimensions in MKS System

The MKS system, or the meter-kilogram-second system, is a fundamental framework in physics for measuring physical quantities. It provides a base dimension system from which other derived dimensions can be established.

In the given exercise, understanding dimensions in the MKS system is vital to solving the problem of determining the dimensions of a variable, denoted as \(b\). Each physical quantity can be expressed in terms of the fundamental dimensions of mass \(\mathrm{M}\), length \(\mathrm{L}\), time \(\mathrm{T}\), and electric charge \(\mathrm{Q}\) within this system.

Breaking down the dimensions helps clarify relationships within physical formulas:

• Mass (M): Often related to gravitation or inertia in physical laws.
• Length (L): The spatial aspect of the physical quantity being measured.
• Time (T): Accounts for duration and temporal evolution of a process.
• Charge (Q): Fundamental to electrical properties.

In our particular exercise, we use dimensional analysis to balance the equation and determine the missing dimension of \(b\). Hence, becoming comfortable with re-arranging and solving dimensional equations using MKS units leads to more profound insights into physical phenomena.