Problem 27 - Master Resource Book in JEE Main Physics

Question

Which of the following combinations have the dimensions of time? \(L-C \cdot R\) represents inductance, capacitance and resistance respectively? (a) \(R C\) (b) \(\sqrt{L C}\) (c) \(R / C\) (d) \(C / L\)

Step-by-Step Solution

Answer

Options (a) RC and (b) \(\sqrt{LC}\) have the dimensions of time.

Step 1: Identify Dimensions of Given Quantities

Identify the dimensions of inductance (\(L\)), capacitance (\(C\)), and resistance (\(R\)) using fundamental units.\- The dimension of inductance is \ \[ [L] = [M][L]^2[T]^{-2}[A]^{-2} \] \- The dimension of capacitance is \ \[ [C] = [M]^{-1}[L]^{-2}[T]^4[A]^2 \] \- The dimension of resistance is \ \[ [R] = [M][L]^2[T]^{-3}[A]^{-2} \]

Step 2: Calculate Dimensions of Each Option

For each option, calculate the dimensions based on the fundamental dimensions of \(L\), \(C\), and \(R\):\- **Option (a) \(RC\):** Dimensions are \ \[ [R][C] = ([M][L]^2[T]^{-3}[A]^{-2})([M]^{-1}[L]^{-2}[T]^4[A]^2) = [T] \] \- **Option (b) \(\sqrt{LC}\):** Dimensions are \ \[ \sqrt{[L][C]} = \sqrt{([M][L]^2[T]^{-2}[A]^{-2})([M]^{-1}[L]^{-2}[T]^4[A]^2)} = [T] \] \- **Option (c) \(R/C\):** Dimensions are \ \[ [R]/[C] = ([M][L]^2[T]^{-3}[A]^{-2})/([M]^{-1}[L]^{-2}[T]^4[A]^2) = [M]^2[L]^4[T]^{-7}[A]^{-4} \] \- **Option (d) \(C/L\):** Dimensions are \ \[ [C]/[L] = ([M]^{-1}[L]^{-2}[T]^4[A]^2)/([M][L]^2[T]^{-2}[A]^{-2}) = [M]^{-2}[L]^{-4}[T]^6[A]^4 \]

Step 3: Identify Options with Dimensions of Time

Review the calculated dimensions from Step 2 to determine which options have the dimensions of time \([T]\).\- **RC** has dimensions [T].\- **\(\sqrt{LC}\)** has dimensions [T].\Both Option (a) and (b) have dimensions of time.

Step 4: Analyze Mismatch Options

Review other options to ensure they clearly do not represent dimensions of time.\- **Option (c)** has dimensions \([M]^2[L]^4[T]^{-7}[A]^{-4}\).\- **Option (d)** has dimensions \([M]^{-2}[L]^{-4}[T]^6[A]^4\).\These options do not simplify to time.

Key Concepts

Inductance

Inductance is like storing energy in a magnetic field. It happens when electrical current flows through a coil. Picture it like a coil in an electric motor or transformer. When current changes, a magnetic field forms around it. This coil tries to keep the current steady by opposing the change.

The SI unit of inductance is the henry (H), which can be broken down in terms of fundamental units:

• Inductance Dimensions: dimension of inductance is defined as dimensions as \([L] = [M][L]^2[T]^{-2}[A]^{-2}\), meaning it involves mass (M), length (L), time (T), and current (A).

When examining problems involving inductance, always keep in mind how it interacts with other components, such as resistors and capacitors. It's all about how the change in current affects the circuit and how the magnetic field opposes those changes.

Capacitance

Capacitance is all about storing electrical charge. It's like a bucket for holding electricity. Capacitors use two plates separated by an insulator. When connected to a voltage, they store charge like a rechargeable battery, but for short-term use.

The fundamental unit of capacitance is the farad (F). Let’s break it down:

• Capacitance Dimensions: The dimensions are given by \([C] = [M]^{-1}[L]^{-2}[T]^4[A]^2\). This tells you it involves inverse mass (-1), inverse length (-2), time to the fourth power (T^4), and current squared (A^2).

Capacitors affect circuits by storing and releasing energy. The quicker the voltage changes, the more effect the capacitor will have. Understanding this process is crucial when analyzing circuits, especially in preventing components from damage due to sudden voltage changes.

Resistance

Resistance is like the difficulty electricity faces when traveling through a wire. Think of it as friction for electric current. It's why your phone gets warm when charging.

Resistors are devices used to control the amount of current in a circuit, with units measured in ohms (Ω). Here's what resistance is made of in fundamental terms:

• Resistance Dimensions: dimensions of resistance are given by \([R] = [M][L]^2[T]^{-3}[A]^{-2}\), showing involvement of mass (M), length (L), inverse time cubed (-3), and inverse current squared (-2).

You can alter resistance by changing the materials or the dimensions of a wire, like using thicker wires for less resistance. Understanding resistance ensures that electrical devices work properly without overheating or breaking.