Problem 39 - Master Resource Book in JEE Main Physics

Question

Column I gives three physical quantities. Select the appropriate units for the choice given in Column II. Some of physical quantities may have more than one choice correct. Column I Column II A. Capacitance (p) Ohm-second B. Inductance (q) Coulmb \(^{2}\)-joule \(^{-1}\) C. Magnetic induction (r) Coulomb (volt) \(^{-1}\) (s) Newton (Ampere meter) \(^{-1}\) (t) Volt second (Ampere) \(^{-1}\) (a) \(\mathrm{q} \quad \mathrm{p} \quad \mathrm{s}\) (b) \(\mathrm{q} \quad \mathrm{r} \quad \mathrm{p}\) (c) \(\mathrm{r} \quad \mathrm{s} \quad \mathrm{p}\) (d) \(\mathrm{r}\) t \(\mathrm{q}\)

Step-by-Step Solution

Answer

None of the options fully match the derived unit correspondences; check for matching errors in setup.

Step 1: Understanding the Units

First, let's analyze the options in Column II and see what physical quantities or units they might correspond to.\( \text{Ohm-second} \) is a unit derived from Ohm's law, typically related to inductance.\( \text{Coulomb}^{2} \text{joule}^{-1} \) is used in the context of electrical potential energy and capacitance.\( \text{Coulomb (volt)}^{-1} \) is a unit of capacitance, where capacitance \( C = \frac{Q}{V} \) (Charge per Voltage).\( \text{Newton (Ampere meter)}^{-1} \) is a unit related to magnetic properties.\( \text{Volt second (Ampere)}^{-1} \) is generally used for magnetic induction and is linked directly with the concept of magnetic flux.

Step 2: Matching Capacitance

Identify the correct unit for Capacitance from the options.Capacitance is measured in Farads, which can be expressed as Coulomb per Volt. Thus, the correct unit is \( \text{Coulomb (volt)}^{-1} \). Therefore, the correct match for Capacitance (A) is (r).

Step 3: Matching Inductance

Identify the correct unit for Inductance from the options. Inductance, measured in Henry, relates time and resistance, so Ohm-second can be a valid unit for this. However, magnetic flux linkage (Volt-second per Ampere) is also used in this context. Therefore, the correct match for Inductance (B) is (p) and (t).

Step 4: Matching Magnetic Induction

Identify the correct unit for Magnetic Induction from the options. Magnetic Induction, or Magnetic Field, is measured in Teslas, which can be expressed in terms of Newton per Ampere-meter. So, (s) is the correct unit. It can also relate to (t) as it involves the interaction of Volt-second.

Step 5: Choosing Correct Answer Set

Based on the matching of physical quantities and units, we have:- Capacitance (A) corresponds to (r).- Inductance (B) corresponds to (p) and (t).- Magnetic Induction (C) corresponds to (s) and (t).Reviewing the answer choices, option (d) \((r)\; t \; (q)\) does not match any correct relationship we've built. Option (c) \((r)\; (s)\; (p)\) does not fit perfectly with any either. Option (b) \((q)\;(r)\;(p)\) seems plausible but doesn't cover our full matches. Therefore, none of the answer options comprehensively cover the answers. Reviewing the matches, option (a): \((q)\; (p)\; (s)\) doesn't fit at all.

Key Concepts

Capacitance

In the fascinating world of physics, capacitance is a key concept that describes an object's ability to store electrical charge. This characteristic is vital in understanding how various electronic circuits function and is measured in Farads, abbreviated as F. As we unravel its definition, capacitance is primarily about the ratio of the electric charge (Q) stored on each conductor to the potential difference (V) between them. This relationship is mathematically expressed as \( C = \frac{Q}{V} \).
Capacitors are the components used to exploit this property. Think of capacitors as energy reservoirs ready to provide charge when needed. They are used extensively in electronic devices for tasks such as power supply stabilization and signal filtering.
If we're delving into units, in our exercise context, capacitance can be beyond Farads, sometimes expressed as Coulomb per Volt, or \( \text{Coulomb (volt)}^{-1} \). This alternative unit offers us a direct glimpse into how capacitors relate to charge and voltage, confirming their fundamental role in electric fields and circuits.

Inductance

Inductance plays an instrumental role in electromagnetism and electronics. It is the property of a conductor by which a change in current flowing through it induces an electromotive force (emf) across itself or nearby conductors. This is thanks to the magnetic field that forms around the conductor when electric current flows through it.
Measured in Henrys (H), inductance is a central feature in coils or inductors, components that are designed to harness this property. These devices are ubiquitous in many electrical applications like transformers, radio receivers, and transmitters.

• Ohm-second is one way to quantify inductance, relating time (second) and resistance (Ohm), owing to their connectedness through electromagnetic induction.
• Alternatively, the unit Volt-second per Ampere reflects how the induced voltage relates to changes in current over time.

Understanding inductance is crucial for designing circuits that rely on magnetic fields to function efficiently and smoothly.

Magnetic Induction

Magnetic induction, also known as magnetic flux density, refers to how a magnetic field generates electric fields, and it's a cornerstone of electromagnetics and several practical applications. It encapsulates the notion of a magnetic field's strength and its ability to induce electric currents in a conductor.
This phenomenon is measured in Teslas (T), emphasizing the intensity of the magnetic field with respect to force and current. We typically represent magnetic induction with the symbol \( B \), and it forms the basis of many technologies, from electric motors to inductive charging pads.

• Units such as Newton per Ampere-meter \( \text{(N (A\cdot m})^{-1}} \) signify how the force is distributed over current and distance, presenting a clear depiction of the field's interaction with electric currents.
• Volt-second per Ampere is another unit reflecting the linkage between electricity and magnetism, often observed in situations involving changes in the magnetic environment.

By grasping how magnetic induction works, we're better prepared to understand and innovate with the magnetic and electric components that power modern technology.