Problem 36 - Master Resource Book in JEE Main Physics

Question

Match the following of Column I with Column II. Column I Column II I. Lorentz force A. \(\oint E \cdot \mathrm{d} \mathbf{A}=\frac{q}{\varepsilon_{0}}\) II. Gauss's law B. \(\quad \mathbf{d B}=\frac{\mu_{0}}{4 \pi} \frac{i \mathbf{d} l \times \mathbf{r}}{r^{3}}\) III. Biot-Savart law C. \(\quad \mathbf{F}=q(\mathbf{E}+(\mathbf{v} \times \mathbf{B}))\) IV. Coulomb's Law D. \(\quad F=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r^{2}}\)

Step-by-Step Solution

Answer

I-C, II-A, III-B, IV-D

Step 1: Understanding Lorentz Force

Recall that the Lorentz force is the force acting on a charged particle moving in an electromagnetic field. Its mathematical representation is given by: \( \mathbf{F}=q(\mathbf{E}+(\mathbf{v} \times \mathbf{B})) \). This correlates with option C from Column II.

Step 2: Understanding Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. Its formula is represented as: \( \oint E \cdot \mathrm{d} \mathbf{A}=\frac{q}{\varepsilon_{0}} \). This corresponds to option A from Column II.

Step 3: Understanding the Biot-Savart Law

The Biot-Savart Law describes how a magnetic field is generated by a steady electric current. The formula for this law is: \( \mathbf{d B}=\frac{\mu_{0}}{4 \pi} \frac{i \mathbf{d} l \times \mathbf{r}}{r^{3}} \). This matches option B from Column II.

Step 4: Understanding Coulomb's Law

Coulomb's Law describes the electrostatic interaction between two charges. Its formula is: \( F=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r^{2}} \). This is associated with option D from Column II.

Key Concepts

Lorentz Force

The Lorentz force is a fundamental concept in electrodynamics that describes how charged particles interact with electric and magnetic fields. It combines both the effects of electric fields and magnetic fields on a charged particle. When a charged particle, like an electron, enters an electromagnetic field, it experiences a force that is determined by the equation: \( \mathbf{F}=q(\mathbf{E}+(\mathbf{v} \times \mathbf{B})) \).
This formula tells us:

• \( q \) is the charge of the particle.
• \( \mathbf{E} \) is the electric field.
• \( \mathbf{v} \) is the velocity of the particle.
• \( \mathbf{B} \) is the magnetic field.

The force exerted by the electric field is \( q\mathbf{E} \), acting in the direction of the field. The magnetic part \( q(\mathbf{v} \times \mathbf{B}) \) depends on the velocity of the particle and is perpendicular to both \( \mathbf{v} \) and \( \mathbf{B} \).
Lorentz force is crucial for understanding the motion of charged particles in fields, which is key in devices like cyclotrons and cathode ray tubes.

Gauss's Law

Gauss's Law is an essential principle for understanding electric fields around charged objects. It links the electric flux passing through a surface to the charge enclosed within that surface. Mathematically, it is expressed as:\[ \oint E \cdot \mathrm{d} \mathbf{A}=\frac{q}{\varepsilon_{0}} \].
In this equation:

• \( \oint \) denotes a closed surface integral.
• \( \mathbf{E} \) is the electric field vector.
• \( \mathrm{d} \mathbf{A} \) is an infinitesimal area element on the closed surface.
• \( q \) is the total enclosed charge.
• \( \varepsilon_{0} \) is the permittivity of free space, a constant.

Gauss's Law simplifies complex electric field calculations, especially in cases with high symmetry such as spherical, planar, and cylindrical symmetries. It's a powerful tool for calculating the resultant electric fields produced by diverse charge distributions.

Biot-Savart Law

The Biot-Savart Law is pivotal in magnetostatics and explains how magnetic fields are generated by currents. It's a way to calculate the magnetic field produced by a small current element. The full expression is:\[ \mathbf{d B}=\frac{\mu_{0}}{4 \pi} \frac{i \mathbf{d} l \times \mathbf{r}}{r^{3}} \].
Here's what these symbols represent:

• \( \mathbf{d B} \) is the differential magnetic field.
• \( \mu_{0} \) is the permeability of free space.
• \( i \) is the current flowing through the wire.
• \( \mathbf{d} l \) is a small segment of the wire carrying current \( i \).
• \( \mathbf{r} \) is the position vector from the current element to the point where the magnetic field is measured.

The Biot-Savart Law helps us understand how currents produce magnetic fields, which is critical for the design of electromagnets, transformers, and various electrical devices.

Coulomb's Law

Coulomb's Law is foundational for electrostatics, providing insight into the forces between charged objects. It defines the force between two point charges and is conveyed through:\[ F=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r^{2}} \].
The components of this formula include:

• \( F \) is the electrostatic force between the charges.
• \( q_{1} \) and \( q_{2} \) are the magnitudes of the two charges.
• \( r \) is the distance between the charges.
• \( \varepsilon_{0} \) is the permittivity of free space.

Coulomb's Law indicates that the force is attractive if the charges have opposite signs and repulsive if the charges have the same sign. Understanding this law is fundamental in developing insights into the behavior of electric fields and potentials resulting from multiple charges. It's essential in fields ranging from chemistry to material science.