7 Most SAQ’s of Moving Charges and Magnetism Chapter in Inter 2nd Year Physics (TS/AP)

4 Marks

SAQ-1 : State and explain Biot – Savart’s law.

Introduction

Biot-Savart’s Law is a fundamental law in electromagnetism that describes the magnetic field generated by a steady electric current. It is crucial for understanding the relationship between electricity and magnetism, particularly in the context of generating magnetic fields through electric currents.

Explanation of Biot-Savart’s Law

Biot-Savart’s Law states that the magnetic field (B) produced by an element of a current-carrying wire is directly proportional to the current (I) and the length of the wire element (dl), and inversely proportional to the square of the distance from the wire element to the point where the magnetic field is being calculated.

The law is mathematically expressed as

$$\mathbf{B} = \frac{\mu_0}{4\pi} \int \frac{I (d\mathbf{l} \times \hat{\mathbf{r}})}{r^2}$$

where

1. μ_0​ is the permeability of free space.
2. I is the electric current.
3. dl is the infinitesimal length of the current-carrying wire.
4. r^ is the unit vector from the wire element to the point of observation.
5. r is the distance from the wire element to the point of observation.
6. The cross product (dl × r^) signifies that the magnetic field is perpendicular to both the direction of the current and the direction from the current to the point of observation.

Summary

Biot-Savart’s Law is a key principle in electromagnetism that aids in calculating the magnetic field created by a current-carrying conductor. It emphasizes the relationship between the magnitude of the current, the geometry of the current path, and the resultant magnetic field, forming the basis for understanding complex magnetic field distributions in various electrical and magnetic systems.

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SAQ-2 : State and explain Ampere’s law.

Introduction

Ampere’s Circuital Law is a cornerstone of electromagnetism, outlining the relationship between an electric current and the magnetic field it generates. This law is fundamental for understanding magnetic fields and their influence on electric charges and currents.

Understanding Ampere’s Circuital Law

Ampere’s Circuital Law states that the line integral of the magnetic field (B) around a closed path is equal to the product of the magnetic permeability of the medium (μ₀​) and the total electric current (I) passing through the area enclosed by that path. The law is mathematically expressed as:

$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$$

Explaining Ampere’s Circuital Law with an Example

Consider a long, straight conductor carrying an electric current and a circular loop (Ampereian loop) centered at the wire. At any point P on this loop, the magnetic field intensity (B) is uniform as every point is equidistant from the wire. By applying Ampere’s Law, the magnetic field at point P can be calculated as:

$$\mathbf{B} = \frac{\mu_0 I}{2\pi r}​$$

Magnetic Fields

A magnetic field (B) is a vector field representing the magnetic influence on moving electric charges, electric currents, and magnetized materials. Charges moving within this field experience a force perpendicular to both their velocity and the magnetic field direction.

Fleming’s Left and Right-Hand Rules

1. Fleming’s Left-Hand Rule: Used to determine the direction of the magnetic force in an electric motor. It illustrates the interaction between an electric current and a magnetic field resulting in motion.
2. Fleming’s Right-Hand Rule: Applies to electric generators, where it is used to ascertain the direction of induced current created by motion in a magnetic field.

Summary

Ampere’s Circuital Law is pivotal in understanding the interplay between electric currents and magnetic fields, a knowledge crucial in the study of electric motors and generators. It emphasizes the importance of considering the magnetic field’s direction in closed loops and utilizing Fleming’s rules for accurately determining the direction of forces and currents in electromagnetic systems.

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SAQ-3 : Find the magnetic induction due to a long current carrying conductor.

Introduction

Magnetic induction caused by a long current-carrying conductor is a fundamental concept in electromagnetism, illustrating how electric currents give rise to magnetic fields. Understanding this principle is crucial for various applications in electrical engineering and physics.

Derivation of Magnetic Induction

1. Ampere’s Circuital Law: To find the magnetic induction, we utilize Ampere’s Circuital Law, which states: $$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$$ Here, B is the magnetic field, μ₀​ is the permeability of free space, and I is the current through the conductor.
2. Considering a Long Straight Conductor: For a long straight conductor carrying a current I, we imagine a circular loop of radius r around the conductor as our path for the integration.
3. Uniform Magnetic Field Along the Loop: The magnetic field is uniform along the circular loop and is perpendicular to the radius at every point. Therefore, the dot product B⋅dl simplifies to Bdl as B and dl are parallel.
4. Application of Ampere’s Law: Substituting into Ampere’s Law, we get: $$\oint B dl = \mu_0 I$$ Since B is constant along the loop, it comes out of the integral, leading to: $$B \oint dl = \mu_0 I$$
5. Calculating the Magnetic Field (B): The integral ∮dl is the circumference of the circular loop, which is 2πr. Therefore, $$B \times 2\pi r = \mu_0 I$$ Simplifying for B, we get: $$B = \frac{\mu_0 I}{2\pi r}$$​

Summary

The magnetic induction B due to a long straight current-carrying conductor is given by:

$$B = \frac{\mu_0 I}{2\pi r}​$$

where r is the perpendicular distance from the conductor. This formula is vital in understanding the magnetic effects of current in wires and is widely used in the design of electrical devices and understanding magnetic fields around conductors.

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SAQ-4 : Derive an expression for the magnetic induction at the Centre of a current carrying circular coil using Biot – Savart’s law.

Introduction

Understanding the magnetic fields produced by current-carrying conductors is a key concept in physics. Biot-Savart’s law provides a method to calculate the magnetic field at any point due to a small current element. This law is particularly useful for determining the magnetic field at the center of a circular coil carrying current.

Understanding Biot-Savart’s Law

Biot-Savart’s Law posits that the magnetic field (dB) due to a small element of a current-carrying conductor is directly proportional to the current (i) and the length of the element (dl), and inversely proportional to the square of the distance (r²) from the point to the element.

Application to a Circular Coil

1. Circular Coil Configuration: Consider a circular loop with current (i) flowing through it. For a small element dl at a distance r from the center of the loop, Biot-Savart’s law gives: $$dB = \frac{\mu_0}{4\pi} \cdot \frac{i \cdot dl \times r}{r^2}​$$
2. Angle Between dl and r: In a circular loop, the angle (θ) between dl and r is 90 degrees, so sinθ = 1.
3. Substituting Values: Substituting into Biot-Savart’s law, we get: $$dB = \frac{\mu_0}{4\pi} \cdot \frac{i \cdot dl}{r^2}​$$
4. Total Magnetic Field (B): To find the total magnetic field at the center of the loop, integrate dB over the entire loop: $$B = \int dB = \frac{\mu_0}{4\pi} \cdot \frac{i}{r^2} \int dl$$ The integral of dl over the loop equals the circumference of the loop (2πr), leading to: $$B = \frac{\mu_0}{4\pi} \cdot \frac{i}{r^2} \cdot 2\pi r$$
5. Simplified Expression for B: Simplifying the expression, we obtain the total magnetic field at the center of the loop: $$B = \frac{\mu_0 \cdot i}{2r}​$$

Summary

By applying Biot-Savart’s Law, we deduce that the magnetic field at the center of a circular coil carrying a current is $$B = \frac{\mu_0 \cdot i}{2r}$$ This expression is vital for understanding the magnetic effects of circular currents and is foundational in electromagnetism studies.

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SAQ-5 : Derive an expression for the magnetic induction (B) at a point on the axis of a current carrying circular coil using Biot – Savart’s law.

Introduction

The derivation of the magnetic induction (B) at a point on the axis of a current-carrying circular coil is an important application of Biot-Savart’s Law in electromagnetism. This concept is crucial for understanding the magnetic fields generated by circular coils.

Biot-Savart’s Law and Its Application to a Circular Coil

Biot-Savart’s Law states that the magnetic field (dB) created by a small segment of current-carrying wire is proportional to the current (I), the length of the segment (dl), and the inverse square of the distance (r) from the segment to the point of observation.

Derivation of Magnetic Induction for a Circular Coil

1. Consider a Circular Coil: Imagine a circular coil of radius R carrying a current I. Let’s derive the magnetic field at a point P located at a distance x from the center of the coil along its axis.
2. Application of Biot-Savart’s Law: According to Biot-Savart’s Law, $$dB = \frac{\mu_0}{4\pi} \cdot \frac{I \cdot dl \times \hat{r}}{r^2}$$
3. Geometry of the Problem: For an element dl of the coil, the distance r from dl to point P is given by $$r = \sqrt{x^2 + R^2}$$ The angle θ between dl and r^ (the line joining dl and P) is 90 degrees.
4. Magnetic Field Contribution: The magnetic field dB due to dl at P is perpendicular to the plane formed by dl and r^. Therefore, we only consider the component of dB along the axis of the coil: $$dB_{\text{axial}} = dB \cdot \cos\phi$$ where ϕ is the angle between dB and the axis of the coil.
5. Calculation of dB_axial​: Using trigonometry, $$\cos\phi = \frac{x}{\sqrt{x^2 + R^2}}$$ Substituting into the equation for dB_axial​, we get: $$dB_{\text{axial}} = \frac{\mu_0}{4\pi} \cdot \frac{I \cdot dl \cdot x}{(x^2 + R^2)^{3/2}}​$$
6. Total Magnetic Field at Point P: Integrating dB_axial​ over the entire coil gives the total magnetic field B at point P: $$B = \int dB_{\text{axial}} = \frac{\mu_0 \cdot I \cdot x}{2(x^2 + R^2)^{3/2}} \int dl$$ Since the integral of dl over the entire coil is the circumference of the coil (2πR), we obtain: $$B = \frac{\mu_0 \cdot I \cdot R^2}{2(x^2 + R^2)^{3/2}}$$

Summary

The magnetic induction (B) at a point on the axis of a current-carrying circular coil is given by:

$$B = \frac{\mu_0 \cdot I \cdot R^2}{2(x^2 + R^2)^{3/2}}$$

This formula, derived using Biot-Savart’s Law, is fundamental in calculating the magnetic fields generated by circular coils in various applications in physics and engineering.

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SAQ-6 : Derive an expression for the magnetic dipole moment of a revolving electron.

Introduction

The magnetic dipole moment associated with an electron revolving around a nucleus is an important concept in physics, particularly in the study of atomic structure and electromagnetic theory. This moment arises due to the orbital motion of electrons and can be understood in terms of classical electromagnetism.

Derivation of Magnetic Dipole Moment for a Revolving Electron

1. Electron as a Current Loop: A revolving electron in an atomic orbit can be considered equivalent to a current loop. This is because the motion of the electron constitutes a circular current.
2. Current Equivalent of Revolving Electron: The current (I) equivalent to the orbiting electron is given by the charge of the electron (e) divided by the time period (T) of its revolution: $$I = \frac{e}{T}​$$
3. Time Period and Angular Velocity: The time period (T) is related to the angular velocity (ω) of the electron by $$T = \frac{2\pi}{\omega}$$ Substituting this into the expression for I, we get: $$I = \frac{e \omega}{2\pi}​$$
4. Area of the Electron’s Orbit: If the radius of the electron’s orbit is r, then the area (A) of the orbit is πr².
5. Magnetic Dipole Moment (μ): The magnetic dipole moment of the current loop is given by the product of the current (I) and the area (A) of the loop: μ = I ⋅ A Substituting the expressions for I and A, we get: $$\mu = \frac{e \omega}{2\pi} \cdot \pi r^2$$
6. Simplifying the Expression: Simplifying, the expression for the magnetic dipole moment becomes: $$\mu = \frac{1}{2} e \omega r^2$$ Alternatively, using angular momentum (L), where $$L = mvr = \frac{1}{2} m \omega r^2$$ (with m being the mass of the electron), the magnetic dipole moment can also be expressed as: $$\mu = \frac{e}{2m} L$$

Summary

The magnetic dipole moment (μ) of a revolving electron is given by:

$$\mu = \frac{1}{2} e \omega r^2$$

or equivalently,

$$\mu = \frac{e}{2m} L$$

This expression highlights the relationship between the electron’s orbital motion and its resultant magnetic dipole moment, playing a significant role in the magnetic properties of atoms and the foundation of quantum mechanics.

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SAQ-7 : What are the basic components of a cyclotron? Mention its uses?

Introduction

A cyclotron is a particle accelerator that propels charged particles to high velocities. It operates on the principle that a charged particle moving perpendicular to a magnetic field undergoes a magnetic Lorentz force, resulting in a circular path.

Principle of a Cyclotron

The cyclotron is based on the principle that a charged particle in a magnetic field experiences a Lorentz force. This force is perpendicular to the particle’s velocity, causing it to move in a circular trajectory.

Construction of a Cyclotron

1. D-Shaped Metallic Cylinders (“Dees”): A cyclotron typically includes two hollow, D-shaped metallic cylinders known as “dees.”
2. Placement of Dees: These dees are positioned within a strong electromagnet, with the magnetic field running along the length of the cylinder.
3. Ion Source and Gap: Between the dees, there’s a gap housing an ion source, and both dees are connected to a high-frequency oscillator.

Working of a Cyclotron

1. Ion Emission: An ion emitted from the source is drawn towards a negatively charged dee.
2. Polarity Change and Acceleration: As the ion exits the dee, the polarity changes, causing the ion to accelerate and move in a larger radius circle.
3. Spiral Path and Ejection: The particle follows a spiral path of increasing radius, and when it nears the cyclotron’s edge, a deflector plate removes it for collision with a target.

Mathematical Derivation

1. Force Balance: The balance between magnetic and centripetal forces is given by Bqv = mv²/r, implying vr = Bq/m is constant.
2. Period and Oscillator Frequency: The time for one semi-circular path is t = πm/Bq. The period T of one revolution and the oscillator is 2t = 2πm/Bq, leading to the oscillator frequency ν = Bq/2πm, known as the resonance condition.

Limitations of Cyclotrons

1. Size Constraint: The size of the cyclotron limits the maximum particle velocity.
2. Uniform Magnetic Field: Creating a uniform magnetic field across the entire dee is challenging.
3. Relativistic Effects: At high speeds, relativistic effects can disrupt the resonance conditions, limiting the cyclotron’s effectiveness.

Summary

The cyclotron is an essential tool in nuclear physics and medicine, built upon principles of electromagnetism and central force motion. While it has its limitations, its ability to accelerate particles is crucial for various scientific and medical applications.