📘 Moving Charges and Magnetism
Physics • Class XII
Introduction to Electricity and Magnetism
Electricity and magnetism have been known for over 2000 years, but their relationship was discovered in 1820 by Hans Christian Oersted. He observed that a current in a straight wire caused a deflection in a nearby magnetic compass needle, indicating that moving charges produce a magnetic field.
Oersted's Experiment
Oersted's experiment demonstrated that: - The compass needle aligns tangentially to an imaginary circle around the wire. - The effect is noticeable when the current is large and the needle is close to the wire. - Reversing the current reverses the needle's orientation. - Iron filings around the wire form concentric circles, confirming the magnetic field's presence.
Historical Developments
In 1864, James Maxwell unified the laws of electricity and magnetism, leading to the realization that light is an electromagnetic wave. The discovery of radio waves by Hertz and their production by J.C. Bose and G. Marconi marked significant advancements in the late 19th century. The 20th century saw remarkable progress in electromagnetism, leading to new technologies for electromagnetic wave production, amplification, transmission, and detection.
Overview of Chapter Topics
This chapter covers: - The forces exerted by magnetic fields on moving charged particles (electrons, protons, current-carrying wires). - How currents produce magnetic fields. - The acceleration of particles in a cyclotron. - Detection of currents and voltages using a galvanometer.
Conventions in Magnetism
In this chapter, the following conventions are used: - A current or field emerging from the plane of the paper is depicted by a dot (•). - A current or field going into the plane of the paper is depicted by a cross (×).
Magnetic Force and Fields
Before discussing magnetic fields, it is important to recall the electric field concept from Chapter 1. The interaction between two charges involves: - A charge Q, which produces an electric field E.
Electric Field and Charge Interaction
The electric field E is defined as E = Q ˆr / (4πε₀)r², where ˆr is the unit vector along r. A charge q experiences a force F given by F = qE = qQ ˆr / (4πε₀)r². The electric field conveys energy and momentum and propagates over time. The principle of superposition applies, allowing fields from multiple charges to add vectorially.
Magnetic Field and Its Properties
Moving charges produce a magnetic field B(r), which is also a vector field. It is defined at each point in space and can depend on time. The magnetic field obeys the principle of superposition, meaning the total magnetic field from multiple sources is the vector sum of the individual fields.
Lorentz Force
The Lorentz force on a point charge q moving with velocity v in electric field E(r) and magnetic field B(r) is given by F = q[E(r) + v × B(r)]. Key features include: - The force depends on charge (q), velocity (v), and magnetic field (B). - The magnetic force is opposite for negative charges. - The force is zero if the charge is not moving. - The direction of the magnetic force is perpendicular to both velocity and magnetic field, determined by the right-hand rule.
Magnetic Field and Its Units
The magnitude of the magnetic field B is defined as 1 SI unit when the force acting on a unit charge (1 C) moving perpendicular to B with a speed of 1 m/s is one newton. The dimensional formula for B is [B] = [F/qv]. The unit of B is Newton second / (coulomb metre), known as tesla (T), named after Nikola Tesla. A smaller unit, gauss, is also used, where 1 gauss = 10^-4 tesla. The Earth's magnetic field is approximately 3.6 × 10^-5 T.
Magnetic Force on a Current-Carrying Conductor
The analysis of magnetic force can be extended to a straight rod carrying current. For a rod with a uniform cross-sectional area A and length l, with mobile charge carriers (electrons), the total number of carriers is nlA, where n is the number density of carriers. The force on these carriers in an external magnetic field B is given by: F = (nlA)q v_d B, where q is the charge value and v_d is the average drift velocity. The current density j is defined as j = nq v_d, leading to the equation: F = Il B, where I is the current and l is a vector representing the length of the rod.
Calculating Magnetic Force
For a straight rod, the magnetic force can be calculated using the equation F = Il B. If the wire has an arbitrary shape, the Lorentz force can be calculated by considering it as a collection of linear strips and summing the forces. This summation can often be converted to an integral.
Examples of Magnetic Force
Example 4.1: A straight wire of mass 200 g and length 1.5 m carries a current of 2 A suspended in mid-air by a uniform horizontal magnetic field. The magnitude of the magnetic field can be calculated using the balance of forces: m g = I l B, resulting in B = (m g) / (I l) = 0.65 T. Example 4.2: When a charged particle moves in a magnetic field, the direction of the Lorentz force can be determined using the right-hand rule. If the magnetic field is parallel to the positive y-axis and the charged particle moves along the positive x-axis, the force on an electron (negative charge) will be along the -z axis, while for a proton (positive charge), it will be along the +z axis.
Motion of a Charge in a Magnetic Field
The motion of a charge in a magnetic field involves a magnetic force that is perpendicular to the velocity of the particle, resulting in no work done and no change in the magnitude of velocity. The direction of momentum may change.
Centripetal Force and Radius of Circular Motion
The magnetic force acts as a centripetal force, given by the equation q v B. The radius (r) of the circular path can be derived from the centripetal force equation: m v²/r = q v B.
Angular Frequency and Frequency of Rotation
The angular frequency (w) is related to the charge and mass of the particle and is independent of velocity or energy. The frequency of rotation (n) is given by n = q B / (2π m).
Helical Motion and Pitch
When there is a component of velocity parallel to the magnetic field, the particle moves in a helical path. The distance moved along the magnetic field in one rotation is called the pitch (p).
Biot-Savart Law
The Biot-Savart Law describes the relationship between electric current and the magnetic field it produces. It is fundamental for understanding how currents create magnetic fields.
Biot-Savart Law Overview
The Biot-Savart law describes the magnetic field produced by a finite conductor carrying current. The magnetic field dB at a point P, at a distance r from an infinitesimal element dl of the conductor, is determined by the current I, the length of the element |dl|, and the distance r.
Magnetic Field Calculation
According to the Biot-Savart law, the magnitude of the magnetic field dB is given by the equation: dB ∝ (I |dl| sin θ) / r² In vector notation, this is expressed as: dB = (μ₀ / 4π) (I dl × r) / r³ where μ₀ is the permeability of free space.
Comparison with Electrostatic Field
The Biot-Savart law has similarities and differences with Coulomb’s law for electrostatic fields: 1. Both laws are long-range and depend inversely on the square of the distance from the source. 2. The electrostatic field is produced by a scalar source (electric charge), while the magnetic field is produced by a vector source (I dl). 3. The electrostatic field is directed along the displacement vector, whereas the magnetic field is perpendicular to it. 4. The angle dependence in the Biot-Savart law is not present in the electrostatic case.
Constants and Relationships
The relationship between the permittivity of free space (ε₀), permeability of free space (μ₀), and the speed of light (c) is given by: c² = 1 / (μ₀ ε₀) In SI units, μ₀ is fixed at 4π × 10⁻⁷ Tm/A.
Example Calculation
Example 4.4 illustrates the calculation of the magnetic field at a point on the y-axis due to a current-carrying element. Given: - Current I = 10 A - Distance r = 0.5 m - Element length ∆x = 1 cm Using the formula: B = (μ₀ / 4π) (I ∆x sin θ) / r² For θ = 90°, sin θ = 1, the magnetic field B is calculated to be 4 × 10⁻⁸ T in the +z-direction.
Magnetic Field on the Axis of a Circular Current Loop
This section evaluates the magnetic field due to a circular coil along its axis, summing the effects of infinitesimal current elements (I dl). The current I is steady, and the evaluation is in free space (vacuum).
Calculation of Magnetic Field (B)
To calculate the magnetic field at point P on the axis of the loop, consider the following steps: 1. Define the circular loop with radius R, centered at origin O, and the distance x from O to point P. 2. Use the Biot-Savart law to find the magnitude dB of the magnetic field due to an infinitesimal element dl: dB = (μ₀/4π) × (I dl × r) / r², where r² = x² + R². 3. The components of dB perpendicular to the x-axis cancel out, leaving only the x-component.
Final Expression for Magnetic Field
The net magnetic field at point P due to the entire circular loop is given by: B = (μ₀/4) × (I R²) / (x² + R²)^(3/2). At the center of the loop (x = 0), the magnetic field simplifies to: B = (μ₀ I) / (2 R).
Direction of Magnetic Field
The direction of the magnetic field lines due to a circular wire can be determined using the right-hand thumb rule: curl the palm of your right hand around the wire with fingers in the direction of current; the thumb points in the direction of the magnetic field.
Magnetic Field Contributions
For straight segments of a wire, the differential length element (dl) and radius (r) are parallel, resulting in dl × r = 0. Therefore, straight segments do not contribute to the magnetic field (|B|). For a semicircular arc, all segments contribute equally, and the direction of B is determined by the right-hand rule. The magnitude of B for a semicircular arc is half that of a circular loop.
Magnetic Field of a Coil
For a tightly wound coil with N turns and radius R carrying current I, the magnetic field at the center can be calculated using the formula: B = (μ₀NI)/(2R). For example, a coil with 100 turns, radius 10 cm, and current 1 A results in B = 4.6 × 10⁻⁷ T.
Ampere's Circuital Law
Ampere's circuital law relates the magnetic field around a closed loop to the current passing through the surface bounded by the loop. The law is expressed as B·dl = μ₀I, where I is the total current through the surface. The integral is taken over the closed loop coinciding with the boundary C of the surface.
Magnetic Field of an Infinite Wire
For an infinite current-carrying wire, the magnetic field at a distance r from the wire can be calculated using B = μ₀I/(2πr). This result is derived from Ampere's law and is useful for evaluating magnetic fields in systems with symmetry.
The Solenoid
A long solenoid is defined as having a length much greater than its radius, consisting of a wire wound in a helix with closely spaced turns. The net magnetic field inside the solenoid is uniform and strong, while the field outside approaches zero.
Magnetic Field Inside a Solenoid
A solenoid can create a large magnetic field by inserting a soft iron core. For example, a solenoid with a length of 0.5 m, a radius of 1 cm, and 500 turns carrying a current of 5 A produces a magnetic field. The number of turns per unit length (n) is calculated as follows: - n = 500 turns / 0.5 m = 1000 turns/m Using the long solenoid formula, the magnetic field (B) is given by: - B = μ₀ n I - B = 4π × 10⁻⁷ T m/A × 1000 turns/m × 5 A = 6.28 × 10⁻³ T.
Force Between Two Parallel Currents
Two parallel current-carrying conductors exert forces on each other due to their magnetic fields. This phenomenon was studied by Ampere. The force experienced by one conductor due to the other can be calculated using: - F_ba = (μ₀/2π) * (I_a * I_b * L / d) Where: - F_ba is the force on conductor b due to conductor a, - I_a and I_b are the currents in conductors a and b, - L is the length of the conductors, and - d is the distance between them. The forces are equal in magnitude and opposite in direction (F_ba = -F_ab), consistent with Newton's third law.
Definition of the Ampere
The magnitude of the force per unit length between two parallel conductors is used to define the ampere (A). The expression for the force per unit length (f_ba) is: - f_ba = (μ₀/2π) * (I_a * I_b / d) This definition establishes the ampere as one of the seven SI base units.
Definition of the Coulomb
The SI unit of charge, the coulomb, is defined in terms of the ampere. When a steady current of 1A flows through a conductor, the charge that passes through its cross-section in 1 second is one coulomb (1C).
Force on a Conductor in a Magnetic Field
When a straight conductor carrying a steady current is placed in a magnetic field, the force per unit length can be calculated using the formula: f = I B sin θ. - For a current flowing from east to west (θ = 90°): - f = I B = 1 × 3 × 10⁻⁵ = 3 × 10⁻⁵ N m⁻¹ - The direction of the force is downwards. - For a current flowing from south to north (θ = 0°): - f = 0, hence no force acts on the conductor.
Torque on a Current Loop in a Magnetic Field
A rectangular loop carrying a steady current I in a uniform magnetic field experiences torque but no net force. - When the magnetic field B is in the plane of the loop: - Forces F₁ and F₂ act on arms AB and CD, respectively, with magnitudes: - F₁ = I b B - F₂ = I b B - The net force is zero, but a torque τ is produced, calculated as: - τ = I A B, where A = ab (area of the rectangle).
Forces and Torque on Current Loop
When forces F1 and F2 act on arms AB and CD of a current-carrying loop, they are equal and opposite, with magnitude F1 = F2 = I b B. However, since they are not collinear, they create a couple, resulting in torque. The torque is less than when the plane of the loop is aligned with the magnetic field due to a decreased perpendicular distance between the forces.
Torque Calculation
The magnitude of the torque on the loop can be expressed as τ = I A B sin θ, where θ is the angle between the magnetic moment and the magnetic field. This can also be expressed as τ = 12 sin(2θ) F1 + F2, illustrating the relationship between the forces and torque.
Magnetic Moment
The magnetic moment of the current loop is defined as m = I A, where A is the area of the loop. The direction of the area vector A is determined by the right-hand thumb rule. The dimensions of the magnetic moment are [A][L²] and its unit is Am².
Equilibrium Conditions
The torque τ vanishes when the magnetic moment m is either parallel or antiparallel to the magnetic field B. Parallel alignment indicates stable equilibrium, while antiparallel alignment indicates unstable equilibrium. A small rotation from the parallel position produces a restoring torque, while from the antiparallel position, it produces an increasing torque.
Multiple Turns in a Coil
For a coil with N closely wound turns, the expression for torque remains τ = m × B, with m = N I A. This accounts for the total magnetic moment of the coil.
Example Problem
Consider a 100-turn circular coil of radius 10 cm carrying a current of 3.2 A placed in a uniform magnetic field of 2T. The coil rotates through an angle of 90°. Key calculations include the magnetic field at the center of the coil, the magnetic moment, and the torques in initial and final positions, as well as the angular speed acquired after rotation.
Moment of Inertia and Angular Motion
According to Newton’s second law, I = d(ω)/d(t), where I is the moment of inertia of the coil. The relationship between angular displacement (θ), angular velocity (ω), and torque can be expressed as: d(ω)/d(t) = (m B/ I) sin(θ). Integrating this from θ = 0 to θ = π/2 is necessary for further calculations.
Current-Carrying Loop in Magnetic Field
A current-carrying circular loop on a smooth horizontal plane cannot turn around itself in a uniform magnetic field because the torque would need to be in the vertical direction. The orientation of stable equilibrium occurs when the area vector (A) of the loop aligns with the external magnetic field, maximizing the magnetic flux.
Magnetic Dipole Analogy
The magnetic field due to a circular current loop behaves similarly to the electric field of an electric dipole. The magnetic field (B) at a distance x from the center of a loop of radius R carrying current I is given by B = (μ₀/4π) * (2I R)/(x²) for x >> R. The magnetic moment (m) is defined as m = I A, where A = πR².
Magnetic Dipoles
A planar current loop is equivalent to a magnetic dipole with dipole moment m = I A. Unlike electric dipoles, which consist of two charges, magnetic dipoles are the most elementary elements in magnetism. Magnetic monopoles have not been observed.
Moving Coil Galvanometer (MCG)
The moving coil galvanometer is used to measure currents and voltages in circuits. It consists of a coil that rotates in a uniform radial magnetic field, aided by a cylindrical soft iron core. When current flows through the coil, a torque τ = NI AB acts on it, where N is the number of turns, I is the current, A is the area of the coil, and B is the magnetic field strength.
Galvanometer Functionality
The galvanometer can detect current flow and is used in Wheatstone’s bridge arrangements. The pointer indicates deflection based on current direction. However, it cannot measure current directly due to its sensitivity and high resistance. To measure current, a shunt resistance (r_s) is connected in parallel to allow most current to bypass the galvanometer.
Galvanometer and Current Sensitivity
The galvanometer is a device used to measure current and can be converted into an ammeter. The current sensitivity of the galvanometer is defined as the deflection per unit current, represented by the equation: φ = NAB / Ik. To increase sensitivity, manufacturers can increase the number of turns (N) in the galvanometer.
Galvanometer as a Voltmeter
The galvanometer can also function as a voltmeter by connecting it in parallel with a circuit section. To minimize disturbance in voltage measurement, a large resistance (R) is connected in series with the galvanometer.
Conversion of Galvanometer
The conversion of a galvanometer to an ammeter requires a shunt resistance (r_s) of very small value in parallel. Conversely, to convert it to a voltmeter, a large resistance (R) is added in series.
Key Equations
1. Lorentz Force: F = q (v × B + E) - The total force on a charge moving in magnetic and electric fields. 2. Force on a Conductor: F = I l × B - A straight conductor of length l carrying current I in a magnetic field B experiences this force. 3. Cyclotron Frequency: νc = (qB / 2πm) - The frequency of a charge in a circular orbit in a magnetic field, independent of speed and radius.
Exercises on Magnetic Fields
The following exercises involve calculating magnetic fields and forces in various scenarios: - Exercise 4.1: Calculate the magnetic field at the center of a circular coil with 100 turns, radius 8.0 cm, and current 0.40 A. - Exercise 4.2: Determine the magnetic field at a point 20 cm from a long straight wire carrying 35 A. - Exercise 4.3: Find the magnitude and direction of the magnetic field 2.5 m east of a wire carrying 50 A. - Exercise 4.4: Calculate the magnetic field 1.5 m below a power line carrying 90 A. - Exercise 4.5: Determine the magnetic force per unit length on a wire carrying 8 A at an angle of 30º to a magnetic field of 0.15 T.
Moving Coil Meters
Two moving coil meters, M1 and M2, have the following particulars: - M1: R1 = 10 Ω, N1 = 30, A1 = 3.6 × 10–3 m², B1 = 0.25 T - M2: R2 = 14 Ω, N2 = 42, A2 = 1.8 × 10–3 m², B2 = 0.50 T Determine the ratio of: - (a) current sensitivity of M2 to M1 - (b) voltage sensitivity of M2 to M1.
Electron in a Magnetic Field
In a chamber with a uniform magnetic field of 6.5 G (1 G = 10–4 T), an electron is shot into the field with a speed of 4.8 × 10⁶ m/s normal to the field. The path of the electron is circular due to the magnetic force acting as a centripetal force.
Counter Torque on a Coil
Exercise 4.13 involves a circular coil of 30 turns and radius 8.0 cm carrying a current of 6.0 A, suspended vertically in a uniform horizontal magnetic field of 1.0 T, with field lines at an angle of 60° to the normal of the coil. Calculate the magnitude of the counter torque required to prevent the coil from turning.