Moving Charges and Magnetism — Class 12 Physics Notes

Topics covered in Moving Charges and Magnetism

1. 1.Introduction of Moving Charges & Magnetism

   Short Answer

   Electricity and magnetism are closely linked, a fact discovered by Hans Christian Oersted in 1820 when he noticed that electric current in a wire affected a nearby compass needle. This discovery led to the understanding that moving charges create magnetic fields around them. James Maxwell later unified the laws of electricity and magnetism, leading to the realization that light is an electromagnetic wave. This chapter explores how magnetic fields interact with moving charges and how currents produce magnetic fields, among other topics.

   Long Answer

   Happy Note: Imagine a world without electricity and magnetism - no lights, no smartphones, no internet! Thankfully, we've understood these forces for over 200 years, making our modern world possible. Let's dive into the fascinating journey of how we discovered the intimate relationship between moving charges and magnetism.

   1. Historical Discovery: In 1820, Hans Christian Oersted made a groundbreaking observation that a straight wire carrying electric current could deflect a nearby compass needle. This indicated that electric current produces a magnetic field in the surrounding space.

   2. Oersted's Experiment: He found that the compass needle aligns tangentially to an imaginary circle centered on the wire, with the plane of the circle perpendicular to the wire. The deflection of the needle was affected by the direction of the current and its magnitude, as well as the proximity of the needle to the wire.

   3. Further Developments: Following Oersted's discovery, scientists like James Maxwell made significant strides in understanding electricity and magnetism. Maxwell's equations unified these forces, revealing that light itself is an electromagnetic wave. This understanding paved the way for the discovery of radio waves and the development of various electromagnetic technologies.

   4. Application in Modern Technology: Today, the principles of electricity and magnetism are foundational to countless technologies. For example, the chapter discusses how magnetic fields can exert forces on moving charged particles, which is the principle behind devices like the cyclotron used in particle physics. It also covers how currents and magnetic fields interact, a concept used in measuring devices like the galvanometer.

   Real-Life Examples:

   • Compass Near a Wire: If you've ever played with a compass and a wire carrying current, you've seen Oersted's discovery firsthand.
   • Medical Imaging: Technologies like MRI (Magnetic Resonance Imaging) rely on our understanding of magnetism to create detailed images of the inside of the human body.
   • Communication Technology: The transmission and detection of electromagnetic waves form the basis of radio, television, and internet technologies.

   Careers and Industries:

   • Physics and Engineering: Understanding moving charges and magnetism is crucial for developing new technologies and scientific instruments.
   • Medical Field: Techniques like MRI scanning require a deep understanding of magnetism.
   • Telecommunications: The principles of electromagnetism are fundamental to the design and operation of communication devices and networks.

2. 2.Magnetic Force

   Short Answer

   Magnetic forces arise from magnetic fields, which are created by moving charges or magnets. The Lorentz force describes how charged particles experience force in the presence of electric and magnetic fields. A current-carrying conductor placed in a magnetic field will experience a magnetic force, dependent on the current's direction, the conductor's length, and the magnetic field's strength and direction.

   Long Answer

   Happy Note: Just like the invisible force that pulls a fridge magnet, magnetic forces are at play all around us, guiding everything from the path of an electron to the direction of a compass needle. Let’s explore this invisible world.

   1. Sources and Fields: Magnetic fields are created by moving electric charges or by the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, called spin. Magnets and currents are primary sources of magnetic fields in our everyday life.

   2. Magnetic Field: A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A magnetic field is represented by the symbol B, and its unit of measurement is the tesla (T).

   3. Lorentz Force: The Lorentz force equation describes the force on a charged particle moving through an electric and a magnetic field. It is given by =(+×)F=q(E+v×B), where F is the force, q is the charge of the particle, E is the electric field, v is the velocity of the particle, and B is the magnetic field. The ×× represents the cross product, indicating that the direction of the force is perpendicular to both the velocity of the particle and the magnetic field.

   4. Magnetic Force on a Current-carrying Conductor: When a conductor carrying an electric current is placed in a magnetic field, it experiences a force. This force is given by =(×)F=I(L×B), where I is the current, L is the length vector of the conductor (in the direction of the current), and B is the magnetic field. The direction of the force is determined by the right-hand rule, which states that if you point your thumb in the direction of the current and your fingers in the direction of the magnetic field, your palm faces in the direction of the force.

   Real-Life Examples:

   • Electric Motors: The principle of a current-carrying conductor experiencing a force in a magnetic field is what makes electric motors work, turning electrical energy into mechanical movement.
   • Magnetic Levitation (Maglev) Trains: These trains float above the tracks, thanks to magnetic forces repelling the train from the track, reducing friction and allowing high speeds.

   Careers and Industries:

   • Electrical Engineering: Designing devices that use magnetic forces, like motors and generators.
   • Transportation: Developing maglev trains or other transportation technologies that use magnetic levitation.
   • Scientific Research: Studying fundamental physics of magnetic fields and forces for new technologies.

3. 3.Motion In A Magnetic Field

   Short Answer

   Motion in a magnetic field involves the circular motion of charged particles when they enter a magnetic field perpendicularly. The magnetic force acts as a centripetal force, causing the particle to move in a circular path. The radius of this path depends on the particle's charge, velocity, mass, and the strength of the magnetic field.

   Long Answer

   When a charged particle, such as an electron or a proton, enters a magnetic field (B) at a right angle to the field, it experiences a magnetic force perpendicular to both its velocity (v) and the magnetic field. This force does not work on the particle (since it's always perpendicular to the direction of motion), but it changes the direction of the particle's velocity, causing it to move in a circular path. The radius (r) of this path can be calculated using the formula:

   =r=qBmv​

   where:

   • m is the mass of the particle,
   • v is the velocity of the particle,
   • q is the charge of the particle,
   • B is the magnetic field strength.

   Numerical Example:

   Let's consider an electron with a charge of 1.6×10−191.6×10−19 Coulombs and a mass of 9.11×10−319.11×10−31 kg, moving at a velocity of 2×1062×106 m/s into a magnetic field of 0.50.5 Tesla.

   1. Objective: Calculate the radius of the path taken by the electron in the magnetic field.

   Using the formula =r=qBmv​, let's plug in the values:

   • =9.11×10−31m=9.11×10−31 kg,
   • =2×106v=2×106 m/s,
   • =1.6×10−19q=1.6×10−19 C,
   • =0.5B=0.5 T.

   Let's calculate the radius.

   Calculation

   Let me calculate the radius of the path for you.

   The radius of the path taken by the electron in the magnetic field is approximately 2.28×10−22.28×10−2 meters, or 2.28 cm.

   Real-Life Example: This principle is utilized in devices like cyclotrons, where charged particles are accelerated in circular paths by magnetic fields. Such mechanisms are foundational in particle physics research, allowing scientists to study the properties of subatomic particles.

   Careers and Industries:

   • Scientific Research: Understanding the motion of particles in magnetic fields is crucial in fields like nuclear physics and astrophysics.
   • Medical Field: Technologies like particle therapy for cancer treatment use magnetic fields to direct charged particles to tumors with precision.
   • Engineering: Designing cyclotrons and other particle accelerators for various applications, from medical treatment to materials science.

4. 4.Magnetic Field Due To A Current Element, Biot-savart Law

   Short Answer

   The Biot-Savart Law provides a way to calculate the magnetic field produced at a point by a small segment of current-carrying wire (current element). The law states that the magnetic field (dB) at point P due to a current element (Idl) is directly proportional to the current (I), the length of the element (dl), and the sine of the angle (θ) between the element and the line joining the point and the element. It is inversely proportional to the square of the distance (r) from the current element to the point P.

   Long Answer

   1. Understanding the Diagram: The image you've shared shows a current I flowing through a small segment of wire (dl), which is at a distance r from a point P where the magnetic field (dB) is being calculated. The angle θ is between the direction of dl and the line from dl to the point P.

   2. Biot-Savart Law: The law mathematically describes the magnetic field contribution from the current element as follows:

      =04⋅⋅⋅sin⁡2dB=4πμ0​​⋅r2I⋅dl⋅sin(θ)​

      where:

      • dB is the differential magnetic field at point P,
      • 0μ0​ is the permeability of free space,
      • I is the current through dl,
      • dl is the length of the current element,
      • θ is the angle between dl and the line joining dl to point P,
      • r is the distance from the current element to point P.

   3. Using the Law: To find the total magnetic field at point P, one would integrate this expression over the length of the wire carrying the current.

      Application of the Biot-Savart Law

      Let's say we have a current I flowing through a straight wire segment of length dl, and we want to find the magnetic field at point P, which is at a distance r from the segment and at an angle θ to the wire segment.

      1. Objective: Calculate the magnitude of the magnetic field (dB) at point P due to the current element Idl.

      2. Using the Biot-Savart Law: We apply the formula:

         =04⋅⋅⋅sin⁡2
         dB=4πμ0​​⋅r2I⋅dl⋅sin(θ)​

         where:

         • 0=4×10−7μ0​=4π×10−7 T·m/A (Tesla meter per Ampere) is the permeability of free space,
         • I is the current in Amperes (A),
         • dl is the length of the current element in meters (m),
         • θ is the angle between the current element and the line joining the element to the point P in radians,
         • r is the distance from the current element to point P in meters (m).

      3. Calculation: We would insert the given values for I, dl, θ, and r into the equation to find the magnitude of dB.

      Hypothetical Numerical Example

      Assume:

      • =2I=2 A (Amperes),
      • =0.1dl=0.1 m (meters),
      • =90°θ=90° (which is 22π​ radians),
      • =0.5r=0.5 m (meters).

      We convert θ to radians because in calculus, we work with radians:

      =90°=2 radiansθ=90°=2π​ radians

      Now we plug these values into the Biot-Savart Law equation.

      Calculation

      Let's calculate the magnitude of the magnetic field at point P due to the current element.

      The magnitude of the magnetic field at point P due to the current element Idl is approximately 8.0×10−88.0×10−8 Tesla.

   Real-Life Examples:

   • Designing Magnetic Sensors: Engineers use the Biot-Savart Law to calculate magnetic fields when designing sensors that detect current flow.
   • Magnetic Field Mapping: In MRI machines, this law helps in calculating the magnetic fields necessary to produce clear images of the body.

   Careers and Industries:

   • Medical Technology: Designing MRI machines which use magnetic fields to image the body.
   • Electrical Engineering: Developing sensors and other devices that rely on understanding magnetic fields.

   This concept is a great example of how abstract scientific laws are applied to create the technology we use in our daily lives, from medical diagnostics to electronic devices.

5. 5.Magnetic Field On The Axis Of A Circular Current Loop

   Short Answer

   The magnetic field on the axis of a circular current loop is a magnetic field created by an electric current flowing in a circular loop. This field is directed along the axis of the loop and its strength decreases as you move away from the loop.

   Long Answer

   Understanding the Concept

   When an electric current flows through a circular loop, it creates a magnetic field around it. This magnetic field has a specific direction and magnitude, especially interesting along the axis of the loop. The axis of the loop is an imaginary line that passes through the center of the loop and perpendicular to the plane of the loop.

   How It's Calculated

   The strength of the magnetic field on the axis of a circular current loop can be calculated using Ampere's law or Biot-Savart law. However, for simplicity, we can use the formula derived from these laws:

   =022(2+2)3/2B=2(R2+x2)3/2μ0​IR2​

   where:

   • B is the magnetic field strength,
   • 0μ0​ is the magnetic constant (4×10−7 N/A24π×10−7N/A2),
   • I is the current flowing through the loop,
   • R is the radius of the circular loop, and
   • x is the distance from the center of the loop along the axis.

   Real-life Example

   Imagine a simple electric fan. The motor inside it uses coils of wire (which act like circular current loops) to create magnetic fields that interact with magnets, causing the fan to spin.

   Activities to Understand Better

   1. Visualizing Magnetic Field Lines: Use a compass and a wire connected to a battery to form a loop. As you move the compass around the loop, observe how its needle aligns with the magnetic field lines created by the current in the loop.

   2. Effect of Distance on Field Strength: Move the compass farther away from the loop along the axis and observe how the needle's movement becomes less pronounced, indicating the decrease in field strength with distance.

   Application in Real Life and Careers

   This principle is widely used in designing electric motors and generators, which are foundational to numerous industries like automotive, manufacturing, and consumer electronics. Understanding magnetic fields is crucial for engineers and physicists who work on developing more efficient electrical machines and devices.

6. 6.Ampere's Circuital Law

   Short Answer

   Ampere's Circuital Law relates the magnetic field in a closed loop to the electric current passing through the loop. It is stated as:

   The line integral of the magnetic field (B) around a closed path (Amperian loop) is equal to the permeability of free space (μ₀) times the current (I) enclosed by the path.

   Mathematically:

   ∮⃗⋅⃗=0∮B⋅dl=μ0​Ienc​

   Long Answer

   Understanding Ampere's Circuital Law

   1. Basic Concept: Ampere's Circuital Law says that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is proportional to the current enclosed in the loop.

   2. Mathematical Formulation: The law is given by the integral of the magnetic field ⃗B dotted with an infinitesimal vector element of the loop ⃗dl, which equals the permeability of free space 0μ0​ times the enclosed current Ienc​:

   ∮⃗⋅⃗=0∮B⋅dl=μ0​Ienc​

   1. Application of the Law: It is used to calculate the magnetic field created by a given current distribution. For simple geometries like a straight wire, solenoid, or toroid, the law simplifies the calculations significantly.

   Real-life Example

   Electricians use Ampere's Law to ensure the magnetic fields generated by the currents in power cables don't interfere with nearby electronics or create excessive magnetic fields that could be harmful.

   Activity

   To see Ampere's Law in action, you can perform an experiment with a compass and a wire carrying current. If you form the wire into a loop and pass a current through it, you can move the compass around the loop and observe how the needle aligns with the magnetic field, illustrating the relationship between current and the magnetic field.

   Application

   This law is fundamental in electrical engineering and physics, especially in the design of electrical devices like transformers, inductors, and electric motors, where understanding the relationship between current and magnetic fields is essential.

   Diagram Explanation

   • Wire Carrying Electric Current: The wire in the center of the diagram represents a conductor through which an electric current is flowing. The direction of the current is indicated by the arrow labeled "current."
   • Magnetic Field Lines: The concentric circles around the wire show the magnetic field lines. According to the right-hand rule, the magnetic field lines circle the wire in the direction shown.
   • Amperian Loop: The closed loop around the wire is the path along which we'll apply Ampere's Circuital Law. The path of integration is indicated by an arrow labeled "path of integration."
   • Integral Path: The arrow around the loop indicates the direction of traversal when performing the line integral of the magnetic field.

7. 7.Torque on a rectangular current loop in a uniform magnetic field.

   Short Answer

   Torque on a rectangular current loop in a uniform magnetic field is the force that causes the loop to rotate. The torque is calculated by the cross product of the magnetic moment of the loop and the magnetic field. It's given by the formula =sin⁡τ=NIABsin(θ), where N is the number of turns, I is the current, A is the area of the loop, B is the magnetic field, and θ is the angle between the normal to the loop and the magnetic field.

   Long Answer

   Understanding Torque on a Current Loop

   A rectangular current loop in a uniform magnetic field experiences a torque that tends to rotate it. This torque can be understood by considering the forces on each side of the loop due to the magnetic field.

   How it works: A current-carrying conductor in a magnetic field experiences a force known as the Lorentz force. For a rectangular loop, this force acts on each side of the loop, and the forces on opposite sides are equal in magnitude but opposite in direction, creating a couple that results in a torque.

   Derivation of the Torque Formula:

   • Consider a rectangular loop of length l and width w, with a current I flowing through it.
   • The magnetic field B is uniform and makes an angle θ with the normal to the plane of the loop.
   • The force on each side of the loop is given by =×( or)×sin⁡F=I×(l or w)×Bsin(θ).
   • The torque τ about the center of the loop due to each side is ×2F×2w​ or ×2F×2l​, depending on whether we consider the length or the width.
   • The total torque is then the sum of the torques from all four sides of the loop.

   For a rectangular loop, the torque is maximum when the plane of the loop is parallel to the magnetic field (=90°θ=90°) and is zero when the plane of the loop is perpendicular to the magnetic field (=0°θ=0°).

   Formula: The general formula for the torque on a current loop in a magnetic field is: =sin⁡τ=NIABsin(θ) where:

   • N is the number of turns in the loop.
   • I is the current in the loop.
   • A is the area of the loop (=×A=l×w).
   • B is the magnetic field strength.
   • θ is the angle between the normal to the loop and the magnetic field direction.

   Real-Life Example: Electric motors work on the principle of torque on a current loop. When current flows through the motor's coils (current loops) in a magnetic field, it creates a torque that rotates the motor's shaft.

   Activity: To demonstrate this, you can create a simple motor with a battery, a magnet, and a coil of wire. When you connect the battery to the coil and place it near the magnet, you'll see the coil start to rotate due to the torque.

   Uses: Understanding the torque on a current loop is essential for designing electric motors and generators, which are used in countless applications from household appliances to industrial machinery. ---- Let's understand the basics of how this works and derive the mathematical expression for the torque:

   Explanation:

   1. Current-Carrying Loop: The diagram shows a loop of wire carrying a current I placed in a magnetic field B. The loop has a width a and a length b, which form the sides of the rectangular loop.

   2. Magnetic Field: The magnetic field is represented by the symbol B and is oriented perpendicular to the plane of the loop.

   3. Rotation Axis: The loop is able to rotate about an axis. In this case, the axis is horizontal and the loop rotates in the plane of the page.

   4. Torque on the Loop: When a current flows through the loop, a torque is generated due to the magnetic forces on the sides of the loop. The sides of length b are in the plane of the magnetic field and do not contribute to the torque. However, the sides of length a are perpendicular to B, and the force on these sides is what causes the torque.

   Mathematical Expression:

   The torque τ experienced by a current loop in a uniform magnetic field is given by the vector cross product of the magnetic moment ⃗m of the loop and the magnetic field ⃗B: ⃗=⃗×⃗τ=m×B

   The magnetic moment ⃗m of the loop is given by the product of the current I, the area A of the loop, and the unit vector normal to the plane of the loop ^n^: ⃗=⋅⋅^m=I⋅A⋅n^

   For a rectangular loop, the area A is the product of its sides a and b: =⋅A=a⋅b

   So the magnetic moment becomes: ⃗=⋅⋅⋅^m=I⋅a⋅b⋅n^

   And the torque is: ⃗=⋅⋅⋅⃗sin⁡τ=I⋅a⋅b⋅Bsin(θ)

   Here, θ is the angle between the normal to the loop and the magnetic field direction. Since the maximum torque occurs when =90∘θ=90∘, the sine function becomes 1, and the expression for the maximum torque is: =⋅⋅⋅τmax​=I⋅a⋅b⋅B

   This torque will cause the loop to rotate about the axis, and in the case of an electric motor, this rotation is harnessed to do work, such as turning a shaft connected to wheels or other mechanical systems.

   In the context of the diagram, τ is directed out of the page when I flows as shown, and B is directed into the page. The brush system ensures that as the loop rotates, the current direction is switched every half turn to maintain continuous rotation in the same direction.

   The practical use of this principle is found in electric motors, where multiple loops (or coils) and a commutator (the brushes in the diagram) are used to create a steady rotational force (torque) to drive various mechanical systems.

8. 8.The Moving Coil Galvanometer

   Short Answer: A moving coil galvanometer is a device used to detect and measure small electric currents. It works on the principle that a current-carrying coil placed in a magnetic field experiences a torque.

   Long Answer: In the moving coil galvanometer, a small and light coil of wire is suspended between the poles of a magnet. When an electric current passes through the coil, it experiences a magnetic force from the magnet around it. This force causes the coil to turn. The amount the coil turns is proportional to the current that is flowing through it. The coil is attached to a pointer that moves over a scale, which shows the magnitude of the current.

   Looking at the diagram, we can see the main parts of the galvanometer:

   • Permanent Magnet: Produces a uniform magnetic field.
   • Coil: A rectangular coil of wire with many turns, where the electric current to be measured flows.
   • Pivot: A small metal pin that allows the coil to rotate.
   • Pointer: Attached to the coil, it moves over the scale to indicate the current's magnitude.
   • Scale: Graduated to measure the current.
   • Soft-Iron Core: Enhances the magnetic field's strength and makes it more uniform.
   • Sp: A spiral spring that provides the controlling torque and also leads the current in and out of the coil.

   Mathematically, the torque τ experienced by the coil is given by: =τ=nIAB where:

   • n is the number of turns in the coil
   • I is the current in the coil
   • A is the area of the coil
   • B is the magnetic field strength

   The deflection θ of the coil (and hence the pointer) is directly proportional to the current I, provided that the magnetic field B, the number of turns n, and the area of the coil A are all constant.

   In real life, galvanometers are used in laboratories to measure small currents or as part of other instruments. In terms of careers, they are important for physicists, electrical engineers, and anyone who works with electrical circuits.