Unveiling the Mysteries of Magnetism: From Fields to Materials

16. Magnetic Fields and Their Interactions

Magnetism is a fundamental force of nature, closely intertwined with electricity. It governs the behavior of compasses, motors, and even the Earth's protective magnetic shield.

16.1 Magnetic Field Lines and Magnetic Flux; Oersted’s Experiment

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. We visualize magnetic fields using magnetic field lines, which are imaginary lines indicating the direction of the magnetic force on a north pole. These lines always form closed loops, never intersect, and are denser where the field is stronger.

Magnetic flux (Φ_B) is a measure of the total number of magnetic field lines passing through a given area. It is calculated as Φ_B = B⋅A⋅cos(θ), where B is the magnetic field strength, A is the area, and θ is the angle between the magnetic field vector and the area vector. The unit of magnetic flux is the Weber (Wb).

Oersted’s Experiment, performed in 1820 by Hans Christian Ørsted, was a groundbreaking discovery. He observed that a compass needle deflected when placed near a current-carrying wire. This demonstrated the direct connection between electricity and magnetism, proving that electric currents create magnetic fields.

16.2 Force on Moving Charge; Force on a Conductor

A moving electric charge experiences a force when it enters a magnetic field. This Lorentz force (F) is given by the equation F = qvBsin(θ), where q is the charge, v is its velocity, B is the magnetic field strength, and θ is the angle between the velocity vector and the magnetic field vector. The direction of the force is perpendicular to both the velocity and the magnetic field, and can be determined by the right-hand rule (for positive charges).

Similarly, a current-carrying conductor placed in a magnetic field also experiences a force. This force is the sum of the Lorentz forces on all the moving charges within the conductor. The magnitude of the force on a conductor of length L carrying current I in a magnetic field B is given by F = BILsin(θ), where θ is the angle between the current direction and the magnetic field. This principle is fundamental to the operation of electric motors.

16.3 Force and Torque on Rectangular Coil, Moving Coil Galvanometer

When a rectangular current loop is placed in a uniform magnetic field, the forces on the opposite sides are equal and opposite, but they may not be collinear, leading to a torque (τ). The torque tends to rotate the coil. The magnitude of the torque is given by τ = NIABsin(α), where N is the number of turns, I is the current, A is the area of the coil, B is the magnetic field, and α is the angle between the magnetic moment of the coil and the magnetic field.

This principle is the basis for the moving coil galvanometer, a sensitive instrument used to detect and measure small electric currents. A coil suspended in a radial magnetic field experiences a torque proportional to the current, causing it to deflect. The deflection is then used to indicate the current's magnitude.

16.4 Hall Effect

The Hall effect describes the production of a voltage difference (the Hall voltage) across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current. This effect is a result of the Lorentz force acting on the charge carriers. It can be used to determine the type of charge carriers (electrons or holes) and their concentration in a material, as well as to measure magnetic field strengths.

16.5 Magnetic Field of a Moving Charge

A moving electric charge generates its own magnetic field. The magnetic field (B) produced by a point charge (q) moving with velocity (v) at a distance (r) from the charge is given by a simplified form of the Biot-Savart law for a point charge: B = (μ₀/4π) * (qvsin(θ)/r²), where μ₀ is the permeability of free space and θ is the angle between the velocity vector and the position vector from the charge to the point where the field is being measured. The direction of the field can be found using the right-hand rule.

16.6 Biot and Savart Law and Its Application

The Biot-Savart Law is a fundamental law in electromagnetism that describes the magnetic field generated by a steady electric current. It states that the magnetic field dB at a point P due to a current element Idl is given by dB = (μ₀/4π) * (Idl x r̂ / r²), where r̂ is the unit vector from the current element to the point P, and r is the distance.

Applications:

• Circular Coil: The magnetic field at the center of a circular coil of radius R carrying current I is B = (μ₀I)/(2R). Along the axis, the field is B = (μ₀IR²)/(2(R²+x²)^3/2).
• Long Straight Conductor: The magnetic field at a distance 'r' from a long straight current-carrying wire is B = (μ₀I)/(2πr).
• Long Solenoid: Inside a long solenoid, the magnetic field is approximately uniform and given by B = μ₀nI, where 'n' is the number of turns per unit length.

16.7 Ampere’s Law and Its Applications

Ampere’s Law provides a simpler way to calculate magnetic fields for situations with high symmetry. It states that the line integral of the magnetic field B around any closed loop (called an Amperian loop) is proportional to the total steady current I_enc passing through the area enclosed by the loop: ∮ B⋅dl = μ₀I_enc.

Applications:

• Long Straight Conductor: Using an Amperian circular loop, Ampere's law readily gives B = (μ₀I)/(2πr), identical to the Biot-Savart result.
• Straight Solenoid: Applying Ampere's law to a rectangular Amperian loop inside and outside a long solenoid confirms that B = μ₀nI inside and B ≈ 0 outside.
• Toroidal Solenoid: For a toroidal solenoid, the magnetic field inside the toroid is B = (μ₀NI)/(2πr), where N is the total number of turns and r is the average radius of the toroid.

16.8 Force Between Two Parallel Conductors Carrying Current - Definition of Ampere

Two parallel current-carrying conductors exert forces on each other due to the magnetic fields they produce. If the currents flow in the same direction, the wires attract; if they flow in opposite directions, they repel. The force per unit length (F/L) between two long parallel conductors separated by distance 'd' and carrying currents I₁ and I₂ is given by F/L = (μ₀I₁I₂)/(2πd).

This force is used to define the Ampere (A), the SI unit of electric current. One ampere is defined as the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one meter apart in vacuum, would produce between these conductors a force equal to 2 x 10^-7 newtons per meter of length.

17. Magnetic Properties of Materials

Materials respond differently when placed in a magnetic field, a property that is crucial for various technological applications, from data storage to medical imaging.

17.1 Magnetic Field Lines and Magnetic Flux (Revisited in Materials Context)

When a material is placed in an external magnetic field, its internal structure (electron spins and orbital motions) interacts with the field, leading to a modification of the magnetic field lines and the magnetic flux within the material. The magnetic field lines will either converge into the material (for paramagnetic and ferromagnetic materials) or diverge from it (for diamagnetic materials).

17.2 Flux Density in Magnetic Material; Relative Permeability; Susceptibility

Magnetic flux density (B) inside a material is the total magnetic field within the material. It includes the applied external field and the field produced by the material's own magnetization.

Relative permeability (μ_r) is a dimensionless quantity that describes the degree of magnetization of a material in response to an applied magnetic field. It is defined as μ_r = μ/μ₀, where μ is the absolute permeability of the material and μ₀ is the permeability of free space. It indicates how much more (or less) effectively a material concentrates magnetic flux lines than a vacuum.

Magnetic susceptibility (χ_m) is another dimensionless quantity that indicates the degree to which a material can be magnetized in an applied magnetic field. It is related to relative permeability by the equation χ_m = μ_r - 1. A positive susceptibility indicates that the material is attracted to magnetic fields, while a negative susceptibility indicates repulsion.

17.3 Hysteresis

Hysteresis is the phenomenon where the magnetization of a ferromagnetic material depends not only on the current applied magnetic field but also on its magnetic history. When a ferromagnetic material is subjected to a varying magnetic field, its magnetization lags behind the field. This lagging forms a hysteresis loop on a B-H curve (magnetic flux density vs. magnetic field intensity). The area of the hysteresis loop represents the energy lost as heat during each cycle of magnetization and demagnetization. Materials with wide hysteresis loops are suitable for permanent magnets, while those with narrow loops are used for transformers and electromagnets.

17.4 Dia-, Para- and Ferro-magnetic Materials

Materials are classified into three main categories based on their magnetic properties:

• Diamagnetic Materials: These materials have a weak, negative susceptibility (χ_m < 0, μ_r < 1). They are weakly repelled by magnetic fields. In these materials, all electron orbits are paired, so there is no net intrinsic magnetic moment. When an external field is applied, it induces a small opposing magnetic moment. Examples include water, copper, gold, and most organic compounds.
• Paramagnetic Materials: These materials have a small, positive susceptibility (0 < χ_m < 1, μ_r > 1). They are weakly attracted to magnetic fields. They possess permanent magnetic dipoles due to unpaired electrons, but these dipoles are randomly oriented in the absence of an external field. When an external field is applied, the dipoles align slightly with the field, leading to a weak magnetization. Examples include aluminum, platinum, and oxygen.
• Ferromagnetic Materials: These materials have a large, positive susceptibility (χ_m >> 1, μ_r >> 1) and are strongly attracted to magnetic fields. They exhibit spontaneous magnetization even in the absence of an external field, due to strong interactions between atomic magnetic moments, forming regions called magnetic domains. When an external field is applied, these domains grow and align, leading to strong magnetization. They also exhibit hysteresis. Examples include iron, nickel, cobalt, and their alloys.