Thermoelectric Effects and Magnetic Fields: An Introduction

Explore the fundamental principles of thermoelectric effects (Seebeck, Peltier) and delve into the fascinating world of magnetic fields, covering key concepts like magnetic flux, forces on charges and conductors, and the applications of Biot-Savart and Ampere's laws.

15. Thermoelectric Effects

Thermoelectric effects describe the direct conversion of temperature differences to electric voltage and vice versa. These phenomena are at the heart of various technologies, from temperature measurement to power generation and refrigeration.

15.1 Seebeck Effect; Thermocouples

The Seebeck effect is the phenomenon where a temperature difference between two dissimilar electrical conductors or semiconductors produces a voltage difference between them. If a circuit is formed, this voltage can drive a current. The magnitude of the Seebeck effect is typically expressed by the Seebeck coefficient (S), which is defined as the voltage produced per unit temperature difference (S = V/ΔT).

Example: Thermocouple

A thermocouple is a device consisting of two different conductors (A and B) that make contact at two junctions, J1 and J2. If J1 is at temperature T1 and J2 is at temperature T2 (and T1 ≠ T2), a voltage will be generated in the circuit. This voltage is proportional to the temperature difference and the Seebeck coefficients of the materials. Thermocouples are widely used for temperature measurement due to their robustness, wide temperature range, and relatively quick response time.

15.2 Peltier Effect: Variation of Thermoelectric e.m.f. with Temperature; Thermopile

The Peltier effect is the inverse of the Seebeck effect: when an electric current flows through a junction between two different conductors, heat is either absorbed or released at the junction. If current flows in one direction, the junction cools; if it flows in the opposite direction, the junction heats up. This effect is used in Peltier coolers (thermoelectric coolers) to create active cooling without refrigerants.

The thermoelectric e.m.f. (electromotive force) generated in a thermocouple varies with temperature. This relationship is often non-linear and can be expressed by polynomial equations. The rate of change of thermoelectric e.m.f. with temperature is related to the Seebeck coefficient.

What is a Thermopile?

A thermopile is a device that converts thermal energy into electrical energy. It consists of several thermocouples connected in series, usually arranged such that alternate junctions are at different temperatures. By connecting multiple thermocouples, the small voltages generated by individual junctions are summed, producing a larger, more easily measurable output voltage. Thermopiles are used in applications like infrared sensors and energy harvesting.


16. Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It is a fundamental concept in physics, crucial for understanding phenomena ranging from compasses to electric motors and generators.

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

Magnetic field lines are a visual representation of a magnetic field. They originate from the North pole and terminate at the South pole outside the magnet, forming continuous loops. The density of field lines indicates the strength of the magnetic field, and the direction of the field at any point is tangent to the field line.

Magnetic flux (ΦB) is a measure of the total number of magnetic field lines passing through a given area. It is defined as ΦB = ∫B ⋅ dA, where B is the magnetic field and dA is an infinitesimal area vector. The SI unit for magnetic flux is the Weber (Wb).

Oersted's Experiment (1820)

Hans Christian Ørsted discovered that electric currents create magnetic fields. He observed that a compass needle deflected when placed near a wire carrying an electric current. This groundbreaking experiment established a direct link between electricity and magnetism, paving the way for the field of electromagnetism.

16.2 Force on Moving Charge; Force on a Conductor

The Lorentz force law describes the force exerted by a magnetic field on a moving electric charge. The force (F) on a charge (q) moving with velocity (v) in a magnetic field (B) is given by: F = q(v × B). The force is perpendicular to both the velocity of the charge and the magnetic field direction. Its magnitude is F = qvBsinθ, where θ is the angle between v and B.

A current-carrying conductor in a magnetic field experiences a force because the current is essentially a collection of moving charges. The force (F) on a conductor of length (L) carrying current (I) in a magnetic field (B) is given by: F = I(L × B), where L is a vector in the direction of the current.

16.3 Force and Torque on Rectangular Coil, Moving Coil Galvanometer

A rectangular coil carrying current in a uniform magnetic field experiences a net force that can be zero, but it experiences a net torque. The torque (τ) on a coil with N turns, area A, carrying current I, in a magnetic field B is given by: τ = N(IA × B) or τ = μ × B, where μ = NIA is the magnetic dipole moment of the coil.

Moving Coil Galvanometer

A moving coil galvanometer is an instrument 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. This torque causes the coil to rotate, and the deflection of a pointer attached to the coil is proportional to the current flowing through it.

16.4 Hall Effect

The Hall effect is 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, pushing them to one side of the conductor. The Hall effect can be used to determine the type of charge carriers (positive or negative) and their density in a material, as well as to measure magnetic field strengths (Hall sensors).

16.5 Magnetic Field of a Moving Charge

A single moving electric charge produces a 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 the Biot-Savart law for a point charge: B = (μ0 / 4π) * (qv × ) / r², where μ0 is the permeability of free space and is a unit vector from the charge to the point where the field is being measured.

16.6 Biot and Savart Law and its application to (i) a circular coil (ii) a long straight conductor (iii) a long solenoid

The Biot-Savart law is a fundamental equation in magnetostatics that describes the magnetic field generated by an electric current. It states that the magnetic field dB at a point due to a current element IdL is given by: dB = (μ0 / 4π) * (IdL × ) / r², where IdL is a vector representing the current element and is a unit vector from the current element to the point of interest.

Applications of Biot-Savart Law:

  • (i) Circular Coil: For a circular coil of radius R carrying current I, the magnetic field at its center is B = (μ0I) / (2R). Along the axis, the field is B = (μ0IR²) / (2(R² + x²)^(3/2)).
  • (ii) Long Straight Conductor: For an infinitely long straight conductor carrying current I, the magnetic field at a distance r from the wire is B = (μ0I) / (2πr). The field lines are concentric circles around the wire.
  • (iii) Long Solenoid: For a long solenoid with n turns per unit length carrying current I, the magnetic field inside the solenoid is approximately uniform and given by B = μ0nI. Outside the solenoid, the field is negligible.

16.7 Ampere’s Law and its applications to (i) a long straight conductor (ii) a straight solenoid (iii) a toroidal solenoid

Ampere's law is another fundamental law in electromagnetism, providing a relationship between the magnetic field and the electric current producing it. In its integral form, it states that the line integral of the magnetic field B around any closed loop is proportional to the total electric current Ienc passing through the loop: ∮B ⋅ dL = μ0Ienc. Ampere's law is particularly useful for highly symmetric current distributions.

Applications of Ampere's Law:

  • (i) Long Straight Conductor: Applying Ampere's law to a circular Amperian loop around a long straight conductor yields B = (μ0I) / (2πr), consistent with the result from Biot-Savart law.
  • (ii) Straight Solenoid: For a long straight solenoid, choosing a rectangular Amperian loop, we can derive the magnetic field inside to be B = μ0nI, where n is the number of turns per unit length.
  • (iii) Toroidal Solenoid (Toroid): For a toroidal solenoid with N turns and radius r, carrying current I, the magnetic field inside the toroid is B = (μ0NI) / (2πr). The field is confined entirely within the toroid.

16.8 Force between two parallel conductors carrying current - definition of ampere

When two parallel conductors carry electric currents, they exert a force on each other. If the currents flow in the same direction, the conductors attract each other; if the currents flow in opposite directions, they repel each other. This is because each wire produces a magnetic field, and the other wire experiences a force due to that magnetic field.

The force per unit length (F/L) between two long, parallel conductors separated by a distance d and carrying currents I₁ and I₂ is given by: F/L = (μ0I₁I₂) / (2πd).

Definition of Ampere

The Ampere (A) is the SI base unit of electric current. Its definition is based on the force between two parallel conductors. Specifically, 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 × 10⁻⁷ Newtons per meter of length.

Tags

Thermoelectric Effect Magnetic Field Electromagnetism
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