Michael Kurniawan 1910403011 University of Cincinnati MAGNETIC FIELDS AND AMPÈRE'S LAW 1. What is a magnetic field? A magnetic field is a vector field that represents the magnetic influence on moving electric charges, electric currents, and magnetic materials. A magnetic field is produced by moving electric charges (currents) and intrinsic magnetic moments of elementary particles associated with a fundamental quantum property called spin. The direction of the magnetic field at a point is the direction of the force that a north pole of a magnet would experience at that point. The strength and direction of a magnetic field are represented by magnetic field lines, with the direction given by the tangent to the lines at any point and the strength indicated by the density of the lines. 2. How do you calculate the magnetic field around a straight currentcarrying wire? The magnetic field 𝐵 around a long, straight current-carrying wire can be calculated using Ampère's law or the Biot-Savart law. Using the Biot-Savart law, the magnetic field at a distance 𝑟 from a wire carrying a current 𝐼 is given by: where 𝜇0 is the permeability of free space (𝜇0=4𝜋×10−7 T⋅m/A). The magnetic field forms concentric circles around the wire, with the direction given by the right-hand rule (if the thumb points in the direction of the current, the fingers curl in the direction of the magnetic field). 3. What is the difference between a magnetic field and an electric field? a. Source: An electric field is produced by electric charges (either stationary or moving), while a magnetic field is produced by moving electric charges (currents) and the intrinsic magnetic moments of particles. b. Force: An electric field exerts a force on electric charges regardless of their motion, while a magnetic field exerts a force on moving charges and current-carrying conductors, but not on stationary charges. c. Direction: The direction of an electric field is the direction of the force experienced by a positive test charge, while the direction of a magnetic field is the direction in which the north pole of a compass needle points. d. Field Lines: Electric field lines begin on positive charges and end on negative charges, whereas magnetic field lines form closed loops, emerging from the north pole of a magnet and entering the south pole. e. Units: Electric field strength is measured in volts per meter (V/m), while magnetic field strength is measured in teslas (T) or gauss (G), where 1T=10,000 G 4. What is the principle of Ampère's law? Ampère's law states that the line integral of the magnetic field 𝐵 around any closed loop is proportional to the total current 𝐼enc passing through the loop. Mathematically, it is expressed as: where ∮𝐵⋅𝑑𝑙 is the line integral of the magnetic field around the closed loop, 𝜇0 is the permeability of free space, and 𝐼enc is the total current enclosed by the loop. Ampère's law is useful for calculating magnetic fields in situations with high symmetry, such as around long straight wires, solenoids, and toroids. 5. How is Ampère's law used to calculate the magnetic field around a solenoid? Ampère's law can be used to calculate the magnetic field inside and outside a long solenoid, which is a coil of wire wound in the form of a helix. For a solenoid of length 𝐿, carrying a current 𝐼, with 𝑁 turns per unit length (turns per meter), Ampère's law gives: Inside the solenoid: a. The magnetic field 𝐵 is uniform and directed along the axis of the solenoid. b. Its magnitude is given by: Outside the solenoid: The magnetic field is negligible (close to zero) if the solenoid is long enough compared to its diameter. 6. What is the Biot-Savart law and how does it relate to Ampère's law? The Biot-Savart law describes the magnetic field 𝐵 generated by a steady current 𝐼 (in amperes) through a small segment of wire or a small current element 𝑑𝑙. It is expressed as: where 𝜇0 is the permeability of free space, 𝐼 is the current, 𝑑𝑙 is the differential element of the current-carrying conductor, 𝑟^ is the unit vector from the current element to the point where the magnetic field is being calculated, and 𝑟 is the distance between the current element and the point. Ampère's law is a generalization of the Biot-Savart law for situations with high symmetry, where the closed-loop integral of the magnetic field around a path equals the total current enclosed. Ampère's law relates the magnetic field to the current enclosed by a closed loop, providing a more straightforward method for calculating the magnetic field in cases of high symmetry, such as long straight wires, solenoids, and toroids.\ 7. How is the magnetic field around a circular loop of wire calculated using Ampère's law? Ampère's law can be applied to calculate the magnetic field around a circular loop of wire carrying a steady current 𝐼: a. Choose a circular path (Amperian loop) of radius 𝑟 centered on the wire loop. b. Ampère's law states: c. The magnetic field 𝐵 at a point on the axis perpendicular to the center of the loop (at distance 𝑥 from the center) is: d. This formula shows that the magnetic field strength decreases as 1/𝑟 with distance 𝑟 from the loop and as 1/𝑥3 with distance 𝑥 from the center along the axis. 8. How was Ampère's law modified by Maxwell to include displacement current? Maxwell's modification of Ampère's law, known as Maxwell-Ampère law, includes the displacement current term to ensure consistency with the laws of electromagnetism and the propagation of electromagnetic waves: where 𝐼free is the free current (the conventional current), 𝜖0 is the permittivity of free space, 𝑑Φ𝐸/𝑑𝑡 is the rate of change of electric flux, and 𝜇0 is the permeability of free space. a. The term 𝜖0 𝑑Φ𝐸/𝑑𝑡 represents the displacement current, which is related to the rate of change of the electric field through a surface. It was introduced by Maxwell to complete the set of four Maxwell's equations and to describe electromagnetic wave propagation. b. Maxwell's modification ensures that the laws of electromagnetism are consistent and applicable to all situations involving time-varying electric fields and currents, including the propagation of electromagnetic waves. 9. How is the magnetic field inside a toroid calculated using Ampère's law? A toroid is a doughnut-shaped coil of wire used to create a magnetic field. The magnetic field inside a toroid can be calculated using Ampère's law as follows: a. Consider a toroid with 𝑁 turns of wire, carrying a current 𝐼, and having a mean radius 𝑅. b. Choose an Amperian loop that is a circle of radius 𝑟 centered on the axis of the toroid and lying entirely within the toroidal coil. c. By symmetry, the magnetic field 𝐵 inside the toroid is tangent to the Amperian loop and uniform in magnitude along the loop. d. Applying Ampère's law: where 2𝜋𝑟 is the circumference of the Amperian loop, and 𝑁⋅𝐼 is the total current enclosed by the loop (since there are 𝑁 turns, each carrying current 𝐼). e. Solving for 𝐵, the magnetic field inside the toroid is: where 𝑟 is the radius at which the magnetic field is being calculated inside the toroid. f. The magnetic field is confined to the region inside the toroid and decreases with increasing distance from the center of the toroid. 10. How does the distribution of the magnetic field in free space compare to that in ferromagnetic materials? The distribution and behavior of magnetic fields differ significantly between free space and ferromagnetic materials due to the materials' magnetic properties. a. In Free Space: • The magnetic field is generated by electric currents or changing electric fields. • The relationship between the magnetic field 𝐵 and the magnetic field intensity 𝐻 is given by 𝐵=𝜇0𝐻, where 𝜇0 is the permeability of free space (𝜇0≈4𝜋×10−7 H/m). • The field lines form closed loops and are continuous in space. b. In Ferromagnetic Materials: • Ferromagnetic materials, such as iron, cobalt, and nickel, have a high magnetic permeability (𝜇), which can be several thousand times greater than 𝜇0 . • The magnetic field inside a ferromagnetic material is much stronger than in free space for the same applied magnetic field intensity 𝐻, due to the material's ability to concentrate magnetic flux. • The relationship between 𝐵 and 𝐻 in ferromagnetic materials is given by 𝐵=𝜇𝐻, where 𝜇=𝜇0 𝜇 𝑟 and 𝜇𝑟 is the relative permeability of the material. • The field lines tend to be concentrated within the material, resulting in a stronger and more confined magnetic field. • Ferromagnetic materials can become magnetized, retaining some magnetization even after the external magnetic field is removed, due to the alignment of magnetic domains within the material.