Gavin Howe
Ampere's law and Biot-Savart law are both fundamental to understanding magnetism and electromagnetism. Ampere's law relates the magnetic field around a closed loop to the electric current passing through that loop, and it is used when the symmetry of the problem permits, i.e., when the magnetic field is constant. Biot-Savart law, on the other hand, is used when there is not enough symmetry to apply Ampere's law. It is a more fundamental approach that evaluates the integral of the current to determine the magnetic field. While Ampere's law is a simplified application of Biot-Savart law in certain scenarios, Biot-Savart law can be used in a broader range of situations, including those with time-varying currents.
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What You'll Learn
Ampere's Law and applications
Ampere's Law, also known as Ampère's circuital law, is a fundamental property of a static magnetic field and a fundamental principle in electromagnetism that relates the circulation of a magnetic field around a closed loop to the electric current passing through that loop. In other words, it specifies the magnetic field that is associated with a given current or vice-versa, provided that the electric field does not change over time.
Ampere's Law can be derived directly from the Biot-Savart law, which is used to calculate the resultant magnetic flux density at a position in 3D space generated by a filamentary current. Ampere's Law simplifies the calculation process by using a certain symmetry. For example, if the problem involves an infinite, straight wire, Ampere's Law can be used to calculate the magnetic field at some radial distance from the wire.
Ampere's Law has many practical applications, the main one being calculating the magnetic field generated by an electric current. This is useful in electromagnets, motors, generators, and transformers.
Ampere's Law can be expressed mathematically as the line integral of the magnetic field surrounding a closed loop being equal to the algebraic sum of the currents passing through that loop. The magnetic field created by an electric current is proportional to the size of that electric current, with a constant of proportionality equal to the permeability of free space.
Ampere's Law is named after André-Marie Ampère, who performed experiments with forces acting on current-carrying wires in the late 1820s.
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Biot-Savart Law and applications
Biot-Savart's law, first introduced in 1802, is a mathematical formula that describes the relationship between force, displacement, and velocity. It is a fundamental principle of magnetostatics, which deals with the relationship between magnetic fields and electric currents. The law is used to calculate the resultant magnetic field at a specific position in three-dimensional space. It is only applicable to steady currents, where the flow of charge is continuous and does not change over time.
The Biot-Savart law is particularly useful for understanding the magnetic field's direction, length, magnitude, and proximity to the electric current. It is represented by the equation:
> B = (μ₀ * I) / (2R)
Where B is the magnetic field intensity, μ₀ is the magnetic permeability of free space, I is the current, and R is the distance from the current-carrying wire.
Some specific applications of the Biot-Savart law include:
- Calculating the magnetic field in an indefinitely long straight wire due to a constant current.
- Determining the magnetic field in the centre of a current-carrying arc or circular coil.
- Calculating the force between two parallel and lengthy current-carrying conductors.
- Finding the magnetic field induction of a circular current loop, both at its centre and along its axis, which is crucial for technologies such as MRI machines and electromagnets.
- Calculating magnetic responses at the atomic or molecular level, such as chemical shieldings or magnetic susceptibilities.
- In aerodynamic theory, it is used to calculate the velocity induced by vortex lines.
In summary, the Biot-Savart law is a fundamental tool in electromagnetism, providing valuable insights into the behaviour of magnetic fields generated by electric currents. Its applications range from understanding basic principles to designing advanced technologies.
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When to use each law
Ampere's Law and Biot-Savart Law are both used to calculate magnetic fields. However, there are some key differences between the two laws that dictate when to use each one.
Ampere's Law is used when the symmetry of the problem permits, i.e., when the magnetic field around an 'Amperian loop' is constant. For example, to find the magnetic field from an infinite straight current-carrying wire at some radial distance. Ampere's Law is derived directly from Biot-Savart Law and works well when there is a path to integrate over which the magnetic field is easy to simplify.
Biot-Savart Law, on the other hand, is used when there is not enough symmetry to use Ampere's Law. It is a more brute-force approach and is used to evaluate the magnetic field at some point along the axis of a current loop. Biot-Savart Law is fundamental to magnetostatics and is valid when the magnetostatic approximation applies. It is also used in situations where the electric field is not given by a gradient, such as when the current in a very long and thin solenoid increases rapidly. Biot-Savart Law can be used to calculate magnetic responses at the atomic or molecular level, provided that the current density can be obtained from a quantum mechanical calculation or theory.
In summary, Ampere's Law is used when there is symmetry in the problem and the magnetic field is constant, while Biot-Savart Law is used when there is less symmetry and a more brute-force approach is needed. Biot-Savart Law is also particularly useful in magnetostatics and for calculating magnetic responses at small scales.
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The relationship between the laws
Ampere's Law and Biot-Savart Law are both fundamental to understanding magnetism and electromagnetism. They are used to calculate magnetic fields, but the relationship between these laws and their applicability depends on the specific situation and the symmetry of the problem.
Ampere's Law relates the integrated magnetic field around a closed loop to the electric current passing through that loop. It is used when there is symmetry in the problem, such as when the magnetic field around an "Amperian loop" is constant. For example, it can be applied to find the magnetic field from an infinite straight current-carrying wire at a certain distance. Ampere's Law is derived directly from the Biot-Savart Law and works well when the path of integration simplifies the calculation.
Biot-Savart Law, on the other hand, is more fundamental and is used when there is insufficient symmetry to apply Ampere's Law. It is valid in magnetostatic situations and consistent with Ampere's Law and Gauss's Law for magnetism. Biot-Savart Law is used to compute the resultant magnetic flux density at a position in 3D space generated by a filamentary current. It is also applicable to situations beyond magnetostatics, such as in aerodynamic theory and at the atomic or molecular level, provided that the current density can be determined.
In summary, Ampere's Law is a specialised application of Biot-Savart Law, useful when there is symmetry in the problem, particularly with infinite wires. Biot-Savart Law is more versatile and can handle a wider range of scenarios, including those beyond magnetostatics, but it is more complex and requires more computational effort.
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The discovery of each law
The Discovery of Biot-Savart Law
In 1820, Hans Christian Ørsted discovered that an electric current generates a magnetic field. This discovery sparked further research into the relationship between electricity and magnetism. Inspired by Ørsted's work, French scientists Jean-Baptiste Biot and Félix Savart conducted experiments in 1820, leading to the formulation of the Biot-Savart Law. This law specifies the magnetic field generated by a steady, distributed current.
The Discovery of Ampere's Law
André-Marie Ampère, a French physicist, furthered Ørsted's discovery by investigating the magnetic force between two current-carrying wires. Ampère experimented with magnets and electric currents, observing that the attraction or repulsion between magnets depended on the direction of the current flow and the orientation of the magnets. Through these experiments and different equations, he formulated Ampère's Law, one of the most fundamental laws of electromagnetism. This law relates the circulation of a magnetic field around a closed loop to the electric current passing through that loop.
In the 1850s, Scottish mathematical physicist James Clerk Maxwell generalized Ampère's work and the results of others into a single mathematical law, known as Maxwell's Circuital Law. Maxwell's work was based on an analogy to hydrodynamics and was published in his 1855 paper, "On Faraday's Lines of Force." However, Maxwell's original law only applied to magnetostatic situations with steady, continuous currents flowing in closed circuits.
To address this limitation, Maxwell introduced the concept of displacement current in his 1861 paper, "On Physical Lines of Force." He added this term to Ampère's law, creating the Ampère-Maxwell Law, which is one of the four Maxwell's Equations. The resulting equation could account for time-varying electric currents and is considered one of the foundations of classical electromagnetism.
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Frequently asked questions
Ampere's Law relates the integrated magnetic field around a closed loop to the electric current passing through that loop. It is used when the symmetry of the problem permits, i.e. when the magnetic field around an 'Amperian loop' is constant.
Biot-Savart Law states that the magnetic field produced at a point in space by a small segment of current-carrying wire is directly proportional to the current, the length of the wire segment, and the sine of the angle between the wire segment and the line connecting the wire segment to the point. It is used when there is not enough symmetry to use Ampere's Law.
Ampere's Law is used when the problem has symmetry, whereas Biot-Savart Law is used when there is no symmetry and is considered a more brute-force approach.
Ampere's Law is used when the magnetic field around a closed loop is constant. Biot-Savart Law is used when there is no symmetry and is applied to calculate the magnetic field in the middle of a current running in a loop.
