PPT No. 18 * Ampère’s circuital law * Applications of Ampère’s circuital law * Ampère Ampère's Circuital Law Ampère's Circuital law relates the magnetic B field to its source, the current density J. Statement of the Law : The line integral of magnetic B-field around any closed path or loop is equal to the Permeability times the net electric current through the area enclosed by the loop. Ampère's Circuital Law For infinitely narrow wire passing current I The line integral of magnetic induction B around a closed curve (or loop) is equal to μ0 times the net current I through the area bounded by the curve Ampère's Circuital Law For wire passing current density J in finite cross-section area S The line integral of magnetic induction B around a closed curve (or loop) is equal to μ0 times the surface integral of the dot product of Current density j and area element dS Ampère's Circuital Law Consider an infinitely long straight conducting wire carrying a steady current I in free space. Its magnetic B field can be represented by The magnetic field lines in form of concentric circular loops surrounding it Consider a circular Amperian loop in the form of a closed circular loop of radius r, with centre on the conductor in the plane perpendicular to the conductor. Magnetic Field of A Straight Conducting Wire Fig. Amperian loop of a straight, Current carrying wire Magnetic Field of A Straight Conducting Wire Constant B-field on the perimeter of Amperian loop Ampère's Circuital Law Every point on the Amperian loop is at the same distance from the conductor. The total length of the path is equal to the circumference of the circle: ℓ = 2 π r The magnitude of magnetic B- field at any point on the loop is given by Biot Savart law B = μ0 I / 2 π r Ampère's Circuital Law Line integral of magnetic induction B around this closed loop is = 2πrB = μ0 I This equation is called as the Ampere's Law. Ampère's Circuital Law In general, when the magnetic field B is always directed along the tangent to the perimeter of a closed curve and has constant value B at all points on the perimeter then B field x Perimeter of closed loop = μ0 x Total current passing through the surface enclosed by the loop Ampère's Circuital Law The equation for the line integral of B is independent of the position relative to the source. It is valid for any closed loop around the wire, the path need not be circular or perpendicular to the wires. It is true for distorted paths or Irregular closed loops or complicated magnetic fields. The circulation of B over a closed loop which does not enclose the current carrying conductor is zero as the current in enclosing loop is zero Ampère's Circuital Law Equation for Ampère's Circuital Law can be generalized for any number of wires carrying any number of currents in any direction. The resultant magnetic field B is due to the net current (algebraic sum = I1 - I2 - I3 + I4) through the surface enclosed by the loop around the wires This is the Principle of Superposition Principle of Superposition The resultant magnetic field B is due to many current carrying wires enclosed by a closed path Ampère's Circuital Law Ampere’s law in Magnetostatics is analogues to the Gauss law in electrostatics in the sense that Ampere’s law connects the line integral of magnetic B-field to enclosed electric currents, while Gauss’ law connects the surface integral of electric field to enclosed charge density Ampere’s law also has differential and integral forms which are equivalent. Ampère's Circuital Law Ampere’s law and Biot Savart’s law can be derived from each other and are equivalent in scientific conent. However, in the case of Biot Savert law, evaluation of complicated integrals is required. It can be avoided by using Ampere’s law. Its power is evident in simplifying physical situations where B-field has some kind of symmetry (e.g. circular, cylindrical etc. symmetry) Applications of Ampere’s Law Magnetic Field of a Solenoid A solenoid is a long wire carrying steady current wound closely in the form of a helix. The wire is coated with insulating material so the adjacent turns are electrically insulated. A solenoid carrying current I, having radius a, length L, Total number of turns N, Number of turns/unit length n = N/L has magnetic field B at a point P well inside the solenoid B = μ0nI B is independent of the length and diameter of the solenoid & uniform over the cross-section of the solenoid Applications of Ampere’s Law Magnetic Field of a Solenoid Applications of Ampere’s Law Toroid: The magnetic field within the volume enclosed by a toroid consisting of N evenly spaced turns of wire carrying current I at a point at distance R from the axis passing through centre and perpendicular to the plane of toroid is given by B is not constant at different points over the cross-section of toroid (unlike solenoid). Applications of Ampere’s Law Magnetic Field of a Toroid The magnetic field within the volume enclosed by a toroid Ampère The ampere is the SI unit of Electric current and is one of the seven SI base units. Its symbol is A. It is named after Andre-Marie Ampere (1775–1836), the French physicist and mathematician, considered as the father of electrodynamics. André-Marie Ampère Ampère The formal definition of ampere is based on the Ampere’s Circuital Law which gives the attractive force between two parallel wires carrying an electric current. One Ampere is the constant current which will produce an attractive force of 2 10–7 newton per meter of length between two straight, parallel, infinitely long conductors having negligible circular cross section placed one meter apart in a perfect vacuum. Ampère Using the SI definition of electric charge, the Coulomb, an electric current of one ampere is one coulomb of charge flowing past a given point /second Charge Q is determined by steady current I flowing for a time t as Q = It. The ampere is a measure of the amount of electric charge passing a point per unit time. Ampère One ampere is equal to around 6.241 1018 (or exactly 6 241 509 479 607 717 888) elementary charge units such as electrons passing a given point each second. Other electrical units are all defined in terms of the ampere, the base unit. For example, One ampere of current results from a potential distribution of one volt per ohm of resistance, or from a power production rate of one watt per volt of potential. Ampère In practice, the name Ampere is often abbreviated as amp. It is likely that in place of the above idealized definition, new definition for Ampere may be adopted at the CGPM (The General Conference on Weights and Measures) meeting in 2011.