Babaria Institute of Technology Subject: Electromagnetics Faculty Guide: Mansi Rastogi Topic: point and line integral form of Maxwell’s Equation MADE BY:- TANUSHA SHARMA(130050111082) PRABMEET SAINI(130050111062) SUKAMAL NARANG(130050111519) ANIRUDH VERMA(130050111094)

Time-varying field:- Moving loop in a time-varying field:- A rectangular loop of metal wire, of width w, moving with constant speed v, is entering a region of uniform B-field. -The B-field is out of the page and is increasing at a constant rate, where B0and  are positive constants. At t = 0, the right edge of the loop is a distance x0into the field, as shown. The motion of the loop and the changing of the B-field.

MOVING LOOP DIAGRAM:-

Faraday’s law in integral and point form:- Electromagnetic induction was discovered independently by Michael Faraday in 1831 and Joseph Henry in 1832. Faraday was the first to publish the results of his experiments. Based on his assessment of recently discovered properties of electromagnets, he expected that when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. He plugged one wire into a galvanometer, and watched it as he connected the other wire to a battery.

Qualitative statement:- The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit. Quantitative:- Faraday's law of induction makes use of the magnetic flux ΦB through a hypothetical surface Σ whose boundary is a wire loop. Since the wire loop may be moving, we write Σ(t) for the surface. The magnetic flux is defined by a surface integral.

where a is an element of surface area of the moving surface ς(t), b is the magnetic field (also called "magnetic flux density"). And B·dA is a vector dot product (the infinitesimal amount of magnetic flux through the infinitesimal area element dA).

Maxwell–Faraday equation:- The Maxwell–Faraday equation is a generalisation of Faraday's law that states that a time-varying magnetic field is always accompanied by a spatially- varying, non-conservative electric field, and vice versa. The Maxwell–Faraday equation is:-

Ampere’s law in integral and point form:- Ampere's law relates magnetic fields to electric currents that produce them. Ampère's law determines the magnetic field associated with a given current, or the current associated with a given magnetic field, provided that the electric field does not change over time. In its original form, Ampere's circuital law relates a magnetic field to its electric current source. The law can be written in two forms, the "integral form" and the "differential form".

In terms of total current, which includes both free and bound current, the line integral of the magnetic B-field around closed curve C is proportional to the total current I passing through a surface S. where J is the total current density (in ampere per square metre, Am−2).

In terms of free current, the line integral of the magnetic H- field (in ampere per metre, Am−1) around closed curve C equals the free current If, enc through a surface S: where Jf is the free current density only. Furthermore is the closed line integral around the closed curve C denotes a 2d surface integral over S enclosed by C is the vector dot product, dl is an infinitesimal element (a differential) of the curve C.

Treating free charges separately from bound charges, Ampere's equation including Maxwell's correction in terms of the H-field is:- where H is the magnetic H field, D is the electric displacement field and Jf is the enclosed conduction current or free current density. In differential form,

Maxwell’s Equation For Point Form:

Maxwell’s equation for line integral

Retarded potential In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past.

Retarded and advanced potentials for time-dependent fields:- For time-dependent fields, the retarded potentials are :- where r is a point in space, t is time,