Question

solve that .............

Answer 1

The four Maxwell's equations (including the Maxwell–Faraday equation), along with the Lorentz force law, are a sufficient foundation to derive everything in classical electromagnetism. Therefore, it is possible to "prove" Faraday's law starting with these equations.

The starting point is the time-derivative of flux through an arbitrary, possibly moving surface in space Σ:



(by definition). This total time derivative can be evaluated and simplified with the help of the Maxwell–Faraday equation, Gauss's law for magnetism, and some vector calculus; the details are in the box below:

Click [show] (right) to see the detailed evaluation and simplification of the time-derivative of flux.Consider the time-derivative of flux through a possibly moving loop, with area Σ(t):

The integral can change over time for two reasons: The integrand can change, or the integration region can change. These add linearly, therefore:



where t0 is any given fixed time. We will show that the first term on the right-hand side corresponds to transformer EMF, the second to motional EMF (see above). The first term on the right-hand side can be rewritten using the integral form of the Maxwell–Faraday equation:





Area swept out by vector element dl of curve ∂Σ in time dt when moving with velocity v.

Next, we analyze the second term on the right-hand side:



This is the most difficult part of the proof; more details and alternate approaches can be found in references.[27][28][29] As the loop moves and/or deforms, it sweeps out a surface (see figure on right). The magnetic flux through this swept-out surface corresponds to the magnetic flux that is either entering or exiting the loop, and therefore this is the magnetic flux that contributes to the time-derivative. (This step implicitly uses Gauss's law for magnetism: Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire.) As a small part of the loop dl moves with velocity vl for a short time dt, it sweeps out a vector area vector dA = vl dt × dl. Therefore, the change in magnetic flux through the loop here is



Therefore:



where vl is the velocity of the curve ∂Σ.

Putting these together,



the result is:



where ∂Σ is the boundary of the surface Σ, and vl is the velocity of that boundary.

While this equation is true for any arbitrary moving surface Σ in space, it can be simplified further in the special case that ∂Σ is a loop of wire. In this case, we can relate the right-hand-side to EMF. Specifically, EMF is defined as the energy available per unit charge that travels once around the loop. Therefore, by the Lorentz force law,



where  is EMF and vm is the material velocity, i.e. the velocity of the atoms that makes up the circuit. If ∂Σ is a loop of wire, then vm=vl, and hence: