A Reformulation Of Faraday’s Law
Consider a particle of charge qo moving around the circular path of The work W done on it in one revolution by the induced electric field is where is the induced emf-that is, the work done per unit charge in moving the test charge
around the path. From another point of view, the work is where qoE is the magnitude of the force acting on the test charge and 27Tr is the distance over which that force acts. Setting these two expressions for W equal to
each other and canceling qo, we find that . w, = 27T/’E. More generally. we can rewrite to give the work done on a particle of charge qo moving along any closed path: W = f F – ds = qo f E· ds. • Arguments of symmetry would also permit the lines of E around the circular path to be radial, rather than tangential. However. such radial lines would imply that there are free charges. distributed symmetrically about the axis of symmetry. on which the electric field lines could begin or end; there are no such charges.
(The circle indicates that the integral is to be taken around the closed path.) Substituting ‘f,qo for W, we find that
This integral reduces at once to Eq. 31-19 if we evaluate it for the special case of , we can expand the meaning of induced emf. Previously, induced emf has meant the work per unit charge done in maintaining current due to a
.changing magnetic flu x, or it has meant the work done per unit charge on a chargedparticle that moves around a closed path in a changing magnetic flux. However, with and , an induced emf can exist without the need of a current
or particle: An induced emf is the sum-via integration-of qua ntities £ . dsaround a closed path, where £ is the electric field induced by a changing magnetic flux and ds is a differential length vector along the closed path.If we combine Eq. 31-21 with Faraday’s law in Eq. 31-6 (‘& = -dcfJ8/dI), we can rewrite Faraday’s law as1. £ . ds = _ dcfJ8
j dt (Faraday’s law).