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Electric Charges And Fields

Question
CBSEENPH12039298

How are em waves produced by oscillating charges?
 
Draw a sketch of linearly polarised em waves propagating in the Z-direction. Indicate the directions of the oscillating electric and magnetic fields.

                                                      OR

Write Maxwell's generalisation of Ampere's circuital law. Show that in the process of charging a capacitor, the current produced within the plates of the capacitor is I = straight epsilon subscript straight o space dϕ subscript straight E over dt, where straight ϕ subscript straight E is the electric flux produced during charging of the capacitor plates. 

Solution

A moving charge produces both electric and magnetic fields, and an oscillating charge produces oscillating magnetic and electric fields. These oscillating electric and magnetic fields with respect to space and time produce electromagnetic waves. 
The propagation of electromagnetic waves can be shown as:


                                                  OR

Maxwell's generalization of ampere's circuital law given by,

contour integral space B with rightwards harpoon with barb upwards on top. space space stack d l with rightwards harpoon with barb upwards on top space equals space mu subscript o space left parenthesis I space plus space I subscript D right parenthesis thin space equals space straight mu subscript straight o space left parenthesis space straight I space plus space straight epsilon subscript straight o space dφ over dt right parenthesis
Consider that a parallel capacitor C is charging in a circuit. 
The magnitude of electric field between the two plates will be, 
straight E space equals space fraction numerator straight q over denominator straight epsilon subscript straight o straight A end fraction , is perpendicular to the surface of the plate. 
Electric flux through the surface will be, 
straight ϕ subscript straight E space equals space straight E with rightwards harpoon with barb upwards on top. space straight A with rightwards harpoon with barb upwards on top space

space space space space space equals space EA space cos space straight theta

space space space space space equals space fraction numerator straight q over denominator straight epsilon subscript straight o space straight A end fraction space straight x space straight A space

space space space space space equals space straight q over straight epsilon subscript straight o

rightwards double arrow space dϕ subscript straight E over dt space equals space fraction numerator straight d space open parentheses begin display style straight q over straight epsilon subscript straight o end style close parentheses over denominator dt end fraction

rightwards double arrow space dq over dt space equals space straight epsilon subscript straight o space dϕ subscript straight E over dt
That space is comma space dq over dt space is space the space rate space of space change space with space time. space

rightwards double arrow space straight I thin space equals space straight epsilon subscript straight o space dϕ subscript straight E over dt