Maxwell - nature of electromagnetic waves. We can make use...

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Maxwell's Equations The reason why the previous section developed the mathematics of waves was so that we could apply it to the understanding of electromagnetic phenomena (to which light pertains). To begin we must review Maxwell's equations which describe the relationship between electric and magnetic fields. Here we will express the equations in terms of the div, grad and curl of vector calculus, however it is worth noting that the equations can also be expressed in integral form. For time- varying electric and magnetic fields and in free space: âàá× = ( - ) + ( - ) + ( - ) = - âàá. = + + = 0 âàá× = ( - ) + ( - ) + ( - ) = μ 0 ε 0 âàá. = + + = 0 These equations tell us that the electric and magnetic fields are coupled: a time varying magnetic field will induce an electric field and a time varying electric field will induce a magnetic field. Moreover, the generated field is perpendicular to the original field. This suggests the transverse
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Unformatted text preview: nature of electromagnetic waves. We can make use of the identity of vector calculus that ( , where is some vector. Hence ( since . , so: 2 We can find a similar result for the magnetic field. From the definition of 2 (the Laplacian), we can write equations of the form: + + = for every component of the electric and magnetic fields. But, comparing this to the differential wave equation we notice the above is just a wave equation in E x , with the velocity equal to v = . Thus every component of the electric and magnetic field propagates through space with this speed. Maxwell deduced this result and found it to be in close agreement with the experimental value for the speed of light! This analysis remains one of the masterpieces of theoretical physics....
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