![]() comonent along the direction of propagation. If E z = 0 and H z = 0, all the components will vanish, therefore it is observed that there must be a z component of either E or H i.e. Distance between two parallel planes equals space fraction numerator vertical line straight d subscript 1 minus straight d. In equation, the components of electric and magnetic fields strengths are expressed in terms of E z and H z. Similarly by using and solving equations 6b and 7a, we getĮquations 8(a,b,c and d) represent the equations of plane waves propagating in +z direction varying sinusoidally between the infinite parallel planes. Substituting value of H y in equation, we have Putting value of Ex in equation 7(b), we have Similarly by substituting equations 5c, d and e in equation 4, we have Similarly expanding equation (2) and equating respective components on both sides, we getįrom assumption (f), as the direction of propagation is along z-direction, the variation of field components can be expressed asĭH z/dg – dH x/dg = d E z/dg = d E x/dg = 0 (5e)īy substituting equations 5a, b and e, we have Ñ x H = d/dx d/dy d/dz = jw ε (E xa x + E ya y + E za z)Ī x d H z/dg – d H y/dz – a y d H z/dx– d H x/dz + a y d H y/dx– d H x/dgĬomparing the respective components on both sides, we get Ñ x E = jwμ H (Faraday’s law of em iduction) (2)Įxpanding equation (1) in rectangular coordinates, we get (d) Planes are of infinite extent in the y and z direction. ![]() (c) Space between planes is perfect dielectric ( 0) of permittivity and permeability. (b) Separation between the planes is ‘a’ meter in x direction. Similarly, if y k, 2the trace is z 4x + k2, which is again a parabola that opens upward. This means that if we slice the graph with any plane parallel to the yz-plane, we obtain a parabola that opens upward. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. If we put x k (a constant), we get z y2 + 4k2. Parallel planes are planes in the same three-dimensional space that never meet. Ñ x H = jw εE (Modified Ampere’s circuital law) (1) Assumptions : (a) Pair of parallel planes are perfectly conducting. If we put x 0, we get z y2, so the yz-plane intersects the surface in a parabola. In general, Maxwell’s equations (Modified Ampere’s Circuital law and Faraday’s law of em induction) in non-conducting region (σ = 0) between the planes are (II) Magnetic field must lie tangentially along the wall surface, that is, the normal component of magnetic field must be zero. Example 1: Finding the Condition for Two Planes to Be Parallel Given that the plane + 2 + 3 4 is parallel to the plane 2 2 3, find the values of and. (I) Electric field must terminate normally on the conductor, that is, tangential component of electric field must be zero. Let us use this property to complete the equations of two planes so that they are parallel. In order to determine the electromagnetic field configuration between parallel planes, Maxwell’s field equation are solved with the following boundary condition : If there is no attenuation, a g = 0 then field variation is expressed as (g) In time varying form, the field variation is expressed as In special case of uniform plane waves, y g reduces to y. Here y gis propagation constant and it is not equal to y(y g ¹ y). (f) Direction of propagation of wave is along z-direction, therefore the variation of all the field component in the z-direction is expressed as e -y g z derivative with respect to y is zero (d/dy =0) (e) As the plane is extended to infinity in the y – direction there are no boundary conditions to be met in this direction, therefore field is uniform in the y- direction i.e. ![]() In the real world, even though you make adjustments to a recipe to accommodate the number of people you need to serve, you sometimes round the amount of an ingredient instead of using an exact amount.(c) Space between planes is perfect dielectric (σ = 0) of permittivity ε and permeability μ. If you have two planes that don’t intersect, they’re parallel. What is the difference in the amount of salt you would need for 30 cupcakes? A parallel plane is a flat, two-dimensional surface. What is the difference in the amount of vanilla extract you would need for 30 cupcakes? How much flour do you need for 30 cupcakes? How much vegetable oil do you need for 30 cupcakes? What number will you need to multiply the amount of each ingredient by to adjust the recipe? How did you determine this number? This recipe serves 10, but you need to serve 30. ![]() You will use the recipe above to answer the following questions:
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