1. (a) In the following, consider Maxwell's equations in vacuum. Use vector calcu- lus notation except where specified a

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1 A In The Following Consider Maxwell S Equations In Vacuum Use Vector Calcu Lus Notation Except Where Specified A 1 (76.41 KiB) Viewed 47 times

1. (a) In the following, consider Maxwell's equations in vacuum. Use vector calcu- lus notation except where specified and provide full details of your deriva- tions. i. State the differential form of Gauss's Law for the electric field. ii. State the definition of the Del operator in cartesian components. iii. Assuming an electric field written in cartesian form, E=E {+Eſ+ Ek, derive an expression for Gauss's theorem in cartesian component form. iv. Derive the cartesian form of the Laplacian operator. V. State the full electrodynamic relationship between electric field and the corresponding potentials. vi. Derive the Poisson and Laplace equations, stating any assumptions you make. (b) State the Divergence theorem and derive the integral form of Gauss's Law for the electric field. (c) State the principle of superposition as applied to electrical fields. (d) With the aid of a diagram and equations, explain the concept of electric flux arising from an electric field penetrating an element of surface area at some angle e. (e) Explain carefully why the use of Gauss's theorem to find the electric field in the following situations is particularly simple, and derive expressions for the electric field as a function of distance from; i. an infinite line of charge, ii. an infinite plane of charge. (1) An electrostatic system for controlling charge particle beams consists of an infinite plane of charge of charge density o, in the r, y plane (that is, the 2 = 0 plane). Above the plane, an infinite line of charge of charge density is placed parallel to the r axis at y=0,2 = 1. Find a relation between o and such that a point charge of magnitude q travelling parallel to the r-axis at y=0,2 = 2 will continue in a straight line. Given this condition is satisfied, is there anywhere else in the system that a particle can travel in a straight line?