Draw a diagram that clearly defines the cylindrical polar coordinates rho , Phi,

Question

Draw a diagram that clearly defines the cylindrical polar coordinates rho , Phi, and z and the associated unit vectors e-^ rho , e-^ Phi and e-^ chi , in terms of the Cartesian coordinate system. State the divergence theorem, being careful to define all of the quantities that appear in it In electrostatics, one of Maxwell's equations for the electric field £ takes the following form: Delta -. E-=n/epsilon 0 where the scalar field n is the charge per unit volume and so is a constant Use this result to prove Gauss's Law: E-. ds-= Qv/epsilon 0 where dS- is a small element of vector area on the closed surface V bounding the volume V and enclosing a net charge Qv. Consider an infinitely long line of charge per unit length lambda , which runs along the z-axis. By symmetry, the electric field E-is everywhere parallel toe-^ Apply Gauss's Law to a hypothetical cylindrical surface coaxial with the z-axis, of length h and radius R. Explain why there is zero contribution to the surface integral from the top and bottom planar surfaces. Show that, due to symmetry, the contribution to the surface integral from the curved surface of the cylinder can be simplified to E-.dS-= E-(R) ds- Hence show that the electric field at a perpendicular distance R from the z-axis is given by E-(r)= lambda/2 pi epsilon o R e-^ rho Above a critical field strength Ecnt air is partially ionised and starts to become conducting Derive a criterion that the line of charge must satisfy (in terms of its radius a and charge per unit length lambda ) to prevent this.

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