In a device called a betatron, charged particles in a vacuum are accelerated by the electric field E that necessarily accompanies a time-dependent magnetic field B(t).
(a) Suppose that, in cylindrical coordinates, the magnetic field throughout the betatron at time t can be approximated by
B = Krnt ez.
where K and n are positive constants. Assuming that the companion electric field takes the form E = E4,(r)eo,
express E in terms of r, K and n. If the electric field in the plane z = 0 is observed from a point on the positive z-axis, do its field
lines point clockwise or anticlockwise? Explain your answer in a sentence or two.
(b) What mathematical feature of the electric field makes it impossible to define an electrostatic potential field in this case? Explain your answer in two or three sentences.