Problem 3 Problem: A point charge with charge +Q is surrounded by a charged conducting shell, with total charge -2Q, of inner radius a and outer radius b. Outside of the shell is an uncharged dielectric shell of inner radius b and outer radius c, with dielectric constant k = 3. -20 IV (a)What is the field in region 1? (b)What is the field in region II? III II (c)What is the field in region III? (d)What is the field in region IV? Air (e)What is the charge on inner surface of the conductor Qe,inner? (f)What is the charge on the outer surface of the con- ductor Qc,outer? Conductor Dielectric (g)What is the bound charge on the inner surface of the dielectric Qd,inner ? (h)What is the bound charge on the outer surface of the dielectric Qd,outer?

A.) i. Define the plane polar unit vectors ŕ and 0 and show that they are orthonormal. Determine dr dê and de de ii. Hence show that for the vector r(t) = r(t)ŕ dr · = rî +rðê, d²r and = - dt dt² († − rė²)ŕ + (2ŕė +rë)ê . iii. If a mass m moves in the plane along r(t) = a and (t) = 5t determine the shape of the motion and find its period. B) Show that the motion of an object under the universal gravitational force μη Fg is governed by the radial and angular equations of motion and μ = 0, 2r0+ rö = 0. Setting r(t) =R for some constant R, solve these equations of motion and determine the period of the motion. C) Consider the following non-linear differential equation *+5= x +4. Find the stationary solutions. Expand around each of the stationary so- lutions that you have found, in order to determine if they correspond to stable or unstable equilibria. D) Show that 1 5 E = 2 - 4x is a conserved quantity. Hence draw the phase-space diagram and de- termine for which values of E oscilations…

Use Stokes' theorem to evaluate SF. dr where F = z²î + y²ĵ + xk and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1). The unit vector normal is upward. Z -3 2 2 X

The figure shows three vectors. Vector A→ has a length of 31.0, vector B→ has a length of 67.0, and vector  C→ has a length of 22.0. Find each magnitude and direction.   |A→×B→|= ?     direction: ?   |C→×B→|= ?