Consider the global law of charge conservation V J d = 0 for a

Consider the global law of charge conservation ∫_∂V J^αd∑_α = 0 for a special choice of the closed 3-surface ∂V: The bottom of ∂V is the ball {t = 0, x² + y² + z² ≤ a²}, where {t , x, y, z} are the Lorentz coordinates of some inertial frame. The sides are the spherical world tube {0 ≤ t ≤ T , x² + y² + z² = a²}. The top is the ball {t = T , x² + y² + z² ≤ a²}.

(a) Draw this 3-surface in a spacetime diagram.

(b) Show that for this ∂V, ∫_∂V J^αd∑_α = 0 is a time integral of the nonrelativistic integral conservation law (1.29) for charge