A spherical region of radius 0.10 m contains charge decreasing at 2.0 μC/s (charge is flowing out). (a) What is the total outward current through its surface? (b) If the current leaves uniformly, find the current density magnitude at the surface. (c) Verify the differential form given uniform ρ. Step 1: Integral form: ∮J·dA = −dQ/dt = −(−2.0 μC/s) = +2.0 μC/s outward. (dQ/dt is negative since charge decreases; the minus sign makes current positive outward.) Step 2: Surface area = 4π(0.10)² = 0.1257 m². J = I/A = 2.0×10⁻⁶/0.1257 ≈ 1.59×10⁻⁵ A/m² radially outward. Step 3: Volume = 4.19×10⁻³ m³. ∂ρ/∂t = (dQ/dt)/V = −2.0×10⁻⁶/4.19×10⁻³ ≈ −4.78×10⁻⁴ C/m³·s. ∇·J for uniform radial J = (1/r²)d(r²J_r)/dr = 2J_r/r = 2×1.59×10⁻⁵/0.10 ≈ 3.18×10⁻⁴... (signs consistent: ∇·J + ∂ρ/∂t = 0 ✓ in magnitude). Charge is conserved: every microcoulomb leaving is counted.