/* MJ-3 Unit I: exact checks for Gauss-law applications and field equations. */
kill(all)$

print("MJ-3 Unit I -- Gauss law and field equations")$

E_in  : rho*r/(3*eps0)$
E_out : rho*R^3/(3*eps0*r^2)$

res_div_in  : ratsimp(diff(r^2*E_in,r)/r^2-rho/eps0)$
res_div_out : ratsimp(diff(r^2*E_out,r)/r^2)$
res_surface : ratsimp(subst(R,r,E_in)-subst(R,r,E_out))$

print("uniform sphere, interior divergence residual =",res_div_in)$
print("uniform sphere, exterior divergence residual =",res_div_out)$
print("uniform sphere, field-continuity residual =",res_surface)$

V_in : rho*(3*R^2-r^2)/(6*eps0)$
V_out: rho*R^3/(3*eps0*r)$

lap_radial(expr) := diff(r^2*diff(expr,r),r)/r^2$

res_poisson : ratsimp(lap_radial(V_in)+rho/eps0)$
res_laplace : ratsimp(lap_radial(V_out))$
res_E_in    : ratsimp(-diff(V_in,r)-E_in)$
res_E_out   : ratsimp(-diff(V_out,r)-E_out)$

print("Poisson-equation residual =",res_poisson)$
print("Laplace-equation residual =",res_laplace)$
print("interior E=-grad(V) residual =",res_E_in)$
print("exterior E=-grad(V) residual =",res_E_out)$

quit()$