This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Voici la résolution de l'Exercice 1 : On a le vecteur position r et sa norme r: r = xe_x + ye_y + ze_z r = sqrt(x^2 + y^2 + z^2) 1. Calcul de grad r Le gradient d'une fonction scalaire f est donné par: grad f = f = ( f)/( x)e_x + ( f)/( y)e_y + ( f)/( z)e_z Pour f = r = (x^2 + y^2 + z^2)^1/2: ( r)/( x) = (1)/(2)(x^2 + y^2 + z^2)^-1/2(2x) = (x)/(sqrt(x^2 + y^2 + z^2)) = (x)/(r) De même: ( r)/( y) = (y)/(r) ( r)/( z) = (z)/(r) Donc: grad r = (x)/(r)e_x + (y)/(r)e_y + (z)/(r)e_z = (1)/(r)(xe_x + ye_y + ze_z) grad r = rr Le gradient de r est le vecteur unitaire radial e_r. grad r = rr 2. Calcul de div r La divergence d'un champ vectoriel A = A_xe_x + A_ye_y + A_ze_z est donnée par: div A = · A = ( A_x)/( x) + ( A_y)/( y) + ( A_z)/( z) Pour r = xe_x + ye_y + ze_z, on a A_x = x, A_y = y, A_z = z. ( x)/( x) = 1 ( y)/( y) = 1 ( z)/( z) = 1 Donc: div r = 1 + 1 + 1 = 3 div r = 3 3. Calcul de rot r Le rotationnel d'un champ vectoriel A = A_xe_x + A_ye_y + A_ze_z est donné par: rot A = × A = e_x & e_y & e_z \\ ()/( x) & ()/( y) & ()/( z) \\ A_x & A_y & A_z rot A = (( A_z)/( y) - ( A_y)/( z))e_x + (( A_x)/( z) - ( A_z)/( x))e_y + (( A_y)/( x) - ( A_x)/( y))e_z Pour r = xe_x + ye_y + ze_z, on a A_x = x, A_y = y, A_z = z. ( z)/( y) = 0 ; ( y)/( z) = 0 ( x)/( z) = 0 ; ( z)/( x) = 0 ( y)/( x) = 0 ; ( x)/( y) = 0 Donc: rot r = (0 - 0)e_x + (0 - 0)e_y + (0 - 0)e_z = 0 rot r = 0