Task 1 Knowing the components of vector A calculate rotA and divA. Data:

Solution Formula Hence, in our case: Conclusion: Sourced field.

Task 2a Data: a) Check if the field is potential. b) If it is, calculate the potential: using the reference point N(1,1,1).

Hints for b) 1) Use relation between vector field and potential: A=grad , (d/dx=Ax….) 2) Use the formula:

Solution a) Checking : Hence conditions for vector field A to be potential are: Conclusion: vector field A is potential.

Solution b) We are looking for function: satisfying equation: N(1,1,1) is reference point (potential is equall to zero) We are looking for function: satisfying equation: Following conditions are formulated:

STEP 1 Integrating respectively equations 1-3 we obtain: Comparing derivative of this function with condition 2) we obtain:

STEP 2 Comparing derivative of the above function with condition 3) we obtain:

Final STEP

This problem may be solved also by use of formula:

Task 3 Calculate the flux of through S. Use: The flux definition Stoke’s theorem 2 4 x y z

Solution a) Hence the flux: Calculation of vector B Remark:Only z-component of B creates the nonzero flux

Solution b) Hence the flux: 2 4 x y z Stoke’s Theorem 2 4 x y z Hence the flux: Remark:Only tangentional components are integrated.

Task 4 Calculate the flux of through the surface of perpendicular. y Calculate the flux of through the surface of perpendicular. From definition. Gauss theorem. 3 x 2 4 z

Solution a) b) Gauss Theorem

Task 5 Evaluate both sides of the divergence theorem for a vector field: y x within the unit cube centered about origin. z

Solution a) b) The closed-surface integral consists of only two terms that are evaluated at x=-0.5 and x=0.5:

Task 6 Evaluate the flux of vector through the area S shown in figure. Use: The flux definition 2 3 x y z

Task 7 Calculate the electric field intensity and inductance around a point charge. Apply a spherical symmetry of the problem.

Task 8 The bar on dielectric is charged with the charge q. Calculate field intensity and potential at point P.

x h a r P

Task 9 For the capacitor shown in Fig.9_1 calculate tte potential if: a) voltage V between plates is known b) charge q is known +q -q a b x V Fig.9_1

Task 10 The charge q is distributed uniformly on the very thin ring with the radius R. Determine the field intensity E in the point P in the distance a from the ring plate. R dE R dEx a x dE P dq Fig.10_1

Task 11 The charge q is distributed uniformly on the circle plate (radius R). Determine the field intensity E in the point P in the distance a from the plate. R dq dE r b dEx a x dE P dq Fig.11_1