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

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

3. 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).

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

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

6. 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:

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

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

9. Final STEP

10. This problem may be solved also by use of formula:

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

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

13. 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.

14. 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

15. Solution a) b)
Gauss Theorem

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

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

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

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

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

21. x h a r P

22. 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

23. 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

24. 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