Given the vector current density J = 10rho2zarho − 4rho cos2 φ aφ mA/m2:
(a) find the current density at P(rho = 3, φ = 30◦, z = 2);
(b) determine the total current flowing outward through the circular band rho = 3, 0 < φ < 2π, 2 < z < 2.8

(a) Current density at P is .

(b) Total current I is 3.257 A

Explanation:

Because question includes symbols and formulas it can be misunderstood. In the question current density is given as below;

where and unit vectors.

(a) In order to find the current density at a specific point (P), we can simply replace the coordinates in the current density equation.  Therefore

(b) Total current flowing outward can be calculated by using the relation,

where integral is calculated through the circular band given in the question. We can write the integral as below,

due to unit vector multiplication. Then,

where . Therefore