The current density across a cylindrical conductor of radius `R` varies in magnitude according to the equation `J = J_0(1 - (r )/(R ))` where `r` is the distance from the central axis. Thus, the current density is a maximum `J_0` at that axis `(r = 0)` and decreases linearly to zero at the surface `(r = R)`. Calculate the current in terms of `J_0` and the conductor's cross - sectional area `A = piR^2`.

Options

A. `(J_(0)A)/(3)`
B. `(J_(0)A)/(6)`
C. `(2J_(0)A)/(3)`
D. `(J_(0)A)/(5)`

Answer

C

Explanation

`J = J_(0) ((r)/(R))`
di = JDA = J `(2pi rdr)`
` i=int_(0)^(R) J_(0)((r)/(R)) 2pirdr`
` i=2piJ_(0) [(R^(3))/(3R)]= (2J_(0A)/(3))`