Total Current in a Cylindrical Conductor

Given:
Radius of the cylindrical conductor, r = 2 mm = 0.002 m
Current density, J = 103e^(-400r) A/m^2

To find:
Total current, T (in mA)

Formula:
Total current (T) = ∫J.dA

Explanation:

1. Finding the expression for total current:
The total current passing through the conductor can be obtained by integrating the current density over the surface area of the conductor.

2. Determining the surface area of the conductor:
The surface area of a cylindrical conductor can be calculated using the formula:
A = 2πrh,
Where A is the surface area, r is the radius, and h is the height of the cylinder.

In this case, since the conductor is infinitely long, the height can be considered as 1 meter. Therefore, the surface area of the conductor is:
A = 2π(0.002)(1) = 0.004π m^2

3. Integrating the current density over the surface area:
The total current can be calculated by integrating the current density over the surface area of the conductor.

∫J.dA = ∫103e^(-400r) dA

Since the current density varies with the distance from the axis, we need to express the surface area in terms of the distance from the axis.

4. Expressing the surface area in terms of the distance from the axis:
The distance from the axis, r, is constant throughout the surface of the cylindrical conductor. Therefore, the surface area can be expressed as:
dA = 2πr.h.dh

5. Substituting the expression for surface area:
∫J.dA = ∫103e^(-400r) (2πr.h.dh)

6. Simplifying the integral:
Since the current density, J, is given as a function of r, we can substitute the expression for J in the integral. Also, the height, h, can be replaced with 1, as the conductor is infinitely long.

∫J.dA = ∫103e^(-400r) (2πr.1.dh)
∫J.dA = 2π ∫103e^(-400r) r.dh

7. Solving the integral:
∫103e^(-400r) r.dh can be integrated with respect to r, treating r as a constant.

∫103e^(-400r) r.dh = 103r.e^(-400r)h

8. Evaluating the integral limits:
Since the conductor is infinitely long, the height, h, can vary from 0 to infinity. Therefore, we need to evaluate the integral between these limits.

∫103r.e^(-400r)h = 103r.e^(-400r)(∞ - 0) = ∞

9. Conclusion:
The integral of the current density over the surface area of the conductor diverges to infinity, indicating that the total current passing through the conductor is infinite.

Therefore, the given answer of "7.5" for the total current is incorrect. The total current cannot be determined based on the provided information.