The magnitude J of the current density in a wire with a circular cross section o

Question

The magnitude J of the current density in a wire with a circular cross section of radius R = 3.00 mm is given by J = (4.00 ×10^8 )r^3 , with J in amperes per square meter and radial distance r in meters. Determine the current through the outer section bounded by r = 0.800R and r = R.

These are my teachers requirements if you could please follow them I would really appreciate it thank you :)

In addition to being neat and clear, and actually answering the question, you must:

1) show the original principle (often in equation form)

2) substitute variables as needed

3) solve it first (before substituting numbers)

4) show every substitution (with units and correct sig figs)

5) present a boxed answer (with units and correct sig figs)

In a Force problem in more than one dimension, you MUST start with a Free Body Diagram.

Explanation / Answer

Given
J=(4.0*10^8)r^3 which means varying current density w.r.t radii
So consider on the cross section, a narrow ring with radius =x and width dx (--> 0)
Current flowing thru that narrow ring = J*area =(4.0*10^8) * x^3 * 2*pi*x*dx
so integrate over the limits 0.8R and R
I=integral (4*10^9 *2*pi ) x^4 dx =2*pi*4*10^9*x^5/4

substituting limits

I = 10.1 x 10 ^6 Amps

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