Answer :
Answer:
(a) [tex]K = \frac{I}{2\pi a}[/tex]
(b) [tex]J = \frac{I}{2\pi as}[/tex]
Explanation:
(a) The surface current density of a conductor is the current flowing per unit length of the conductor.
[tex]K = \frac{dI}{dL}[/tex]
Considering a wire, the current is uniformly distributed over the circumferenece of the wire.
[tex]dL = 2\pi r[/tex]
The radius of the wire = a
[tex]dL = 2\pi a[/tex]
The surface current density [tex]K = \frac{I}{2\pi a}[/tex]
(b) The current density is inversely proportional
[tex]J \alpha s^{-1}[/tex]
[tex]J = \frac{k}{s}[/tex] ......(1)
k is the constant of proportionality
[tex]I = \int\limits {J} \, dS[/tex]
[tex]I = J \int\limits \, dS[/tex] ........(2)
substituting (1) into (2)
[tex]I = \frac{k}{s} \int\limits\, dS[/tex]
[tex]I = k \int\limits^a_0 \frac{1}{s} {s} \, dS[/tex]
[tex]I = 2\pi k\int\limits\, dS[/tex]
[tex]I = 2\pi ka[/tex]
[tex]k = \frac{I}{2\pi a}[/tex]
substitute [tex]J = \frac{k}{s}[/tex]
[tex]J = \frac{I}{2\pi as}[/tex]